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apr factorization numbers finite rank abelian marius march abstract short note give formula factorization number finite rank abelian extends result previous work msc primary secondary key words factorization number subgroup commutativity degree function finite abelian introduction let finite group subgroup lattice two subgroups said factorized expression said factorization denote factorization number number factorizations starting point discussion given papers computed certain classes finite groups obtained explicit formulas factorization numbers elementary abelian rank abelian based connection subgroup commutativity degree see namely obviously applying inversion formula equality one obtains particular abelian consequently formula used following calculate factorization number rank abelian first recall theorem due hall see also permits compute explicitly function finite theorem let finite order unless elementary abelian case also need know total number subgroups finite rank abelian determined using different methods theorem total number subgroups able give main result note theorem following equality holds quantities given remark equality leads formula theorem simplifies particular case corollary however even case explicit formula factorization number difficult written small values examples proof theorem unique elementary abelian subgroup order say frattini subgroup moreover elementary abelian subgroups contained denote subgroups order subgroups order every quotient isomorphic maximal subgroup type type type similarly every quotient isomorphic subgroup index contains type type type way equality becomes view theorem since one obtains notation theorem leads desired formula references farrokhi factorization numbers finite abelian groups int group theory hall contribution theory groups order proc london math soc hampejs subgroups finite abelian groups rank three annales univ sci sect hawkes isaacs function finite group rocky mountain math explicit formula number subgroups finite abelian rank commun korean math soc saeedi farrokhi factorization numbers finite groups glasgow math subgroup commutativity degrees finite groups algebra addendum subgroup commutativity degrees finite groups algebra factorization numbers finite accepted publication ars number subgroups given exponent finite abelian group accepted publication publ inst math beograd marius faculty mathematics cuza university romania tarnauc
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gabidulin codes feb weijun fanga daqing wanc chern institute mathematics lpmc nankai university tianjin china email nankaifwj institute information engineering chinese academy sciences beijing china data assurance communications security research center chinese academy sciences beijing china university chinese academy sciences beijing china email wangliping department mathematics university california irvine usa email dwan abstract paper determine class deep holes gabidulin codes rank metric hamming metric moreover give necessary sufficient condition deciding whether word deep hole gabidulin codes study error distance two special classes words certain gabidulin codes keywords gabidulin codes rank metric deep holes covering radius error rank distance introduction let power prime integer let fnqm vector space finite field fqm paper consider case let basis fqm let map fqm coordinate basis fnqm associate matrix rank weight vector defined rank associated matrix denoted thus define rank distance two vectors refer details codes rank distance integers rank code length dimension fqm subspace dimension fnqm embedded rank metric minimum rank distance code denoted minimum rank weight vectors matrix called generator matrix rows span code rank code length dimension fqm called maximum rank distance separable mrds code error rank distance word fnqm defined min error rank distance plays important role decoding rank codes maximum error rank distance max fnqm called covering radius rank distance word code achieves covering radius code word called deep hole code covering radius deep holes linear code embedded hamming metric studied extensively mds codes generalized codes standard codes projective codes explored deeply gabiduli codes first introduced gabidulin independently delsarte gabidulin codes seen codes furthermore gabidulin codes mrds codes last decade increased interest gabidulin codes mainly relevance network coding covering radius gabidulin codes also studied however little known deep holes codes paper determine class deep holes gabidulin codes rank metric hamming metric meanwhile also obtain covering radius method moreover give necessary sufficient condition deciding whether word deep hole gabidulin codes study error rank distance special class words certain gabidulin codes note refer rank metric hamming metric explicitly pointed paper linearized polynomials gabidulin codes exploit linearized polynomials instead arbitrary polynomials recall results linearized polynomials polynomial fqm defined polynomial form fqm called denoted degq note constant term one easily check fqm name stems particular induces endomorphism space fqm set polynomials fqm denoted fqm ordinary product linearized polynomials need linearized polynomial however composition also linearized polynomial set fqm forms ring operations composition ordinary addition also lemma let fqm fqs smallest extension field fqm contains roots set roots forms vector space fqs let subspace fqm polynomial called lemma let subspace fqm polynomial fqm let fqm denote moore matrix furthermore basis one write det constant fqm clearly addition notion polynomials let fqm fqm linearly independent define matrix without ith column polynomial respect defined det fqm det proposition polynomial polynomial fqm proposition let fqm exists fqm space spanned deep holes gabidulin codes let fqm linearly independent also implies let space spanned gabidulin code fnqm defined linear block code generator matrix using isomorphic matrix representation interpret matrix rank distance defined code gabidulin code length dimension fqm minimum rank distance mrds code gabidulin code also defined follow fnqm fqm degq note interpretation code used throughout rest paper generalized code let fqm left ideal generated element ring fqm respect composition product particular additive subgroup follows space define evaluation map fqm fnqm given following property proposition defined map space isomorphism proof first since polynomial vanishes every second exists proposition third show surjective given fnqm polynomial satisfying proposition result proved polynomial follows element fqm written uniquely form fqm smaller algorithm inqthe ring fqm spaces quotient thus represented polynomials less fqm fqm degq using isomorphism identify word fnqm unique polynomial fqm degq degq easy see error distance definition thus degq provide following upper lower bounds theorem let fqm degq let fnqm corresponding word degq degq furthermore suppose monic degq exists degq subspace fqm degq proof applying algorithm write unique degq implies right hand inequality holds lower bound let fqm consider map defined clear map surjective ker root set roots dimfq ker root degq dimfq dimfq ker degq thus min rank min dimfq degq degq min degq degq degq last equality holds since degq degq degq furthermore proof know degq dimfq root dimfq ker degq degq degq equivalent degq subspace theorem proved theorem immediately deduce following two corollaries corollary gabidulin code covering radius corollary also obtained corollary set degq fqm deep holes gabidulin code number deep holes least according definition may also study gabidulin codes hamming metric showed gabidulin codes mds codes use denote hamming distance vectors hamming distance word respectively similarly following theorem theorem let fqm degq let fnqm corresponding word degq degq furthermore suppose monic degq exists subset gidegq fqm degq proof proof theorem let definition hamming distance exists fqm degq git indices comparing sides deduce degq proves degq furthermore monic equality degq holds case obtain git theorem proved theorem deduce results corollaries still hold hamming metric error distance special class words certain gabidulin codes hope obtain deep holes gabidulin codes consider monic degq theorem write let fqm basis degq equivalent det det denotes matrix deleting row result exist linearly independent elements det det denotes degq theorem deep hole thus deep hole proposition let exist linearly independent elements det det denotes matrix without row similar discussion get result hamming metric case theorem let exist distinct elements det det denotes deep hole proposition let hamming metric exist distinct elements det det denotes matrix without row following discuss error distance special class words certain gabidulin codes using results give two lemmas lemma let finite field characteristic trinomial irreducible trfq finite field function defined lemma even let trfq number solutions equation first consider finite field let let set consider two cases case odd implies number solutions equation lemma since solution number solutions equation lemma thus since element square also obtain corresponding case even implies lemma reducible written thus number solutions also get since element square also get discussion get following result proposition let gabidulin code dimension let deep hole two elements particular deep hole odd proof note two nonzero elements linearly independent thus proposition deep hole two distinct nonzero elements particular discussion deep hole odd cases least therefore desired result obtained remark proposition may hold case hamming metric proposition since possible solutions although always solutions proposition let gabidulin code fqm odd linearly independent set dimension let fqm deep hole proof suppose deep hole proposition two linearly independent elements thus since odd gcd implies linearly dependent contradicts assumption finally consider deep holes gabidulin codes large dimension proposition let gabidulin code fqn linearly independent set dimension let linearized polynomial fqn degree less equals deep hole proof suppose deep hole proposition linearly independent elements det det matrix aij denote matrix aqij note thus det det easy see det det matrix finite field characteristic thus det det det det contradicts thus deep hole proposition holds remark indeed according process proof conclusion fqn axq tion still holds polynomial fqn propositions still hold hamming metric similar analysis conclusions paper study gabidulin codes hamming metric rank metric general results hamming metric case see theorem proposition depend choice set results rank metric case see theorem proposition depend subspace fqm spanned particular latter case depend choice since equals whole space fqm hence problem gabidulin codes hamming metric seems complicated rank metric case hand generalized codes proved problem determining received word deep hole gabidulin codes problem seems complicated although give necessary sufficient condition problem state conjecture conjecture deciding deep holes gabidulin code references bartoli giulietti platoni covering radius mds codes ieee transactions information theory vol cheng murray deciding deep holes codes lecture notes computer science vol cheng wan list bounded distance decodability codes siam journal computing vol cohen karpovsky mattson schatz covering recent results ieee transactions information theory vol cohen lobstein sloane results covering radius codes ieee transactions information theory vol delsarte bilinear forms finite field applications coding theory journal combinatorial theory series vol gabidulin theory codes maximum rank distance problemy peredachi informatsii vol gadouleau yan packing covering properties rank metric codes ieee transactions information theory vol gadouleau yan properties codes rank metric proc ieee globecom san francisco graham sloane covering radius codes ieee transactions information theory vol guruswami vardy decoding codes proceeding soda helleseth klove mykkeltveit covering radius binary codes ieee transactions information theory vol kuijper gabidulin decoding via minimal bases linearized polynomial modules mathmatics keti wan deep holes codes based dickson polynomials finite fields applications vol kschischang coding errors erasures random networking coding ieee transactions information theory vol liao codes chinese annals mathematics lidl niederreiter finite fields cambridge university press cambridge london loidreau properties codes rank metric http silva kschischang approach error control random network coding ieee transactions information theory vol vasantha suresh babu covering radius codes gaita sandesh wan error distance codes science china vol hong deep holes standard codes science china mathematics vol zhang liao new deep holes generalized codes scientia sinica vol zhang wan deep holes projective codes international symposium information theory zhang wan explicit deep holes codes zhuang cheng determining deep holes generalized reedsolomon codes ieee transactions information theory vol
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asymptotic bounds globally optimal positions orthogonal stiffeners rectangular plates elastostatic bending nathan school mechanical engineering faculty engineering tel aviv university ramat aviv tel aviv israel abstract present paper treats problem finding asymptotic bounds globally optimal locations orthogonal stiffeners minimizing compliance rectangular plate elastostatic bending essence paper utilization method analysis orthogonally stiffened rectangular plates first presented mazurkiewicz obtained herein closed form several special cases approximation stiffeners zero torsional rigidity asymptotic expansions expressions deflection field stiffened plate used derive globally optimal stiffening layouts highly flexible highly rigid stiffeners central result obtained work analytical proof fact array flexible enough orthogonal stiffeners number stiffening rectangular plate subjected lateral loading best put form exactly two orthogonal stiffeners one direction keywords elastic plate bending orthogonal stiffeners fredholm kind integral equation asymptotic analysis globally optimal positions introduction paper addresses problem optimal positioning straight uniform orthogonal given number integrally attached thin rectangular isotropic homogeneous plate subjected lateral elastic quazistatic loading layout minimizing total work done loads formulation resembling earlier version dems herein majority previous works assumed neutral lines stiffeners coincide neutral surface plate analysis optimal design problems gained substantial attention literature early papers nowacki mazurkiewicz solved analysis problem assuming stiffeners positioned symmetrically neutral surface plate enforcing geometric compatibility plate stiffeners well static equilibrium stiffeners equation plate equation work mazurkiewicz case plate treated influence stiffeners viewed effective distributed line loads expressed double sine series procedure employed mazurkiewicz bidirectional stiffening beams torsional rigidity produced infinite sets linear algebraic equations cut solved simultaneously proof convergence algebraic systems given fletcher thorne within context analysis unstiffened plates arbitrary boundary conditions work dems equilibrium equations plate stiffeners replaced use green functions bending plate beams compatibility equation formulated fredholm integral equation kind solved numerically assuming stiffeners attached plate finite number points effective point loads corresponding geometric compatibility points computed simultaneous solution cut infinite linear algebraic equations various aspects analysis quazistatic bending free vibrations buckling orthogonally stiffened rectangular plates using double sine series representations examined konchkovskii savin grigolyuk andrianov kalamkarov sometimes called inverse optimization problem optimal positioning stiffeners gained less extensive coverage work dems locations stiffeners varied decrease work done given quazistatic lateral loading optimization procedure utilizes optimality criterion derived variationally work grayhack mahar consider highly flexible stiffeners positioned layout increasing minimum buckling load obtain analytic form equations determining optimal locations objective present paper derivation expression elastic strain perchico energy simply supported thin rectangular plate stiffened single set parallel stiffeners subjected harmonic loading asymptotic expansion elastic strain energy respect nondimensional parameter representing ratio bending rigidity stiffeners plate analytical globally optimal minimization compliance respect locations stiffeners addition closed form globally optimal bound optimal layout asymptotically flexible stiffeners derived case bidirectional stiffening essence results presented herein explicitness simplicity global optimality expressions optimal layouts highly flexible highly rigid stiffeners present work contrast procedure followed dems geometric compatibility equation plate stiffeners formulated fredholm kind integral equation unlike kind equation solved analytically case expressible green kernel function method equivalent method used nowacki mazurkiewicz exception closed form expression obtained mainly solely due restriction unidirectional stiffening thorough review solutions contact problems plate shell bending reformulation governing equations kind fredholm equations found work grigolyuk different approach described books andrianov kalamkarov method homogenization applied finding anisotropic plate structurally equivalent ribstiffened isotropic plate widely used development finite elements topological optimization analysis uniformly stiffened panels although obviously advantageous aforementioned cases method homogenization necessarily suitable purpose finding globally optimal locations asymptotically rigid stiffeners discussion certain mathematical aspects method given andrianov review applications method carried kalamkarov outline present paper follows section deflection field stiffened plate derived assuming continuous interfaces plate stiffeners resulting closed form expression unidirectionally stiffened simply supported plate subjected bisinusoidal loading series general loading cases section solution generalized account bidirectional stiffening section torsional rigidity taken account simpler case unidirectional stiffening section basis results sections asymptotic explicit globally optimal bounds optimal locations highly flexible highly rigid bidirectional stiffeners without torsional rigidity obtained analytically analysis unidirectionally stiffened plates section outlines procedure derivation linear elastic deflection field thin rectangular homogenous isotropic simply supported plate uniform thickness subjected prescribed lateral loading stiffened straight uniform parallel beams perpendicular one sides plate locations specified parametrically graphical representation problem follows fig denote planar cartesian coordinates dimensions plate distribution lateral loading locations stiffeners bending rigidities plate stiffeners denoted respectively fig representation problem lower half plate stiffened symmetrically respect neutral surface model used describe bending plate stiffeners assumed behave beams positioned symmetrically respect neutral surface plate deflection neutral surface along longitudinal axes influence ith stiffener plate represented distributed force per unit length defined positive acting upwards stiffeners plate assumed boundary conditions case considered present section beams simply supported beam assumed subjected distributed force per unit length defined positive acting downwards shown fig fig distributed force plate applies ith stiffener distributed force ith stiffener applies plate solution procedure presented based infinitesimal deformation assumption according deflection plate small relatively thickness thickness turn small relatively say geometric average lengths edges plate according assumption bending plate uncoupled stretch therefore described single force balance equation static equilibrium deflection neutral surface usually middle surface plate appropriate boundary conditions paper treats problem elastic bending thin simply supported plate means plate treated line hinges two adjacent edges line moving supports say moving bearings two edges accordingly deflection field plate would follow assumptions derivation elastic infinitesimal deflection field follow linear superposition wqi total deflection stiffened plate point coordinates denoted namely deflection plate alone due external load alone deflection unstiffened plate due unit lateral force acting function describing distribution given external lateral load vector coordinates stiffening beams vector distributed forces per unit length representing forces acting plate stiffeners due external load system equilibrium components corresponds stiffeners equation assumes everywhere small enough deflection plate small relatively thickness according newton third law qib superscript represents fact distributed force applied beams representing stiffeners due geometric compatibility representing fact stiffeners considered integrally attached plate deflection fields stiffeners equal deflection field stiffened plate along interfaces namely wib next assuming stiffeners treated beams one write qib wib substituting equations relation results next substituting equation equation produces equation deflection field stiffened plate turned standard fredholm integral equation kind solved using standard procedure either differentiating performing integration parts four times consecutively use made green function bending plate obtained navier szilard sin sin sin sin abd uniform beams assumed next taking transform loading sin sin pnr introducing auxiliary function sin deflection field unidirectionally stiffened simply supported plate general loading becomes sin sin sin sin sin pnr elastic strain energy stored deformed structure consistent energy stored stiffened plate energy stored stiffeners equal work done external loads obtained accordingly next one assumes sinusoidal loading distribution pgh sin sin pkr deflection field stiffened plate becomes pgh sin sin pgh sin sin sin sin sin pgh sin sin sin hence compact form deflection field stiffened plate obtained first using trigonometric identity sin sin cos cos double sine sums written difference two single cosine sums summing cosine series analytically relating function produces known sine series following result obtained matter addressed section cos sin sinh abx sinh cosh sinh integrating substituting result expression infinite sums obtained defining following quantities indices related numbers stiffeners jir cosh cosh cosh sinh sinh cosh sinh sinh sinh sinh cosh pgh sin cosh cosh sinh sinh cosh sinh sinh sinh sinh jih cosh cosh cosh sinh sinh cosh sinh sinh sinh sinh deflection field bisinusoidal loading closed form although totally explicitly solution finite linear system equations required becomes pgh sin sin sin wgh deflection field corresponding bisinusoidal loading deflection field general loading expressible double sine series derived superposition pgh sin sin sin numerical verification presented method omitted equivalent method presented nowacki mazurkiewicz analytical summations performed improve convergence help avoid gibbs phenomena however present paper operations needed order obtain asymptotic expansions rather order produce better numerical analysis module therefore comparison deflections stresses stiffened plates various loading cases calculated employed method commercial software tools given generalization analysis bidirectional stiffening treatment bidirectional stiffening much different unidirectional one deflection field obtained using linear superposition wqi distributed forces per unit length exerted plate stiffeners total number stiffeners number aligned stiffeners number aligned stiffeners denoting interaction forces applied aligned stiffeners aligned stiffeners fij one write following relations beam beam first equality sign due assumption neutral axes stiffeners coincide neutral surface plate second due beam theory uniform beams third obtained summing forces exerted aligned stiffener first term third equality sign distributed force per unit length exerted plate aligned stiffener arises newton third law function denotes dirac delta function aligned stiffeners one write using arguments fij instead fij time due newton third law next deriving substituting one owing fact influence function bending plate symmetric insensitive interchange arguments one learn last two terms cancel therefore deriving compatibility equation bending orthogonally stiffened plate one disregard interaction forces stiffeners cancel final governing equation consequently governing equation becomes hence using expressions orthogonality sine functions following strategy solution equations mentioned section one obtains two sets compatibility equations distributed forces stiffeners plate one stiffening direction making use introducing following quantities sin vis sin pnr pms sin jirn sin jims sin jim sin analytically decoupling two sets compatibility equations mentioned one gets single system defined quantities respectively vector matrix infinite sets representing final form compatibility equation bidirectionally stiffened simply supported thin rectangular plate equation infinite inhomogeneous linear system equations every equation finite system equations number equations equal number stiffeners direction parallel axis obviously relation hold therefore infinite system solved exactly one truncate order solve thus system would solved would order solve system necessary write finite systems equations one beneath resulting total number mtot unknowns equations one stiffeners direction parallel axis taking harmonics fourier series representation green influence function bending plate one would write first equations equations forth eventually would equations rows representing matrix similarly vectors one would components components forth resultant vectors components corresponding columns representative matrix naturally one could written equations corresponding stiffener first bunches bunches however better keep equations corresponding different stiffeners bunches truncation error affect similarly possible stiffener represented worse course still remains order within bunch smaller effect order anyway solution system equations achieved gauss elimination last equations smallest error first solved would beneficiary put first important harmonics end reverse order described anyhow defining gmf every component vectors column vector every component matrix matrix results equation unit matrix solution equation would produce series approximate values finite number necessary integrals evaluation deflection field matrix regular system convergent solution written obviously formal writing solution system equations achieved taking inverse large matrix system solved matlab solver based factorization gauss elimination next question convergence addressed way first proof fletcher thorne guarantees system equivalent convergent may solved desired degree accuracy unlike say asymptotic system solved certain limited degree accuracy second system solved author realistic problem parameters showing monotonic convergence upon increase number considered harmonics correlating well fem solution whereas solution unidirectionally stiffened plate presented section considered analytical exact solution bidirectionally stiffened plate may considered exact semianalytical exact due convergence series since exactness solution one ability desired degree possible present case due truncation infinite series procedure one follow order obtain solution discussed thoroughly shortly one solve substitute solution using definitions thus obtaining third row second row one substitute solution along definitions following relation obtained analytical decoupling unidirectional stiffeners torsional rigidity one account torsional rigidities stiffeners relevant especially case stiffeners closed torques stiffeners apply plate related twisting angles plate along interfaces follows stiffener one employ constitutional relation linear elastic shaft denoting torque applied plate shear modulus geometric torsion rigidity parameter respectively coordinate changing along stiffener perpendicular assuming small twisting angles superscript implies quantity related stiffener newton law shaft law superimposing deflection field simplicity unidirectionally stiffened plate becomes wqi wti substituting one gets following equation deflection field deflection due unit couple yet determined betti theorem deflection elastic structure due unit couple point equal rotation angle direction couple perpendicular stiffeners present case slope small rotation angles due unit force point namely substituting yields cos sin sin sin substituting produces fully defined equation deflection field naturally similar steps taken second perpendicular stiffening array present treating obtained equation manner similar one described previous sections introducing following quantities referring definition cos cos one obtains first set compatibility equations use made definitions sin equation obtained calculating distributed deflections platestiffeners interfaces side calculating distributed deflections interfaces results second set compatibility equations following definitions made pns sin cos one decouple two systems analytically defining following auxiliary quantities kronecker delta following infinite set equations obtained naturally order solved system truncated system convergent assume assumed systems paper relying proof convergence fletcher thorne approximate solution desired degree accuracy obtained next applying argumentation section denoting truncated vector tsf quantities one write last form compatibility equation rectangular simply supported plate stiffened single set uniform stiffeners parallel one sides plate taking torsional rigidity stiffeners account follows system would solved numerically defining auxiliary quantities introduced hence denoting corresponding symbol upper bar truncated vectors constructing truncated matrix manner similar one employed equation section obtain second set unknowns required solution solution becomes sin sin pns sin cos abd last order substituted solutions separated appropriate way similar way constructed asymptotic analysis globally optimal layouts highly flexible highly rigid stiffeners section concerned determination optimal layout stiffeners limit case stiffeners much less much rigid ground structure examination limit case important estimation correct asymptotic behavior optimal locations stiffeners respect rigidities determination bounds optimal positions realistic stiffeners derivation assumes thin isotropic homogenous rectangular plate elastostatic bending subjected lateral pressure stiffened two sets straight parallel uniform stiffeners highly flexible bidirectional stiffeners torsional rigidity order asymptotics linear elastostatic strain energy case derived using results sections follows introducing small parameter deriving auxiliary quantities required calculation strain energy asymptotic case one gets recalling expression deflection given explicitly section easily derived sin abpns sin sin abd integrating multiplied half loading thus obtaining expression work strain energy pns abpns sin vis sin substituting one gets explicit expression strain energy stiffened plate abpns pns sin sin min pns pns sin sin sin abp pns sin asymptotic expansion obtained energy consists constant term two terms subtracted naturally energy attains minimum terms subtracted constant attain maxima respect domain unconstrained obviously subtracted sums squares attain maxima squares sums attains maximum first sum rightmost part symmetric respect second sum symmetric respect clearly globally optimal solution words energy bidirectionally stiffened plate attains global minimum respect locations two orthogonal sets straight parallel uniform infinitesimally rigid stiffeners case general lateral loading distribution stiffeners two sets located place knowing optimal layout one calculate energy global optimum substituting min min pns sin pns sin energy corresponds plate stiffened one horizontal stiffener one vertical stiffener respectively following bending rigidities therefore one certain amount material form two orthogonal sets parallel highly flexible stiffeners integrally attached given plate minimize work done external loads globally optimal way use material split exactly two orthogonal beams special case symmetric loading square plate result given shows flexible enough stiffeners stiffening two orthogonal beams energetically equivalent stiffening single beam twice bending rigidity analytically obtained result coincides numerical observation concerning matter given work dems power result globally optimal stiffening layout determined analytically bidirectional stiffening arbitrary number stiffeners arbitrary parameters general loading case limitation result relevant highly flexible stiffeners say actual bending rigidities true relative bending rigidities cease true happens optimal positions stiffeners stiffeners higher bending rigidities reason shortcoming fact first order asymptotics respect stiffeners capture split stiffeners optimal stiffening topology relative stiffness first order approximation valid high enough optimal stiffeners positioned separately following section shows least special case bisinusoidal loading unidirectional stiffening two beams positioned symmetrically respect plate even second order asymptotic expansion energy stiffened plate respect relative stiffness beams degenerated layout given still globally optimal comparison first second order expansions energy enables one determine actual rigidities still optimal place stiffeners together highly flexible unidirectional stiffeners torsional rigidity order asymptotics repeating procedure shown preceding subsection retaining terms order substituting expression work done external loads given one gets abpns pns sin sin yin pns sin sin fijn ijs pns sin sin dijnk ijsr next recalling rearranging results following order asymptotic expansion sin sin sin sin sin sin sin sin sin pns sin pms sin sin sin pns sin pnq sin pns sin prq sin pmk sin pns point result anyway asymptotic one several assumptions made order get analytic form solution optimal stiffening problem first loading assumed bisinusoidal equal lengths sides plate second simplicity clarity illustration unidirectional stiffening assumed two stiffeners positioned symmetrically respect center plate priori symmetry stiffening layout seems justified view symmetry loading assumptions work done loading becomes abp sin sin sin sin sin sin sin sin sin sin distance stiffeners center plate divided length stiffeners employing trigonometric identities recalling definition one obtain simpler expression elastic strain energy loaded structure consistent plate stiffeners cos cos actual energy obtained form next series expressed terms fourier cosine series coshz zsinhz linear function thence substituting functions instead fourier series representations one obtains cos cos coth coth coth coth cos cosh cosh cosh sinh sinh sinh sinh cosh cosh cosh sinh sinh sinh sinh differentiating respect one gets sin coth coth coth coth sin sin sinh cosh cosh sinh sinh sinh sinh cosh cosh sinh sinh sinh cos cosh cosh sinh sinh sinh cosh sinh sinh cosh sgn sinh order determine whether placing stiffeners together optimal one could examined whether second derivative energy respect changes sign increases change sign would indicated unified stiffeners cease correspond global minimum energy start correspond local maximum rigidity increased would imply second order asymptotic expansion energy enables one find critical relative rigidity stiffeners best split critical relative rigidity small enough second order approximation good one however second derivative respect second order expansion energy respect undefined due sign function therefore instead one check whether first derivative vanishes anywhere besides increases fact enough check sign lim sign positive every positive placing stiffeners together never corresponds local maximum second order expansion energy section shows placing stiffeners together globally optimal limit small positive therefore happens local minimum energy every range validity second order asymptotics another global minimum elsewhere due continuity energy respect local maximum two local minima numerical examination shows high enough ceases global optimum globally optimal solution corresponds continuously increasing therefore limit case infinitesimally higher critical value energy would three extrema two local minima local maximum located line segment length tending zero would imply infinite derivatives therefore contradict fact energy continuous vicinity zero one show expanding taylor series around expanding rather one eliminates functions contradiction avoided assumption wherever local minimum energy also global minimum order global minimum local maximum also function continuous vicinity zero behaves power function near zero thus three extrema within line segment starting zero arbitrarily small length hence proving lim positive every second order expansion energy valid one proves placing stiffeners together globally optimal within abovementioned range rigidities next expanding hyperbolic functions using identities hyperbolic trigonometry dividing computing limits recalling sinh sgn sinh following result obtained coth coth coth coth obviously limit positive negative every positive asymptotic expansions show indeed case limit cases coth coth coth coth coth coth numerical verification shows negative positive finite values therefore limit positive thus small enough positive finite values globally optimal stiffen simply supported thin rectangular plate elastostatic bending lateral bisinusoidal loading two stiffeners positioned place geometric center plate parallel one edges plate fact global optimality stiffening layout holds finite values least specified loading due fact result true first second order asymptotic expansions elastic energy stiffened plate essence present subsection structural strain energy corresponding globally optimal layout discussed obtained second order approximation respect setting opt coth coth coth coth first second order approximations energy hold small enough values formally higher orders contribute negligibly first second order approximations energy degenerated stiffening layout proven herein globally optimal general bisinusoidal loadings respectively recalling definition relative bending rigidity assuming blade stiffeners width static height plate thickness poisson ratio plate stiffener young modulus one obtain approximation maximal static height stiffener still globally optimal split hence assuming width stiffener much smaller length one might expect beam namely making approximation based fact quantities smaller unity compliant additional restrictions values say last taking typical structural materials one obtains following approximation taking limit one obtains result short stiffener critical static height hmax square plate hmax stiffener relatively long static height thickness plate would still flexible enough globally optimal split result impractical since stiffener assumed placed symmetrically respect plate thus static height equal thickness plate young modulus plate stiffener stiffener matter whether split however stiffener longer length makes flexible static height larger optimality stiffening single stiffener yet hold example follows case critical static height hmax hmax hmax simple calculation shows least long stiffeners degenerated stiffening layout globally optimal rather realistic parameters one could presume stiffeners absolutely negligible bending rigidity results obtained globally optimal layout corresponds thin plate thin stiffeners fact bending rigidity stiffeners normalized bending rigidity plate small imply stiffeners realistic plate thick since stiffener width height order thickness plate much smaller bending rigidity plate appears definition relative rigidity proportional ratio width stiffener breadth plate usually small number thus derivations contradict assumptions first order plate bending theory following subsection examines opposite case stiffeners highly rigid compared plate flexible stiffeners asymptotically optimal layout combined form single stiffener turns highly rigid stiffeners asymptotically optimal layout optimally spread tend rigidities increase highly rigid unidirectional stiffeners torsional rigidity order asymptotics present subsection examines case stiffeners highly rigid respect plate means first order asymptotic analysis respect flexibilities stiffeners shown rigid enough stiffeners best positioned separately locations determined functionality loading shown optimal positions stiffeners approach asymptotic limits rigidities stiffeners approach infinity functionality approach given analytically bisinusoidal loading equal lengths edges plate unidirectional stiffening two stiffeners positioned symmetrically respect plate condition stiffeners relatively long substituting expression deflection field case given making use employing definitions used previous subsections defining one obtains strain energy stiffened plate follows sin sin evaluating expression abovementioned assumptions dividing coefficients defining matrix readily obtained cosh cosh cosh sinh sinh sinh sinh cosh cosh cosh sinh sinh sinh sinh cosh cosh cosh sinh sinh sinh sinh considering large enough spectral norm much smaller unity expanding expression energy first order taylor series relative flexibility stiffeners one obtains following asymptotic relation sin sin sin sin noting symmetry sine function inverting matrices analytically introducing auxiliary variables bij sinh one gets following representation energy sinh sin taking taylor series expansion respect bij components small retaining first powers results substituting one gets asymptotic representation energy due high order expansion fair approximation sinh asymptotic limit optimal location long rigid stiffeners obtained minimization limit finitely long infinitely rigid stiffeners sin arg min lim arg min next substituting definition performing order asymptotic expansion respect around zero differentiating respect noting derivative vanish one obtains asymptotic expansion case becomes linear function finding single root abovementioned derivative produces expression globally optimal location stiffener one closer origin coordinate system nondimensional form function follows cot opt opt opt opt opt cot cot illustration result obtained one might consider following stiffening parameters seems acceptable due high order expansion respect imply according corresponding optimal position therefore established globally optimal stiffening layout even moderately long stiffeners consists single stiffener finite range rigidities certain value relative rigidity stiffener split two two rigid stiffeners optimal layout distance stiffeners increase hyperbolically rigidities approaches asymptotic limit rate increase would case optimal layout approximately proportional power aspect ratio plate defined length stiffeners divided length edge plate perpendicular stiffeners highly flexible unidirectional stiffeners torsional rigidity order asymptotics linear elastostatic strain energy case derived using results sections follows introducing small parameters deriving auxiliary quantities required calculation strain energy asymptotic case one gets substituting expression deflection field given gives sin sin pns sin cos abd integrating multiplied half loading thus obtaining expression external work total structural strain energy pns pns sin cos recalling definitions produces following asymptotic expression abn pns sin npns cos mpmq cos expression curved brackets attains global minimum certain reason different values equal one wishes find global minimum point true whenever additional equality constraint variables principle separation variables holds minimum sum sum minima due linearity always holds order asymptotics minima attained point due fact function argument every hence flexible enough stiffeners small enough bending rigidity necessarily implies small enough torsional rigidity since usually order magnitude closed sections approximately proportional smaller dimension third power larger dimension cross section optimal layout since minimal structural compliance single stiffener splitting stiffeners critical bending rigidity supplied phenomenon shown bidirectional stiffening neglecting torsional rigidity unidirectional stiffening accounting reasonable believe least approximately true bidirectional stiffening beams torsional rigidity torsional rigidity respect bending rigidity moreover order emphasize obtained result absolutely trivial noted order expansion section showed coupling variables implying objective function written case examined obvious stiffeners split however subsection proves still true even argument shown delicate reasoning combination results thus makes unreasonable believe realistic bidirectional stiffening closed torsional rigidity may still optimal use two orthogonal major stiffeners various loading distributions beyond abovementioned numerical analysis stiffened plates performed use commercial software showed effect torsional rigidity least blade stiffeners examined section corresponds significant change total work done external loading numerical optimization also showed optimal layout consists one two perpendicular stiffeners true various accounting neglecting effect torsional rigidity last order illustrate result pointed symmetric loading makes term maximal absolute value third term zero coincides understanding symmetric loading would twist therefore stiffener placed would apply torque torsional rigidity would contribute strain energy structure conclusions paper presented several results investigation subject optimal stiffening rectangular plates elastostatic bending previously published method analysis stiffened plate obtained compact forms sections herein utilized derive globally optimal stiffening layouts parameters shown optimal layout two orthogonal sets parallel highly flexible stiffeners negligible torsional rigidity consists two orthogonal beams increased rigidities positioned perpendicularly addition asymptotically globally optimal layouts long stiffeners obtained highly flexible highly rigid stiffeners orders approximation former latter respectively effect interaction perpendicular stiffeners shown cancel stiffeners negligible torsional rigidity effect torsional rigidity examined perspective ability influence optimal layout asymptotically flexible stiffeners found insufficient least order asymptotic expansion structural compliance respect rigidities stiffeners special case unidirectional stiffening addition mentioned although shown structural compliance minimization problem solved beams different parameters producing result higher bending rigidity stiffeners respect plate unidirectional stiffening becomes optimum noteworthy results correlating present work found works schade clarckson rozvany fuchs samsonov fuchs brull cheng olhoff leary harari lam santhikumar lagaros perchikov fuchs szczepanik essence contribution work existing body knowledge utilization method solution enforcing continuous interfaces plate set stiffeners including derivation compact form strain energy structure based method purpose derivation globally optimal positions uniform stiffeners least asymptotic cases highly flexible highly rigid stiffeners simplifying assumptions loading harmonic stiffeners least relatively long obtained results seem give insight structural behavior widely used structural elements stiffened panels globally optimal layouts hard obtain either geometric topological optimization even comes determination symbolic dependence optimal layouts parameters asymptotic analysis together closed form solution elastostatic bending problem seems might shed additional light area although asymptotic approach optimal stiffening problem employed specific combination assumptions results presented herein seem marginal contribute understanding subject another subject deflection field thin rectangular plate subjected arbitrary boundary conditions general loading given fletcher thorne discussed goriupp timoshenko solution procedure given fletcher thorne produces deflection field plate arbitrary boundary conditions given deflection field plate subjected prescribed loading clearly one supply solution procedure given authors deflection field stiffened plate instead unstiffened one paper effect torsional rigidity stiffeners accounted way similar way done mazurkiewicz compact expressions entities representing solution case torsional rigidity obtained using analytic summation series section manner done sections work fletcher thorne acknowledgments present work partly supported israeli science foundation grant references andrianov lesnichaya manevich metod usrednienia statikie dinamike rebristikh obolochek moskva nauka cheng olhoff investigation concerning optimal design solid elastic plates int solids struct clarkson elastic analysis flat grillage cambridge university press cambridge dems szelag optimal design disks plates int solids struct fletcher thorne bending thin rectangular plates proceedings national congress applied mechanics ann arbor michigan fuchs substitute function methods structural optimization application continuous beams dissertation technion fuchs brull new strain energy theorem use optimum design continuous beams comput struct goriupp die dreiseitig gelagerte rechteckplatte arch appl mech grayhack mahar buckling plates asymptotic approach siam appl math grigolyuk tolkachev kontaktnyie zadachi teorii plastin obolochek moskva mashinostroenie kalamkarov composite reinforcement elements construction wiley new york konchkovskii plity staticheskie raschety moskva stroiizdat lagaros fragiadakis papadrakakis optimum design shell structures stiffening beams aiaa lam santhikumar automated rib location optimization plate structures struct multidisc opt mazurkiewicz bending buckling rectangular plate reinforced transversely ribs variable rigidities bull acad polon sci ser sci tech mazurkiewicz buckling rectangular plate obliquely reinforced ribs variable flexural rigidity bull acad polon sci ser sci tech rozvany gin optimal design structures variable support conditions optim theory applic nowacki statecznosc plyt prostokatnych wzmocnionych zebrami arch mech stos nowacki zagadnienia statyki dynamiki plyt wzmocnionych zebrami arch mech stos leary harari finite element analysis stiffened plates comput struct perchikov fuchs optimal layouts stiffeners plates bending topology optimization approach paper presented european conference computational mechanics solids structures coupled problems engineering lnec lisbon june samsonov optimal location thin rib elastic plate izv acad nauk ussr mtt mechanics solids savin fleishman plastinki obolochki rebrami zhestkosti kiev naukova dumka schade orthogonally stiffened plate uniform lateral load appl mech szczepanik optimization topology stiffener locations structures using evolutionary methods paper presented european conference computational mechanics solids structures coupled problems engineering lnec lisbon june szilard theory analysis plates classical numerical methods prentice hall timoshenko theory plates shells edn mcgraw hill new york
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dec rasa open source language understanding dialogue management tom bocklisch rasa tom joey faulkner rasa joey nick pawlowski rasa nick alan nichol rasa alan abstract introduce pair tools rasa nlu rasa core open source python libraries building conversational software purpose make based dialogue management language understanding accessible software developers terms design philosophy aim ease use bootstrapping minimal initial training data packages extensively documented ship comprehensive suite tests code available https introduction conversational systems becoming pervasive basis human computer interaction seek natural ways integrate automation everyday life examples conversational include apple siri amazon alexa microsoft cortana conversational systems becoming widespread platforms like facebook messenger opening chatbot developers common tasks conversational systems include scheduling booking customer support modern open source libraries held high standard professionalism extends implementations machine learning algorithms large amount work involved maintaining widely used project code produced research groups often falls short rasa nlu core aim bridge gap research application bringing recent advances machine learning want implement conversational systems introduce rasa nlu core easy use tools building conversational systems since statistical dialogue system intended rasa already used thousands developers worldwide many conversational systems tools split natural language understanding rasa nlu dialogue management rasa core section describes code architecture outline developer experience demonstrate example application meekan see https klms bluebot see https see http see https nips conversational workshop long beach usa related work rasa takes inspiration number sources rasa api uses ideas focus consistent apis strict inheritance keras consistent apis different backend implementations indeed libraries optional components rasa application text classification loosely based fasttext approach sentences represented pooling word vectors constituent token using word embeddings glove trained intent classifiers remarkably robust variations phrasing trained examples intent braun show rasa nlu performance compares favourably various solutions custom entities recognised using conditional random field rasa core approach dialogue management similar takes different direction many recent research systems currently support learning natural language understanding state tracking dialogue management response generation jointly learned dialogue transcripts rasa language understanding dialogue management fully decoupled allows rasa nlu core used independently one another allows trained dialogue models reused across languages language generation encourage developers generate variety responses authoring multiple templates response currently easier reliable using example neural network generate grammatically coherent semantically correct responses rasa core also account uncertainty voice transcription nlu would typically achieved partially observable markov decision processes pomdp support training via reinforcement learning currently alpha emphasise new users rather implementing reward function simulated user immediately placing humans loop encourage developers train dialogue policy interactively see section recent work williams follows similar machine teaching approach one difference work dialogue policy directly exposed surface form user utterances whereas rasa core dialogue policy receives recognised intent entities related software libraries mentioned introduction large body research statistical dialogue systems last decades translated widely used software libraries notable contribution end pydial recently released toolkit dialogue research compared pydial rasa core emphasises needs software developers researchers field open toolkits include opendial ravenclaw number natural language processing nlp libraries widespread use necessary mention also number online services natural language understanding nlu term understood dialogue research community converting short user messages dialogue acts comprising intent set entities like online services rasa nlu hides implementation details new users advantage slightly experienced users fully customise nlu system language understanding performed number components implementing common api therefore easily configurable suit needs particular project description code rasa architecture modular design allows easy integration systems example rasa core used dialogue manager conjunction nlu services rasa nlu code implemented python services expose http apis used easily projects using programming languages architecture dialogue state saved tracker object one tracker object per conversation session stateful component system tracker stores slots well log events led state occurred within conversation state conversation reconstructed replaying events user message received rasa takes set steps described figure step performed rasa nlu subsequent steps handled rasa core figure message received passed interpreter rasa nlu extract intent entities structured information tracker maintains conversation state receives notification new message received policy receives current state tracker policy chooses action take next chosen action logged tracker action executed may include sending message user predicted action listen back step actions frame problem dialogue management classification problem iteration rasa core predicts action take predefined list action simple utterance sending message user arbitrary function execute action executed passed tracker instance make use relevant information collected history dialogue slots previous utterances results previous actions actions directly mutate tracker executed may return list events tracker consumes events update state number different event types slotset allslotsreset restarted etc full list documentation https natural language understanding rasa nlu natural language understanding module comprises loosely coupled modules combining number natural language processing machine learning libraries consistent api aim balance customisability ease use end pipelines sensible defaults work well use cases example recommended pipeline processes text following components first text tokenised parts speech pos annotated using spacy nlp library spacy featuriser looks glove vector token pools create representation whole sentence classifier trains estimator dataset default mutliclass support vector classifier trained component trains conditional random field recognise entities training data using tokens pos tags base features since components implements api easy swap say glove vectors custom word embeddings use different machine learning library train classifier components handling words many customisation options advanced users detailed documentation https policies job policy select next action execute given tracker object policy instantiated along featurizer creates vector representation current dialogue state given tracker standard featurizer concatentates features describing last action intent entities recent user message slots currently defined featurization slot may vary simplest case slot represented single binary vector element indicating whether filled slots categorical variables encoded binary vector take continuous values specify thresholds affect featurisation simply passed featurizer float hyperparameter specifies number previous states include featurisation default states stacked form array processed recurrent neural network similar sequence model practice find problems value works well usage training data formats rasa nlu core work training data formats rasa nlu requires list utterances annotated intents entities specified either json structure markdown format markdown syntax especially compact easy read rendered many text editors web applications like github intent show chinese cuisine restaurants json format slightly cumbersome read whitespace sensitive suitable transmission training data applications servers text show chinese restaurants intent entities start end value chinese entity cuisine rasa core employs markdown specify training dialogues aka stories greet inform location rome price cheap inform cuisine spanish inform people six story starts name preceeded two hashes choice name arbitrary helpful debugging body story sequence events separated newlines event inform location bombay price expensive user utterance annotated dialogue act general format intent entities pairs separated commas system actions also events specified lines starting dash end story denoted newline machine teaching addition supervised learning rasa core supports machine teaching approach developers correct actions made system find practical approach generating training data exploring space plausible conversations efficiently example user engaged machine teaching restaurant recommender system described section user presented following history bot bot user said whose intent greet currently slots cuisine none people none price none location none bot wants due intent correct yes intent right action wrong intent wrong export current conversations stories quit user inputs indicating action wrong provided prompt lists possible actions probability assigned dialogue policy next action bot choosing correct action creates new training data point rasa core partially trains dialogue policy moves conversation forward next step completed trained model persisted newly generated training data saved file visualisation dialogue graphs rasa core also capability visualise graph training dialogues story graph directed graph actions nodes edges labeled user utterances occur execution two actions user interaction two consecutive actions edge label omitted graph initial node called start terminal node called end note graph capture full dialogue state possible walks along edges necessarily occur training set simplify visualization heuristic used merge similar nodes generated graph running simplification shown figure simplification two nodes merged replacing single node inherits rasa conversational system referred bot incoming outgoing edges removing duplicates process makes resulting graphs easier interpret nodes merged two conditions met represent action previous turns identical dialogues leading nodes start greet start greet greet inform location rome price cheap inform cuisine spanish inform location rome price cheap inform cuisine spanish inform cuisine spanish inform location rome price cheap inform cuisine spanish inform location rome price cheap end end example story graph without simplification example story graph simplification figure minimal example illustrating story graphs simplified training data contains two stories first interaction nodes therefore considered equivalent merged deployment production environment repositories rasa nlu core contain dockerfiles producing static virtual machine images aids reproducibility ease deployment variety server environments web servers running http api support parallelism allowing handle large request volumes production environment demonstration demonstrate usage rasa core use babl dialogue dataset simple exercise system asked search restaurant fill several slots able perform successful search system may ask user preference slot available slots location number people cuisine price range interesting dataset due inherent problem multiple ways get information single correct action every case accuracy precision therefore appropriate metrics evaluating dialogue policy instead consider system chooses actions depending information already available attempt fill slots empty figure see probabilities rasa core attaches action given slots already knows see core follows rough pattern asking cuisine followed location followed number people training data however recognises could also ask unfilled slots attributing probability one filled slots lower diagonal grid given vanishingly small probability illustrates rasa core use contextual clues learn conversations outlook rasa nlu core active development serve platform making applied research conversational usable developers never finished number topics active development including improved support reinforcement learning making nlu robust typos slang supporting languages also plan figure plot probabilities choosing actions babl example section sequentially inform system correct cuisine location number people price system chooses next action based information already listed box top left system favours asking slots informed slots filled searches restaurant release datasets comparing performance different models authors welcome external contributions project specifics found repositories github acknowledgements authors indebted users libraries providing invaluable feedback creating supportive community around tools special acknowledgement owed external contributors libraries lists contributors may viewed https https references bohus rudnicky ravenclaw dialog management framework architecture systems comput speech july bordes weston learning dialog corr braun matthes langen evaluating natural language understanding services conversational question answering systems proceedings annual sigdial meeting discourse dialogue pages chollet keras joulin grave bojanowski mikolov bag tricks efficient text classification arxiv preprint lafferty mccallum pereira conditional random fields probabilistic models segmenting labeling sequence data lison kennington opendial toolkit developing spoken dialogue systems probabilistic rules acl page pedregosa varoquaux gramfort michel thirion grisel blondel prettenhofer weiss dubourg machine learning python journal machine learning research oct pennington socher manning glove global vectors word representation proceedings conference empirical methods natural language processing emnlp pages ultes barahona vandyke kim casanueva budzianowski wen gasic pydial statistical dialogue system toolkit proceedings acl system demonstrations pages wen gasic mrksic vandyke young semantically conditioned natural language generation spoken dialogue systems arxiv preprint wen vandyke mrksic gasic ultes young trainable dialogue system arxiv preprint williams asadi zweig hybrid code networks practical efficient dialog control supervised reinforcement learning arxiv preprint williams liden demonstration interactive teaching dialog control hybrid code networks proceedings annual sigdial meeting discourse dialogue pages young thomson williams statistical spoken dialog systems review proceedings ieee
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senate permissionless byzantine consensus protocol wireless networks zhiyuan bhaskar sheng zhisheng mar zhiyuan niuzhs tsinghua university beijing china bkrishna university southern california los angeles usa blockchain technology achieved tremendous success open permissionless decentralized consensus employing pow variants whereby unauthorized nodes gain disproportionate impact consensus beyond computational power however powbased systems incur high delay low throughput making ineffective dealing applications hand byzantine bft consensus algorithms better delay throughput performance employed closed permissioned settings avoid sybil attacks paper present wireless network coordinate based byzantine consensus senate based conventional bft consensus framework yet works open systems wireless devices faulty nodes may launch sybil attacks senate legislature quota senators per state district constant irrespective population state senators senate selected participating distributed nodes based wireless network coordinates wnc fixed number nodes per district wnc space elected senators participate subsequent consensus reaching process broadcast result thereby senate proof sybil attacks since pseudonyms faulty node likely adjacent wnc space hence fail elected index consensus sybil attack wireless network permissionless blockchain distance geometry introduction recent years witnessed explosive development digital cryptocurrency academic fields financial markets behind tremendous success key enabling technology digital cryptocurrency blockchain combines several judiciously designed techniques facilitate trusted distributed ledgers intermediary eliminated transactions particular bitcoin blockchain ingeniously adopts pow mining among purposes deal identity attacks sybil attacks open permissionless systems wherein identities participating nodes assumed known priori specifically node whether authentic pseudonym must solve cryptographic puzzle participate block generation process obtain rewards mining therefore impact mining figure application scenarios autonomous driving systems wherein vehicles pedestrians intersections based distributed consensus left terminals drones sensors actuators act based coordinated synchronized behavior right node directly tied computational power irrespective number identities according necessity prior identity authorization procedure blockchain technologies categorized permissionless permissioned blockchains permissionless blockchains bitcoin ethereum applied open systems wherein faulty nodes may apply sybil attacks counteraction usually involves enforcing strict coupling consensus impact node computational power pow bitcoin resources ethereum casper node despite robustness sybil attacks price payed permissionless blockchains usually suffers high processing delay blocks recommended bitcoin amounts one hour low throughput transactions per second bitcoin many existing works try remedy issue however inherent mining based probabilistic consensus reaching technique key limiting factor hand permissioned blockchains hyperledger fabric need wary sybil attacks since participating nodes gone explicit authentication process trusted adopting long line existing research byzantine bft protocols processing delay throughput permissioned blockchains dramatically improved nice comparison traditional bft protocols permissionless blockchains presented paper dense wireless network scenario considered likely encountered future deployments well table omparisons blockchain technologies open system delay permissionless blockchain yes large permissioned blockchain small senate yes small context vehicle networks intelligent transportation figure come solution bft consensus offers benefits permissioned permissionless blockchains table details bft consensus permissionless systems proposed protocol namely sybilproof wireless network coordinate wnc based byzantine consensus senate consists three major phases sortition senator selection byzantine agreement senate thwarts sybil attack exploiting fact even faulty node forge wireless channel nodes unique wireless fingerprint leveraged identify nodes fully decentralized approach proposed achieve notations throughout paper use boldface uppercase letters boldface lowercase letters lowercase letters designate matrices column vectors scalars respectively transpose matrix denoted denotes entry element matrix vector respectively nuclear norm matrix denoted respectively vector consisting diagonal entries square matrix denoted diag trace matrix denoted matrix entries one denoted likewise zero matrix denoted related work idea allowing selected nodes participate bft consensus reaching shared neo algorand neo delegated bft consensus based blockchain small number servers statically configured run consensus behalf larger open network similar senate algorand counteracts sybil attacks adopting sortition phase difference random verifiable function based solution combining pos leveraged algorand whereas senate based underlying wireless channels senate employing pos tied digital currency thus applied broadly achieving consensus wireless systems moreover avoids unfairness introduced pos intentionally favors participants resources pol wireless networks adopted dasu replace pow faster transactions pols generated authorities wireless network operators hence notion centralization introduced senate also uses concept pol whereas high level nodes generate pols manner without trusted authority work wireless channel fingerprints utilized protect sybil attacks survey approach however existing work relies trusted authority verify wireless channels nodes encrypted keys wireless hoc network commodity devices considered view selection policy based signalprint observations proposed work considers fully decentralized wireless network novel wnc based protocol senate proposed compared existing work senate much lower running time better understandability hence favourable implementations system model consider wireless network consisting geographically distributed nodes full connectivity namely pair nodes network within radio communication range system open permissionless sense node join system without prior identity authentication objective good nodes reach valid definition validity addressed later consensus set values certain time period deterministic concerted actions carried meantime subject malicious behaviors faulty nodes note considered scenario distinguishes state machine replication wherein log proposed client different nodes agree log record different nodes may different set initial values sensory data environment hence reasonably good valid consensus needs reached unlike existing work byzantine consensus mainly adopts internet overlay network wireless overlay considered regard behavior faulty node clarified specifically following assumptions made paper objective faulty node rig consensus reaching process benefit rather halting process achieve purpose possible malicious behaviors include byzantine node namely comply protocol report arbitrary messages sybil attack generate pseudonyms gain inappropriate power process reaching consensus overlay wireless network faulty node block interfere nodes transmissions messages first assumption describes motive faulty node therefore implications two assumptions second assumption simply states message level restriction behavior faulty node perspectives message content identity message sender existing byzantine agreement protocols internet overlay third assumption also implied limits malicious behavior node whereas wireless networks broadcast nature electromagnetic waves assumption implications meaning faulty node assumed comply communication protocol instance faulty node would transmit another node scheduled consensus protocol assumption stems large part first assumption since messing communication protocol transmitting high power thus blocking nodes leads retransmissions hence halting consensus reaching process besides following two reasons also justify assumption attack becomes trivially devastating without assumption namely faulty node sufficient transmit power block transmissions time prevent reaching consensus node complying communication protocol obviously malicious easy spot work assume nodes obtain ranging estimations based others pilot signals however focus specific methods obtain distance estimations based receive signal strength rss time arrival toa approaches studied extensively net effect ranging estimations considered statistical model dij nij dij denotes geographic distance hence dji dij distance estimation denoted estimation error introduced multiplicative additive random coefficients nij respectively wireless localization literature usually assumed ranging estimations nij modeled gaussian distributed variable ranging estimations shadow fading modeled assumed gaussian distributed often termed shadow fading taking logarithm sides considering ranging estimation error observed thereby approach effective short distances since multiplicative error component approach applies wide range distances although may require central node calibrate clocks terminals ensure synchronization aloha game critical requirement every good player node knows total number users including faulty nodes system determine action considered scenario requirement poses challenge presence faulty nodes namely faulty nodes let good nodes believe fewer nodes system good nodes may behave conservatively hence benefit faulty nodes prevent chorus procedure proposed leverages unique pattern channel power delay profile pdp estimate total number participating nodes key observation faulty node forge components mpcs hence feature utilized especially los transmission environment estimate node quantity chorus procedure consisting time slots node supposed randomly select one time slot receive hand transmit pilot symbols remaining time slots analysis section shows even presence faulty nodes procedure robust receive time slot nodei estimates pdp receive signal calculates number mpcs los environment measurement gives accurate estimation number transmitters system moreover even faulty node generate multiple mpcs given location analogous let nodes perform chorus first every node estimation population based nodes unique timbre procedure lasts time slots since assume nodes operate mode otherwise procedure shortened one time slot wherein every node transmit receive simultaneously detailed procedure well analysis appropriate given section aloha game based sortition given node estimations total number nodes system let every node selfish aloha game prevent faulty nodes gaining advantages methodology essentially identical blockchain technology allows miners compete opportunity register block particular aloha random access game selfish users implemented roughly described follows overview senate consists three major phases sortition senator selection byzantine agreement sortition sortition process objective prevent faulty nodes generate arbitrarily many pseudonyms note mean eliminate sybil attack sortition completely key achieve developing aloha game selfish users one cheat based nash equilibrium arguments every node selfish sense want transmit soon possible time slot however successful transmission good node would stop competing whereas faulty node might keep transmitting launch sybil attacks node successfully transmits time slot selected candidate denotes successful transmission aloha game transmits pilot signal ranging estimations immediately afterwards definition shown section nash equilibrium exists based every node adopts transmit time slot probability one benefit changing strategy unilaterally quorum candidates reached sortition phase terminates candidates going senator selection next phase note sybil attack still possible faulty node may occupy several seats among candidates senator selection phase dedicated removing pseudonyms generated faulty nodes ranging estimations among nodes fully distributed manner candidates selected longer follow random access protocol instead candidate assigned unique time slot transmit frame time slots phase since every candidate transmitted pilot signal every candidate obtained distance estimations candidates distance estimations denoted vector estimated euclidean distance matrix edm squared norm hence distance feedback symmetry verification candidate feeds back dedicated time slot afterwards every candidate obtains distance estimations pair candidates network note dij dij thereby every node remove suspicious distance feedback based checking estimated edm assume feedback perfect constant related nij case long hold removed since tell lying robust wnc generation despite fact symmetry verification extent depending distance estimation error eliminate untruthful distance feedback faulty node still launch refer shout attack definition shout attack shout attack faulty node pretends away nodes synchronously adding distance estimations nodes particular ranging estimations faulty node purposely transmit pilot signals later supposed accordingly feed back tampered likewise whisper attack defined faulty node pretends nearer nodes ease exposition use shout attack illustration henceforth larger distance estimations ranging estimations faulty node purposely amplify pilot signal power accordingly feed back tampered larger distance estimations definition shout attack detected symmetry verification gives faulty node arbitrarily many fake geographical locations arbitrarily far real one purpose shout attack hence create pseudonyms facilitate sybil attack causes severe challenge senate since senate uses location information sybil protection thwart shout attack introduce seesaw test based following intuition real world space increasingly unlikely faulty node launches shout attack away nodes proportionally number nodes grows analogous placing elastic sticks pair nodes system lengths sticks given distance estimations circumstance faulty node launching shout attack space illustrated figure related sticks bent dramatically hence elastic force levers screened seesaw test like lighter side seesaws argument mathematically formalized theorem states local error proportional number good nodes good node node faulty node node node node figure seesaw test faulty node launching shout attack would levered iterative wnc calculation seesaw tests proposed essence illustrated follows round iteration every node uses distance estimation another push resp pull distance estimation larger resp smaller predicted distance two nodes current wnc moves accordingly end round node largest local error related seesaws bent identified faulty node therefore removed note termination criterion specified later added wnc calculation terminates system small prediction error clustering obtaining coordinates candidates clustering algorithm applied coordinates candidates divided clusters representatives one cluster selected senators prevents sybil attacks pseudonyms one faulty candidate likely fall cluster byzantine agreement senators run byzantine agreement protocol reach consensus primarily consider median validity consensus defined follows assume single consensus value reached upon denote sorted array initial values good nodes among nodes nodes faulty assumed hence definition median validity call value medianvalid holds thereby adopt jack algorithm ensures following properties long number faulty senators satisfies agreement every selection input values every selection faulty senators good senators decide value termination every good senator decide value finite time median validity decision upon agreement every senator broadcasts consensus value every good node system adopts majority value since consensus reached senate majority value reflect consensus value ensures safety agreement among nodes algorithm sortition input output chorus procedure every node selects transmits pilot symbol time slot else receives signals estimates number transmitting nodes based receive pdp estimation denoted total number nodes estimated aloha game based sortition every node transmit every time slot candidate selected new successful transmission happens corresponding node afterwards transmits pilot signal return faulty nodes conjure always transmitting thereby sortition sortition phase described algorithm elaborate several details follows analysis chorus duration node receives signals assuming estimation number transmitting nodes correct probability good node transmitting time slot good node therefore unbiased estimate denotes estimation transmitting node receive time slot optimal attack faulty node launch let good nodes believe nodes system transmission strategy latter would conservative therefore worst effect nevertheless basically bft protocol plugged senate phase works well open system since achieved previous phases bernoulli random variable parameter letting sufficiently large compared denotes infinitesimal lte system time slot nodes chorus procedure lasting yields nash equilibrium aloha game aloha game described follows every node participates game time slot node successfully transmits node receives payoff denotes transmission cost leaves faulty node may otherwise stay game keep playing time slot collision happens every transmitting node receives payoff detailed payoff function described table every node goal maximize payoff single time slot game repeated based game setting prove existence nash equilibrium theorem exists nash equilibrium every node adopts transmit probability time slot algorithm senator selection input output validsenate distance estimation feedback based pilot signals received sortition phase estimtes distance candidates feeds back distance vector edm symmetry verification proof proof based see appendix details remark theorem indicates allowing every node selfish symmetric equilibria exists namely even faulty node improve payoff changing transmission probability unilaterally specifically increases transmission probability would collisions payoff decreases due cost decreases chance success chance also decreases transmission cost per time slot clearly plays critical role denotes relative cost per transmission compared one successful transmission practice propose aside power resource cost due wireless transmissions economic approach applied whereby small fee charged every sortition transmission enhance robustness process every element dij invalidvalue robust wnc generation every terminal generate wnc simultaneously based following procedure terminate false terminate false dij invalidvalue maxerror maxi errindex arg maxi maxerror sei remove corresponding entries else terminate true senator selection phase candidates transmit fashion detailed algorithm description phase presented algorithm explanations follow robust wnc generation rationale robust wnc generation follows face edm estimation error introduced faulty nodes denote entries arbitrarily large considering malicious behavior goal recover towards end two structures exploited although faulty nodes cause arbitrarily large error error sparse terms entries majority still good edm stems space limited hence mature tools distance geometry utilized verify authenticity thereby considering space edm written diag use space ease exposition paper however generalization considered straightforward dij clustering validsenate false else kmeans validsenate true return validsenate geographical location coordinates candidates formulate wnc generation problem follows exploiting sparse error property minimize subj rank norm based formulation notoriously nonconvex fact based compressive sensing theory therefore norm relaxed norm often exhibits near optimal performance minimize subj rank adopt method solve based estimation update based gradient objective function kxi corresponds algorithm also note algorithm keep track local error array whose element represents squared distance error related much levered seesaw test figure therefore take account fact candidate small error updated based location candidate large error latter likely faulty node based argument remove candidate largest error end round error evenly distributed among candidates means error introduced ranging estimation instead faulty nodes case selected senators reach quorum algorithm returns validsenate false intriguingly method similar spring network based method pair nodes connected spring objective works minimize elastic potential energy system equivalent total square error tse distance prediction given current lengths springs distance estimations placing nodes distances among nodes rest lengths springs plane although objective minimizing tse presented method turns similar vivaldi algorithm except faulty detection analysis seesaw test seesaw test based rationale faulty node implementing shout attack detected resultant location would space question arises accordingly faulty node given certain strength shout attack moreover effect number good nodes question important answer quantitatively characterize effectiveness seesaw test forged locations seeking concise illustrative answer consider simplified scenario one faulty node without loss generality located trying launch shout attack good nodes located concretely consider faulty node adds arbitrary independent real error vector entries edm related note general shout attack whereby error added synchronously arbitrary error written based note ranging estimation error considered subsection focus synthetic error faulty node level faulty node measured min min rank denotes reconstructed coordinates nodes given tempered edm words level quantified minimum squared euclidean distance reconstructed coordinate space projection space given faulty node implements attack minimizes distance essential note sequence minimization meaning faulty node first chooses error closest space selected since quantity affected locations good nodes noting closer good nodes produce stronger lever force seesaw test given strength shout attack expectation taken given location distribution following theorem adopt gaussian distribution ease exposition theorem assume faulty node attack strength good nodes coordinates generated based gaussian distribution zero mean variance error independent min proof see appendix remark shown faulty node conceal lie noting scales attack strength comparable squared distance measurement whereby attack becomes quite obvious addition effect amplified approximately times intuitive since becomes increasingly difficult lie good nodes form concrete space another note long least nodes determine space remark theorem assumes error matrix independent coordinates good nodes requires faulty node aware coordinates nodes assumed mostly good nodes sortition phase reasonable coordinate information accessible sortition phase distance feedback occurs corollary space min byzantine consensus sybil attack senate figure probability valid consensus reached nodes remark corollary generates effectiveness seesaw test higher dimensions scenarios applications drone swarms byzantine agreement phase since removed sybil nodes selected senators great extent basically byzantine agreement protocol implemented among senators particular adopt jack scheme proposed consists following two major stages setup stage senator broadcasts initial value receives senators initial values thereby senator broadcasts acceptable values proposed value jointly considering senators initial values search stage rotating among predetermined leaders round leader receives proposals senators accordingly proposes value based acceptable values agreement reached based proposals leader would propose agreed value proved long one leader among leaders good node valid agreement would reached therefore faulty nodes allowed phase validity assured median validity specified section termination agreement properties also proved fact protocol achieves optimum safety optimum median validity simulations run computer simulation test performance senate ranging estimations nodes assumed perfect information exchange simulated direct modifications data arrays nodes figure nodes senate selects candidates senators number faulty nodes shown performance consensus probability shown obtained running algorithms episodes node locations randomly generated square side length meter episode faulty nodes assumed always launch sybil attacks propose randomly generated values deviating true values specifically good faulty nodes initial values uniformly randomly generated interval respectively figure also simulate jack algorithm without faulty nodes launching sybil attacks comparisons jack algorithm scenario attack assumptions ensure consensus number faulty nodes reach majority otherwise direct consequences design jack algorithm definition median validity observed senate perform conventional bft protocol sybil attack verifies senate also observe senate perform better faulty nodes reach majority senate randomly selects senators probability selected senators dominated good nodes jack algorithm also perform least well senate sortition phase added effect shown figure line original jack algorithm conclusion senate distributed bft protocol applicable systems without prior identity authentication order prevent malicious nodes generate arbitrary number pseudonyms senate leverages wireless signals transmitted nodes identities manner based fact pseudonyms likely adjacent geographically thereby selected nodes senators participate final consensus reaching process computer simulations show senate comparing consensus probabilities systems wherein faulty nodes launch launch sybil attacks respectively acknowledgement work sponsored part nature science foundation china china postdoctoral science foundation hitachi ltd appendix proof theorem first invoke following lemma ensures existence nash equilibrium lemma finite symmetric game symmetric equilibrium finite symmetric game lemma denotes game wherein every player finite action set payoff received player given action players actions identical irrespective specific player mixed strategy contrast pure strategy latter employs fixed action time whereas former viewed mix randomized strategy latter proof lemma based brouwer fixed point theorem omitted brevity provided existence symmetric equilibria ready derive transmission probability every node due symmetry suffices consider single node whose game choice shown table table payoff functions nodes silent transmit silent node transmits rank rank rank rank addition eigenspace spanned eigenspace span seen following equation expected payoff success collision note assume independent power error probability event based principle indifference expected payoffs zero yields concludes proof appendix proof theorem based edm without error written diag attack implemented edm derive resultant coordinate covariance tampered attack based first notice eigenspace eigenvalues zeros rank inequality since error assumed independent coordinates gaussian coordinate vectors uniformly directed space constant share error power leaked coordinate eigenspace spanned considering objective faulty node minimize leakage power beyond space case equivalent maximizing power space given total power fixed best attack faulty node implement concentrate error power linear space spanned eopt note solution also satisfies conditions however results negative eigenvalue violates theorem proves important property edm exists least three distance measurements violating triangle inequality words shout attack may put faulty node space whereas whisper attack would lead violation triangle inequality much easier spot let consider minimization inside expectation optimal approximation problem whose solution well known dominant singular space eigenspace denoting singular value decomposition svd singular values always arranged order opt wherein denote first two columns respectively contains two dominant singular values follows minimum projection error min rank opt coordinate power leakage beyond space due tampered edm solve quantity adopt orthogonalization process set given optimal attack derived based symmetry gaussian distributed clear direction third vector relevant hence replaced power brevity detailed process omitted orthogonal basis vectors expected leakage power given min min guerraoui next bft protocols proceedings european conference computer systems ser eurosys new york usa acm quest scalable blockchain fabric bft replication open problems network security camenisch eds cham springer international publishing online accessed https dasu kanza srivastava unchain blockchain proc symposium foundations applications blockchain vol newsome shi song perrig sybil attack sensor networks analysis defenses proceedings international symposium information processing sensor networks ser ipsn new york usa acm faria cheriton detecting attacks wireless networks using signalprints proceedings acm workshop wireless security new york usa acm demirbas song scheme sybil attack detection wireless sensor networks proceedings international symposium world wireless mobile multimedia networks washington usa douceur sybil attack international workshop systems springer patwari ash kyperountas hero moses correal locating nodes cooperative localization wireless sensor networks ieee signal process vol jul conclusion follows immediately mackenzie wicker selfish users aloha gametheoretic approach ieee veh tech vol references dokmanic parhizkar ranieri vetterli euclidean distance matrices essential theory algorithms applications ieee signal process vol nov nakamoto bitcoin electronic cash system tschorsch scheuermann bitcoin beyond technical survey decentralized digital currencies ieee communications surveys tutorials vol thirdquarter macqueen methods classification analysis multivariate observations proc fifth berkeley symposium mathematical statistics probability vol oakland bentov lee mizrahi rosenfeld proof activity extending bitcoin proof work via proof stake sigmetrics performance evaluation review vol gilad hemo micali vlachos zeldovich algorand scaling byzantine agreements cryptocurrencies proceedings symposium operating systems principles new york usa acm eyal gencer sirer renesse bitcoinng scalable blockchain protocol usenix symposium networked systems design implementation nsdi santa clara usenix association stolz wattenhofer byzantine agreement median validity international conference principles distributed systems opodis compressive sampling proc international congress mathematicians vol madrid spain dabek cox kaashoek morris vivaldi decentralized network coordinate system sigcomm comput commun vol nash equilibrium points games proceedings national academy sciences vol cachin architecture hyperledger blockchain fabric workshop distributed cryptocurrencies consensus ledgers fen cheng reeves vorobeychik wellman notes equilibria symmetric games international workshop game theoretic decision theoretic agents gtdt castro liskov practical byzantine fault tolerance osdi berkeley usa usenix association gower euclidean distance geometry math sci vol ongaro ousterhout search understandable consensus algorithm usenix annual technical conference usenix atc philadelphia usenix association
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ieee transactions signal processing estimation mitigation channel massive mimo jun orod raeesi student member ieee ahmet gokceoglu member ieee mikko valkama senior member ieee duplex tdd based massive mimo systems rely reciprocity wireless propagation channels calculating downlink precoders based uplink pilots however effective uplink downlink channels incorporating analog radio base station user equipments ues exhibit due behavior individual transmit receive chains downlink precoder aware channel nrc system performance significantly degraded due nrc induced interference terms work consider general massive mimo system mismatches ues well mutual coupling mismatch system coexist induce channel nrc based signal models first propose novel iterative estimation method acquiring side nrc matrices also propose novel downlink precoder design utilizes obtained estimates furthermore efficient pilot signaling scheme ues introduced order facilitate executing proposed estimation method precoding technique practical systems comprehensive numerical results indicate substantially improved spectral efficiency performance proposed nrc estimation precoding methods adopted compared existing methods index channel channel state information mismatch linear precoding massive mimo mutual coupling time division duplexing tdd ntroduction assive mimo one key potential technologies upcoming systems base stations bss deploy large antenna arrays several tens hundreds antenna units per array facilitate high beamforming spatial multiplexing gains systems feasible transmit downlink pilots antenna order estimate corresponding spatial channels user equipments ues feedback channel state information csi amount overhead approach proportional number antennas side massive mimo raeesi gokceoglu valkama department electronics communications engineering tampere university technology tampere finland work supported finnish funding agency technology innovation tekes project evolution take wireless communication networks academy finland projects tut graduate school work submitted ieee possible publication copyright may transferred without notice version may longer accessible systems thus envisioned primarily deploy duplex tdd based radio access rely reciprocity physical uplink downlink channels obtaining csi turn requires substantially smaller pilot reference signal overhead proportional number ues common assumption tdd systems physical propagation channels reciprocal within coherence interval impacts side transceiver analog effective downlink uplink channels reciprocal hardware induced phenomenon often referred channel nrc problem typically mismatches frs side radio transmit receive modes seen main cause nrc another source nrc considered literature differences mutual coupling antenna units associated transceivers transmit receive modes impacts nrc achievable system performance studied various works recent literature end provides downlink analysis general mimo system types precoding nrc due mismatch specifically focusing massive mimo systems study achievable downlink transmission mrt precoding schemes demonstrating significant performance degradation practical values nrc parameters also large amount work reported literature addressing estimation mitigation nrc tdd based mimo systems studies divided three main categories follows carries using reference antenna help additional circuitry method capable estimating side nrc carries without additional circuitry mutual coupling antennas utilized exchanging pilot signals reference antenna similar also method estimates side nrc also commonly neglects mutual coupling mismatch iii transmits specific pilot signals ues ues send back received signals certain properly precoded forms facilitate side nrc parameter estimation often referred ota approach work focus estimation mitigation nrc massive mimo system context deploying mrt precoding novelty contributions paper summarized follows ieee transactions signal processing return real imaginary parts complexvalued arguments respectively element row column matrix represented vij whereas element main diagonal diagonal matrix shown circularly symmetric gaussian distribution variance denoted finally denote identity matrices respectively consider generalized nrc induced coexisting mismatches associated radio transceivers sides well mutual coupling mismatches side antenna system unlike many earlier works consider mismatch ystem odel roblem ormulation respect reports similar consider tdd based downlink modeling however proposed mitigation scheme transmission scenario large number suitable mainly small scale mimo systems antenna units denoted transmits ues antennas simultaneously resource address estimation mitigation nrc signal system models written sources sides unlike many arbitrary subcarrier underlying orthogonal frequency works address side nrc division access wave shown popular assumption form ifft fft downlink demodulation pilots side nrc sides respectively major cause performance degradation multiin ideal tdd massive mimo system effective uplink user massive mimo systems thus strongly motivating downlink channels consist reciprocal physical incorporate effects nrc estimation channels building downlink transmission done mitigation processes beamforming downlink data based unlike massive mimo nrc mitigation works estimated channels uplink pilot sequences length assume availability downlink symbols work assume procedure pilots side consider appealing massive downlink transmission however consider mimo scenario downlink pilots generalized uplink downlink effective channel models thus ues rely statistical properties due radio mismatches beamformed channels decode received downlink respect uplink model channel signals estimation phase corresponding downlink received demonstrate performance proposed scheme signal model beamformed data transmission phase imperfect uplink csi unlike works effective channels expressed commonly rely perfect uplink csi assumption uplink training organization paper follows fundamental downlink transmission signal models considered massive mimo system denotes precoded user data whereas mrt precoding schemes nrc first effective uplink downlink multipresented section downlink user mimo channels respectively elaborated precoding approach formulated given nrc estimates detail later section processed noise matrix section iii novel pilot signaling method denotes side thermal noise ues introduced followed proposed novel vector assumed consist elements iterative estimation side nrc matrices average signal noise ratios snrs uplink results empirical performance evaluations terms downlink denoted respectively basic achievable system spectral efficiency presented section system framework largely based following seminal incorporating proposed scheme towork marzetta reciprocal channels gether existing nrc assumed methods reference finally conclusions drawn section effective relative uplink downlink channels notations throughout paper vectors matrices illustrated fig complete description denoted lower upper case bold letters respectively uplink downlink effective channels appearing vector matrix superscripts expressed indicate transposition hermitian transpose pseudo inverse operations pft respectively expectation operator shown represents trace operator diag operator transforms vector diagonal matrix elements joint diagonal vice versa reads diagonal elements matrix ues frequencythe input matrix column vector work response matrix mutual coupling matrix mitigation channel massive mimo systems mimo propagation channels ktx accounts deviation diagonal entries ideal reciprocal response detailed modeling entries matrices based practical nrc modeling introduced denoting variance diagonal elements corresponding variance diagonal elements denoted power elements controlled input reflection coefficients variance characterization given generally referred literature channel ideal reciprocal channel model special case base station mutual coupling lnrx mnn base station lntx channel estimation beamforming nrc mutual coupling mimo propagation channels first shortly address influence nrc downlink transmission carried without processing nrc precoding adopted respect required downlink channel estimate obtained orthogonal uplink training signals observation model given already first line complemented lmmse channel estimator described yields formally krx fig basic system models uplink downlink transmission reception including physical propagation channels transceiver frequency responses antenna mutual coupling devices estimated downlink uplink effective channels respectively using estimated downlink effective channel user data vector assumedh power normalization form precoded reciprocal physical channel subscripts denote transmitting receiving modes respectively note matrices linear precoding matrix reads diagonal mutual coupling matrix general mrt diagonal entries general effective channels assumptions modeling clearly due differences modes radio without loss generality scalar chosen array responses satisfy unit average transmit power constraint hence effective uplink downlink channels described relative agt received signal nrc received downlink signal vector given general diagonal matrix diagonal second line plugging precoded symbol vector entry denoted corresponds expression received signal user ratio modes following corresponding element written similar use decomposition form diagonal matrix measures deviation unity ratio diagonal entry denoted full matrix incorporates denote column row vectors responses mutual coupling side following precoder effective downlink channel matrices respectively notational convenience use decomposition notice denoting column uplink effective ieee transactions signal processing channel matrix effective downlink channel towards user expressed general conventional mimo systems employ downlink pilots acquire downlink csi detection purposes however massive mimo systems shown generally assumed ues employ properties beamformed channel namely downlink csi decode received signal assumption justified law large numbers implies commonly known channel hardening concept utilizing approach acquiring downlink csi ues eliminates need sending downlink pilots directly reduces downlink overhead building plugging received signal nrc general form zsi ziui zsi iui ziui given zsi gkt buk ziui gkt bui based clearly observed nrcblind precoder constructed based estimated uplink effective channel take account nrc effects results increased interference levels thus reduced downlink spectral efficiency illustrated elementary system spectral efficiency evaluation fig detailed evaluation assumptions described section noticed particular precoder case precoding results substantial performance degradation hence strongly motivating develop efficient nrc estimation mitigation techniques downlink precoding principle shown section mrt precoders applied naively without accounting nrc additional iui terms substantially degrade quality received signal side introduce novel nrc mitigation approach called precoding seeks cancel effects nrc properly modifying precoder assuming already estimates nrc matrices denoted precoding approach transforms basic linear precoders given unrc note special case nrc estimation method capable estimating side nrc reduces unrc fig spectral efficiency downlink snr system spectral efficiency performance nrcaware precoder assuming ideal nrc estimates shown fig observed precoder achieves ideal system performance performance fully reciprocal channels evaluation setup details spectral efficiency calculations described section iii roposed stimation nrc atrices nrc mitigation method precoder described section requires knowledge matrices information matrices readily available hence calling efficient estimation approaches thus section propose novel iterative ota estimation framework acquiring accurate estimates based novel pilot signaling concept ues general nrc variances corresponding realizations elements depend hardware characteristics operating conditions temperature vary slowly time thus nrc characteristics corresponding realizations assumed stay constant many propagation channel coherence intervals therefore sufficient perform nrc estimation infrequently every minutes day makes estimation overhead negligible compared signaling pilot overhead commonly rises channel estimation procedures proposed pilot signaling order estimate matrices propose following pilot signaling approach transmits orthonormal pilot matrix xnrc upon reception without decoding ues send back conjugated versions received signals based scheme received signal matrix side written rnrc hxnrc mitigation channel massive mimo systems csi nrc estimation test csi acquisition uplink pilots ttra test nrc acquisition downlink pilots uplink pilots downlink uplink downlink uplink fig assumed radio frame structure incorporating csi nrc estimation well actual data transmission phases downlink snr multiuser receiver noise matrix entries tilde sign used follows distinguish variables actual data transmission pilot signaling phases corresponding received signal ues sending back conjugated version reads ynrc processed received signal corresponding channel estimate available multiple parallel based radio system hence iterative estimation scheme carried per subcarrier manner well furthermore mentioned transceivers behavior modeled transfer functions therefore reasonable assume nrc matrices largely similar set adjacent subcarriers csc typically csc whereas subject variations depending frequency selectivity propagation channels based assumptions estimates obtained averaging per subcarrier estimates csc neighboring subcarriers uplink snr receiver noise matrix entries effective noise matrix seen denoted duration described overall csc pilot signaling symbols uplink downlink channels assumed fixed coherence time csc physical channels typically order several hundred csc symbol intervals determined mostly mobility csc ues system hence assume scenario coherence time least denotes subcarrier index next present symbols taking account pilot signaling actual proposed methods obtain estimates uplink channel estimation mentioned previous simplify notation drop subcarrier index section matrices expected change slowly compared channel coherence time hence assumed values fixed pilot signaling proposed estimation fig illustrates overall assumed radio frame subas described earlier iteratively refined using current frame structure considered massive mimo tdd system estimate proposed estimator builds solving including proposed nrc estimation phase matrix equation based minimizing frobenius norm criterion setting refined estimate formulated overall estimation framework initial step estimating processes argmin received signal ynrc ynrc nrc since pilot matrix xnrc property nrc xnrc subscript frobenius norm processed signal expressed next denoting agt following identity processed noise matrix given nrc target estimate assuming uplink channel estimate respect denoting estimates iteration denote column propose following iterative estimation framework respectively since term sum depends initialize obtain estimate minimizing total sum equivalent separately substitute obtain estimate minimizing term thus estimation successively refine estimates fixing matrix eventually simplified estimation current value one solving column independently used initialization since deviation mentioned earlier nrc matrix incorporates matrix practice small notice mutual coupling side ieee transactions signal processing solution given red corresponding antenna unit fig illustration sparsity threshold rectangular antenna grid antenna spacing power mutual coupling two different antenna units related physical distance thus elements become smaller distance two corresponding antenna units increases estimating nrc matrix treat entries corresponding two antennas distance larger threshold called sparsity threshold zeros yielding sparse matrix structure also define maximum number coupled neighboring antenna elements fig example rectangular antenna layout antenna spacing neighboring shown different values namely measured multiples assumed coupling whereas central antenna elements coupled closest neighboring antenna elements note antenna elements close edges grid coupled less number antenna units illustrated fig bottom left antenna element assumed coupled antennas following estimator build assumption sparse structure number row entries column denoted satisfies assumed index entries known directly determined antenna system architecture geometry assumed coupling threshold discussed assumptions define reduced vector dimension bred contains entries row kept constructing bred similarly column kept construct tred based formulate estimation columns reduced system equations red red argmin bred solved obtained straightforwardly appending zeros appropriate rows note agt matrix ggh positive matrix rank rank obtained corresponding minimum expression depend corresponding values column space tred higher dimensionality larger thus fixed larger one solve red yields smaller values tred proposed estimation next given refined estimate formulated based minimizing frobenius norm criterion argmin diagonal solution obtained diag iik vector given defining matrix diag column given proof see appendix umerical valuations nalysis basic evaluation settings performance measures section using extensive computer simulations evaluate performance proposed nrc estimation mitigation scheme also compare performance performance two existing schemes literature namely based least squares known argos generalized neighbor latter optimized version generalized method presented shown best performance amongst several existing nrc estimation methods based methods estimate nrc means mutual coupling antennas depend downlink pilots compensate nrc side expectation different nrc realizations channel coherence intervals length downlink pilots symbols denoted total number symbols channel coherence interval sinr instantaneous signal interference noise ratio sinr written based sinr scaling useful signal term available receiver context proposed nrc estimation mitigation method csi used hence proposed estimation method contrary two estimation methods utilizes downlink pilots csi acquisition described relevant performance metric normalized mean squared error mse nrc estimation defined side diag diag side proposed method proposed method proposed method spectral consider spectral efficiency key performance metric defined sinr nrc estimation normalized mse mitigation channel massive mimo systems reciprocal channel proposed method proposed method proposed method nrc mitigation mrt fig nrc estimation normalized mse system spectral efficiency different values sparsity input reflection coefficients variance threshold baseline simulation scenario consider equipped infinitely thin dipole antennas square layout spacing illustrated fig input mutual impedances computed based subcarriers csc nrc realizations assumed ghz assumed constant finally variances transceivers impedances assumed side assumed modulated signal bandwidth much smaller baseline carrier frequency uplink channel matrix assumed simulation settings parameter values elements serves also varied evaluations ues simultaneously resource either mrt precoding assume scenario coherence interval contains numerical results ofdm symbols number uplink pilots sent effect sparsity distance threshold coherence interval equal number scheduled study effect normalized mse ues uplink snr phase assumed system spectral efficiency respect fig illustrates scenarios ues rely downlink normalized mse nrc estimation pilots decoding purposes based baseline system settings value varied generalized neighbor methods number downlink seen choice estimating pilots coherence interval set diagonal elements yields lowest mse nrc snr equal downlink snr data transmission estimation note proposed nrc estimation method phase assumed snr choice influences side estimation well since coupling channel two neighboring antennas set estimated iteratively described section two mentioned nrc mitigation methods hand highest nrc estimation accuracy uplink downlink snrs pilot signaling achieved whereas higher proposed nrc estimation framework set estimation accuracy obtained respectively proposed method following spectral efficiencies plotted fig estimated nrc matrices averaged neighboring illustrate combined effect nrc estimation proposed method proposed method proposed method number ues nrc estimation normalized mse ieee transactions signal processing nrc estimation normalized mse iteration number fig nrc estimation normalized mse nrc estimation iteration number spectral sufficient amount iterations convergence obtained commonly order iterations illustrated specifically next effect number iterations fig illustrates reduction nrc estimation normalized mse nrc estimation iteration steps shown fig even proposed method high nrc levels proposed method proposed method iteration rounds sufficiently good proposed nrc mrt nrc mitigation estimator converge therefore continuation set number iteration rounds number ues effect number scheduled users fig nrc estimation normalized mse system spectral fig nrc estimation normalized mse system spectral efficiency efficiency plotted number scheduled ues number ues fig shows based worst performance proposed method high seen highest spectral efficiency achieved best option estimating nrc accuracy mse order based generalized neighbor normalized mse side nrc shown mentioned fixed nrc characteristics additional downlink pilot signaling per coherence fig evaluates normalized estimation mse system spectral efficiency different values interval used together side estimation number scheduled ues fig shows higher side nrc acquisition however detailed description nrc estimation accuracy achieved provided actual pilot signal structure actual whereas number scheduled users estimation method corresponding system spectral efficiency performances exceeds choice yields highest evaluated shown fig proposed nrc nrc estimation accuracy nrc estimation estimation mitigation scheme clearly outperforms directperformances largely similar three choices path based generalized neighbor methods following fig illustrates spectral efficiency difference performance proposed method perspective optimum sparsity distance threshold two methods increases grows remarkably thus continuation difference proposed method used settings two methods already order respectively discussed previous section another advantage utilizing proposed nrc estimation used estimation scheme optimum number ues kopt process rank therefore higher number defined number scheduled ues maximizes increases accuracy nrc estimation spectral efficiency higher compared two nrc proposed method facilitates estimation nonestimation methods instance precoding kopt diagonal elements higher values proposed method whereas noted cases fig based methods around opt fig proposed iterative nrc estimator executed spectral nrc estimation normalized mse mitigation channel massive mimo systems based generalized neighbor proposed method proposed method proposed method generalized neighbor based mrt number ues fig spectral efficiency input reflection coefficients variance spectral proposed method generalized neighbor based mrt number ues fig nrc estimation normalized mse system spectral efficiency number ues spectral mrt precoding effect input reflection coefficient fig shows impact variance input reflection coefficients achievable spectral efficiency proposed estimation mitigation method outperforms two based methods difference proposed method two methods increases grows due ability proposed method estimate elements nrc matrix noted used obtaining results fig still room improving performance proposed method adaptively selecting optimum according level shown already fig summary obtained results overall observed extensive numerical evaluations various scenarios proposed nrc estimation method outperforms two methods selected technical aspects summarized follows proposed method generalized neighbor based snr fig spectral efficiency downlink snr effect downlink snr fig system spectral efficiency plotted downlink snr results show clear advantage employing proposed method estimating nrc snr values proposed estimation mitigation method outperforms based methods low high snr regions especially performance difference visible high snr region employing proposed nrc estimation method eliminates need send downlink demodulation pilots since proposed ota framework facilitates estimating side side nrc characteristics base station therefore resources allocated coherence interval actual downlink data transmission purposes improves spectral efficiency proposed nrc estimation method superior two reference methods number scheduled ues grows reason increasing forcing two nrc estimation methods spend time downlink pilot transmission coherence interval larger number scheduled users improves accuracy proposed nrc estimation method due ability estimate also elements nrc matrix difference performance proposed nrc estimation method two methods increases power antenna mutual coupling mismatch grows ieee transactions signal processing onclusion work proposed efficient nrc estimation mitigation framework massive mimo tdd networks compensate jointly coexisting side nrc general even relatively modest nrc levels cause significant performance loss achievable spectral efficiency standard mrt downlink precoding employed novel approach incorporating dedicated pilot signaling small pilot overhead together efficient iterative estimation techniques proposed acquisition nrc matrices unlike existing methods proposed nrc estimation method acquires transceiver nrc well transceiver nrc rely downlink pilot transmission actual data transmission phase compensate nrc side therefore efficiently employed massive mimo systems rely statistical knowledge beamformed downlink channels terminals data decoding low system pilot overhead extensive computer simulations showed practical values nrc levels snrs number spatially multiplexed users proposed estimation mitigation method clearly outperforms existing methods terms system spectral efficiency ppendix roof estimation let diag using diag therefore solution obtained diag iik argmin since convex solved partial derivative equation finally yields solution given acknowledgment estimation building work presented proof appendix result discussion authors ngo larsson whose technical guidance greatly acknowledged eferences boccardi heath lozano marzetta popovski five disruptive technology directions ieee communications magazine vol february shepard anand marzetta yang zhong argos practical base stations proceedings annual international conference mobile computing networking ser mobicom new york usa acm larsson edfors tufvesson marzetta massive mimo next generation wireless systems ieee communications magazine vol february bourdoux come khaled transceivers systems impact mitigation radio wireless conference rawcon proceedings aug zou raeesi wichman tolli valkama analysis channel due transceiver antenna coupling mismatches tdd precoded downlink ieee vehicular technology conference sept petermann stefer ludwig wubben schneider paul kammeyer ofdm tdd systems transceivers ieee transactions communications vol september wei wang reciprocity mutual coupling tdd massive mimo systems wireless communications signal processing wcsp international conference oct zhang ren pan chen lamare dai antenna systems hardware mismatch achievable rates analysis calibration ieee transactions communications vol april athley durisi gustavsson analysis massive mimo hardware impairments different channel models european conference antennas propagation eucap may rogalin bursalioglu papadopoulos caire molisch michaloliakos balan psounis scalable synchronization reciprocity calibration distributed multiuser mimo ieee transactions wireless communications vol april vieira rusek tufvesson reciprocity calibration methods massive mimo based antenna coupling ieee global communications conference dec wei wang zhu wang sun mutual coupling calibration multiuser massive mimo systems ieee transactions wireless communications vol jan wei wang wang tdd reciprocity calibration massive mimo systems iterative coordinate descent science china information sciences vol guillaud slock knopp practical method wireless channel reciprocity exploitation relative calibration proceedings eighth international symposium signal processing applications vol august zou raeesi valkama efficient estimation compensation transceiver precoded tdd mimoofdm systems ieee vehicular technology conference sept guillaud kaltenberger towards practical channel reciprocity exploitation relative calibration presence frequency offset ieee wireless communications networking conference wcnc april mitigation channel massive mimo systems raeesi gokceoglu zou valkama performance analysis massive mimo downlink channel imperfect csi submitted ieee transactions communications online available http yang marzetta performance conjugate beamforming antenna systems ieee journal selected areas communications vol february jose ashikhmin whiting vishwanath channel estimation linear precoding multiuser tdd systems ieee transactions vehicular technology vol jun jose ashikhmin marzetta vishwanath pilot contamination precoding tdd systems ieee transactions wireless communications vol august larsson poor joint beamforming broadcasting massive mimo ieee transactions wireless communications vol april ngo larsson marzetta massive downlink tdd systems linear precoding downlink pilots communication control computing allerton annual allerton conference oct hochwald marzetta tarokh channel hardening implications rate feedback scheduling ieee transactions information theory vol sept schelkunoff friis antennas theory practice new york john wiley sons ngo larsson channel estimation multicell multiuser mimo systems large antenna arrays ieee international conference acoustics speech signal processing icassp march
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concept formation dynamics repeated inference deep generative models dec yoshihiro naganoa ryo karakidab masato okadaa graduate school frontier sciences university tokyo kashiwanoha kashiwa chiba japan artificial intelligence research center national institute advanced industrial science technology aomi tokyo japan brain science institute riken hirosawa wako saitama japan research fellow japan society promotion science chiyoda tokyo japan abstract deep generative models reported useful broad applications including image generation repeated inference data space latent space models denoise cluttered images improve quality inferred results however previous studies qualitatively evaluated image outputs data space mechanism behind inference investigated purpose current study numerically analyze changes activity patterns neurons latent space deep generative model called variational vae kinds inference dynamics vae demonstrates noise added input data identified vae embeds dataset clear cluster structures latent space center cluster multiple correlated data points memories referred concept study demonstrated transient dynamics inference first approaches concept moves close memory moreover vae revealed inference dynamics approaches abstract concept extent uncertainty input data increases due noise demonstrated increasing number latent variables trend inference dynamics approach concept enhanced generalization ability vae improved corresponding author email addresses nagano yoshihiro nagano ryo karakida okada masato okada preprint submitted neural networks december keywords deep generative models variational inference concept formation introduction research deep generative models extract essential features unlabeled dataset currently active area deep generative models reported useful broad range applications generating images movies text particular conventional bidirectional network structure recognition generation images made possible eliminate noise cluttered images smoothly interpolate different images detail recognition process mapping data point latent variable generation inverse process several studies qualitatively highlighted importance repeated inferences data space latent space present study repeated inferences defined process deep generative model repeats recognition generation images shown using images initial values deep generative models eliminate noise repeating recognition generation several times moreover compared generating output image latent space smoothly morph one image another repeating inferences several times improves quality output image however studies qualitatively evaluate output data fill gap literature quantified dynamics repeated inferences latent space investigate repeating inferences effective wide range applications study focused dynamics repeated inferences variational vae typical type deep generative model first images presented initial inputs vae denoises images generates clean outputs using repeated inferences various factors noise real environments cause data deviate original distribution according manifold hypothesis data natural images widely used typical applications likely concentrate vicinity much lower dimensionality rather high dimensional space data actually presented therefore suggested inference begins outside training data noise added initial inputs however little works paid attention activity patterns neurons drawn original initial point latent space understand patterns repeated inferences used dataset clear cluster structures could intuitively grasped eyes numeric characters examine vae expresses cluster structures latent space utilized mathematical notion known concept first introduced study associative memory models concept referred centroid multiple correlated data points memories analytically demonstrated spontaneously evolve equilibrium state phenomenon called concept formation furthermore dynamics neural activity patterns also studied terms associative memory model multiple correlated memories revealed dynamics neural activity patterns first approach concept move toward memory pattern summary four major findings study first consistent reports demonstrated dynamics repeated inferences drawn unique memory via corresponding concept center cluster latent space second averaging clusters latent space defined abstract concept definition memories concepts abstract concept hierarchically related ascending order found inference dynamics approaches abstract concept extent uncertainty input data increases due noise result suggested model selects appropriate inference strategies accordance fraction noise added input data third identified approximate necessary number latent variables map memories latent space number latent variable increases internal representations clusters tend become orthogonal makes dynamics repeated inferences approaches corresponding concept finally checked generalization error vae result demonstrated generalization performance model improved extent concept observed attract dynamics repeated inferences method variational vae generative model consisting two neural networks namely encoder decoder encoder sends mapping data natural images latent variable space decoder gives inverse mapping objective function vae obtained finding variational lower bound log training data following parameter maximizes log data point considered using latent variable conditional probability distribution taking variational lower bound gives following objective function log log equation prior distribution latent variable dkl qkp divergence probability distributions first term objective function corresponds regularization second term corresponds reconstruction error vae models conditional distributions using respective neural networks optimize parameters backpropagation samples generated method called reparameterization trick encoder latent variable modeled decompose random variable deterministic variables giving sample standard gaussian distribution eliminates need complicated integral training assumed conditions expected reconstruction error log approximated sample average rewritten log log outputs encoder parameter determining parameter decoder trained gradient ascent method maximize output decoder set probability bernoulli distribution expectation conditional probability namely second term objective function approximated average samples separate neural network used encoder decoder mentioned number units middle layer set number samples calculating reconstruction error set activation function set tanh adam used parameter optimization algorithm learning rate reduced descending order number units latent variable set unless otherwise noted modified national institute standards technology mnist database consists handwritten images training data test data used dataset training data considered cluster structures consisting types labels namely study noisy mnist data inferred using trained network according following procedure time evolution latent variable obtained first noise added image training dataset pixels probability selected pixels image intensities selected pixels swapped image set next data variable step taken finally generation recognition repeated times according following two equations obtain time evolution data variable latent variable dynamics numerically analyzed shown deep learning framework keras version theano backend version running cuda cudnn nvidia tesla used numerical simulations results dynamics inference trajectory approach concept first analysis demonstrate transient dynamics inference latent space vae first attracted concept center memorized patterns moves memory representation latent space vae captures cluster structures hidden behind minist data vae displays corresponding clusters latent space fig details expression latent space vae given appendix since cluster structure exists data latent variable vae reflects time evolution inference also seems reflect cluster structure center cluster numerical label namely num nnum num therefore defined concept num means activity pattern latent variable training data label num num num definition concept based two studies associative memory model relationship time development inference concept label represented mnist data numerically analyzed following sections figure consecutive samples data space left right one row image noise applied used initial value image generated concept shown right numerically demonstrated transient dynamics activity pattern first attracted concept center memorized patterns moves memory figure expresses consecutive samples data space time development activity pattern data space aligned left right one row upper left image corresponds initial value image noise applied used initial value image generated concept shown right figure vae removes noise contained image first steps gradually shifts specific image qualitatively suggested result vae inference approaches concept gradual changes output images data space quantitatively evaluated latent space time evolution euclidean distance namely distance step figure time development distance concepts labels mnist data shades represent standard error mean trials figures generated noise fraction neural activity patterns concepts every label mnist data fig latent space evaluated distance concept different initial images calculated figure corresponds label used initial input vae expresses time step repeated inference expresses euclidean distance clarified trajectory vae inference first rapidly approaches concept moves away result qualitatively consistent labels minist data vae makes latent space activity patterns orthogonal relation another cluster correlated cluster lowdimensional space relationships concepts label shown fig activity patterns latent variable space numerical concept shown fig heat map expresses activity pattern neuron corresponds latent variable represents neuron index represents label neurons contribute information representation many neurons pruned active according observation neurons active dimensions latent space examined using cumulative contribution ratio determined principal component analysis cumulative contribution ratio principal component training images given vae shown fig variance latent space explained entirely index input label cumulative ratio components figure activity pattern latent variable space concept represents neuron index latent variable represents label heat map shows activity pattern neuron cumulative contribution ratio principal components cosine similarity activity patterns label dimensions explained nine dimensions examine relationships concepts cosine similarity activity patterns concept fig obtained definition cosine similarity concepts label shown diagonal terms figure hand cosine similarity concepts different labels terms minimal namely near zero figure shows concepts label orthogonal latent space suggested activity patterns corresponding different training data label correlated different label orthogonal latent space vae relationship data hierarchy inference previously arbitrarily determined amount noise added initial input images second analysis examine effect noise dynamics repeated interferences modulated amount noise since noise input images causes data deviate original distribution created another class abstract concept averaged labels concepts addition concept memory respectively measuring distance trajectory neural activity pattern corresponding classes latent space one classes attracts respective neural activity patterns identified abstract concept defined num three classes memories concepts abstract concept hierarchical relationship detailed coarse information order num num calculated minimum distances respective neural activity patterns corresponding classes min figure shows minimum distances according noise fraction figure represents noise fraction probability image intensities pixels swapped every noise fraction minimum distances firing pattern hierarchical concepts calculated changing initial image times dots figure express mean minimum distance bars standard error mean trials parameter regions divided three stages iii correspond minimum distance firing pattern hierarchical concepts iii mint noise fraction figure minimum distances concepts according noise fraction respectively stage firing activity closest original pattern small amount noise interestingly closest concept moderate noise stage activity came close concept stage iii stages memory successfully retrieved inference path close cluster input data belongs however stage iii model could determine original cluster recall failed discovered noisy environment makes recognizing objects difficult neural activity pattern wanders around center memories accordingly model achieves inference dynamics depending input uncertainty space num eric spa concept figure schematic diagram firing patterns latent state space shown section vae extracts cluster structures inherent mnist data infers images center cluster dynamics inference shown schematic diagram fig considered adding noise image corresponds moving initial value direction orthogonal original data space results first analysis suggest inference starts position orthogonal space expressing mnist data activity patterns first approach corresponding concepts high speed transitions corresponding memories approaching concept repeated inferences indicates concept formation occurs latent space vae neuroscience activities visual cortex macaque monkey human brain measured meg reported process global information detailed information previous studies associative memory models explain behaviors spontaneous stabilization concept effect memory retrieval vae also recalled concept memory pattern inference phase results vae consistent findings studies associative memory models cerebral cortex effect model architecture internal representations third analysis mechanism trajectory vae inference approaches concept clarified follows first following question must answered memory structure mentioned affect time evolution inference words memory structure changed controlling hyperparameter vae verified trajectory inference five circumstances examined respectively fig time evolution distance concept different model hyperparameters compared figs number neurons latent variable controlled order condition step step step step step figure time development distance concept number elements latent variable written cosine similarity memory patterns concept corresponding model trajectory approached concept whereas condition approach concept also trajectory gradually approached corresponding concept figure omitted cosine similarity concepts label parameters shown figs similarity term approximately zero hand number latent variables decreases orthogonality concept decays results third analysis suggest orthogonality concepts necessary trajectory inference drawn concept since number latent variables decreases necessary express data fewer dimensions orthogonality concepts lost reduction number latent variables considered cause instability memory patterns corresponding training data stabilize concepts result trajectory inference goes straight stable point also numerically assessed whether labels confused repeated inferences vae although inference starts label incorrectly attracted concept associated label result assessment shown appendix engineering significance concept latent representation fourth analysis engineering significance attraction concept explained follows figure generalization performance model according performance model evaluated using variational lower bound test mnist data parameters minimize generalization error epoch total nine conditions selected learning rates size generalization error minimum value vicinity change significantly fourteen latent variable neurons express training data condition fig number neurons minimize generalization error consistent result results suggest latent neurons required express mnist data network structure used study moreover vicinity cluster structure appears representation latent variable space trajectory inference drawn concept results suggest possible judge generalization performance model without computing generalization error validation loss figure generalization error number elements latent variables represents variational lower bound test data orthogonality internal representations simply observing dynamics repeated inference conclusion future work found vae extracts cluster structures inherent mnist infers images via center cluster results first analysis suggest inference starts point far away original data distribution repeated inferences first approach concept high speed slowly move toward memory pattern learning inference multiple memory patterns widely studied using associative memory models associative memory model embedded multiple correlated patterns centroid correlated patterns spontaneously evolves fixed point time evolution activity patterns approaches concept results first second analyses qualitatively consistent findings suggesting mechanism underlying dynamics repeated inferences vae related traditional associative memory model originally matsumoto proposed model explain dynamics neural activities macaque monkey visual cortex studied sugase although vae investigated study model proposed matsumoto experiment conducted sugase different architectures nature results infer share universal working principle abstract level behavior repeated inferences vae qualitatively consistent time evolution firing patterns observed neuroscience literature current study used vae study simple hierarchical deep generative model taking account vae exhibits patterns similar biological activities future work examine dynamics repeated inferences introducing biologically plausible mechanisms plasticity common noise spontaneous firing based weight future investigation useful neuroscience engineering applications previously several studies demonstrated repeated inferences successfully denoise improve quality inferred images study suggests dynamics repeated inferences approaching center cluster inherent data leads denoising improving quality output images quantitatively observed data space critical take sufficient number latent variables precisely represent concept inherent data number latent variables sufficient cluster structures realized latent space concept hardly identified results suggest stage fig appears number latent variables sufficiently large number latent variables qualitatively changes dynamics repeated inferences study introduced hierarchical concepts num num reflect hierarchical structure mnist dataset previous works discussed relationship hierarchy data deep neural networks example deep neural networks claimed express abstract information deeper layers particular bengio stated deep layers speed mixing markov chains using ability manifest abstract information hand saxe analytically showed deep neural networks learn data order large small modes internal representations branch accordingly hypotheses previous studies pointed representations learning dynamics deep neural networks reflect data hierarchy study suggests inference process deep generative model also related hierarchy data recently researchers actively working models capture features inherent data forms internal representations vae used study embeds data points simple isomorphic gaussian distribution next step expand works using deep generative models aim investigate factors influence behavior repeated inferences approaching concept likewise analyze dynamics repeated inferences another model used training datasets various hierarchies acknowledgements work supported jsps fellows grant number research activity grant number japan society promotion science jsps appendix pca embedding latent representations show vae extract cluster structure hidden behind data activity patterns latent variable using principal component analysis pca shown fig results principal component analysis using data shown fig result using three labels shown fig color point represents label data mnist considered cluster structure consisting types labels figures show latent variables vae extract cluster structure figure pca embedding representations latent variable space color point represents label data pca embedding data corresponding three labels appendix verifying effect moving numbers possibility labels confused repeated inferences vae numerically tested showed trajectory inference approaches concept orthogonality representation latent space hand also conceivable escape concept caused attraction another cluster eliminate possibility discriminative neural network constructed separately vae final state inference vae classified following analysis model structure constructed discriminative neural network kernel size convolution set three size pooling two probability dropout set order input side relu used activation function model recorded discrimination ability test data included mnist dataset result classifying final state inference using discriminative neural network shown fig represents trial inference various initial images represents number label heat map indicates classification probability number label trials prob number step figure result classification final state inference image time evolution distance concept condition excluding trials activity pattern switched different numbers expressed red condition containing trials expressed gray image used initial value inference discriminator classified final state trials trials consider effect labels cause neural activity patterns approaching mismatched concepts taking example label first measured distances neural activity pattern concept two conditions included neural activity patterns reminded inside cluster one condition neural activity patterns latent space averaged distances condition compared means average trajectories compared fig red shows average trial final state identified gray shows average trials shown figure neural activity patterns conditions approached concept moving corresponding memories result suggests presence labels cause neural activity patterns move away concept references amari neural theory association biological cybernetics amit gutfreund sompolinsky models neural networks physical review arulkumaran creswell bharath improving sampling generative autoencoders markov chains arxiv preprint bengio mesnil dauphin rifai better mixing via deep representations proceedings international conference machine learning brincat connor dynamic shape synthesis posterior inferotemporal cortex neuron chollet keras https goodfellow mirza ozair courville bengio generative adversarial nets advances neural information processing systems higgins matthey pal burgess glorot botvinick mohamed lerchner learning basic visual concepts constrained variational framework proceedings international conference learning representations hopfield neural networks physical systems emergent collective computational abilities proceedings national academy sciences karakida igarashi nagata okada correlation network common noise journal physical society japan katori otsubo okada aihara stability analysis associative memory network composed stochastic neurons dynamic synapses frontiers computational neuroscience kingma adam method stochastic optimization proceedings international conference learning representations kingma welling variational bayes proceedings international conference learning representations kullback leibler information sufficiency annals mathematical statistics lee grosse ranganath convolutional deep belief networks scalable unsupervised learning hierarchical representations proceedings annual international conference machine learning monroe shi ritter jurafsky adversarial learning neural dialogue generation conference empirical methods natural language processing liu harris kanwisher stages processing face perception meg study nature neuroscience matsumoto okada yamane neuronal mechanisms encoding information inferiortemporal cortex journal computational neuroscience murata otsubo nagata okada oscillations spurious states associative memory model synaptic depression journal physical society japan nagano karakida watanabe aoyama okada input response neural network model lognormally distributed synaptic weights journal physical society japan nickel kiela embeddings learning hierarchical representations advances neural information processing systems okada notions associative memory sparse coding neural networks radford metz chintala unsupervised representation learning deep convolutional generative adversarial networks arxiv preprint rezende mohamed wierstra stochastic backpropagation approximate inference deep generative models arxiv preprint rifai dauphin vincent bengio muller manifold tangent classifier advances neural information processing systems saito matsumoto saito temporal generative adversarial nets singular value clipping ieee international conference computer vision saxe mcclelland ganguli exact solutions nonlinear dynamics learning deep linear neural networks proceedings international conference learning representations serban sordoni lowe charlin pineau courville bengio hierarchical latent variable model generating dialogues aaai sugase yamane ueno kawano global fine information coded single neurons temporal visual cortex nature theano development team theano python framework fast computation mathematical expressions arxiv url http tomczak welling vae vampprior arxiv preprint vondrick pirsiavash torralba generating videos scene dynamics advances neural information processing systems walker doersch gupta hebert uncertain future forecasting static images using variational autoencoders european conference computer vision zhang wang seqgan sequence generative adversarial nets policy gradient aaai
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optimal choice among class nonparametric estimators jump rate markov processes may romain inria est team bigs institut cartan lorraine france abstract markov process stochastic process whose behavior governed ordinary differential equation punctuated random jumps occurring random times focus nonparametric estimation problem jump rate stochastic model observed within long time interval ergodicity condition introduce uncountable class indexed deterministic flow recursive kernel estimates jump rate establish strong pointwise consistency well asymptotic normality propose choose among class estimator minimal variance unfortunately unknown thus remains estimated also discuss choice bandwidth parameters methods keywords jump rate kernel method nonparametric estimation piecewisedeterministic markov process mathematics subject classification contents introduction problem formulation definition notation assumptions estimation procedure inference times optimal estimation jump rate choose bandwidth parameters simulation study estimation algorithm process bacterial motility fatigue crack propagation ergodicity invariant measures proof theorem sketch proof remainder term martingale term proof proposition electronic electronic address corresponding author address introduction markov processes pdmp abbreviated form introduced literature davis general class stochastic models suitable modeling deterministic phenomena randomness appears point events motion pdmp may defined three local characteristics flow jump rate transition measure starting initial value process evolves deterministic way following first jump time occurs either flow reaches boundary state space fashion non homogenous rate cases location process time governed transition distribution motion restarts new point family stochastic models tackling various problems arising example biology neuroscience reliability indeed applications involving deterministic motion punctual random events may modeled pdmp typical examples models composed deterministic growths followed random losses example size cell grows exponentially time next cell divides two offsprings whose size half size parent cell proposing efficient statistical methods class stochastic models therefore great interest nevertheless particular framework involving deterministic motion punctual random jumps imposes consider specific methods instance authors shown wellknown multiplicative intensity model developed aalen estimating jump rate function directly apply pdmp alternative approaches thus proposed present paper focus recursive nonparametric estimation jump rate pdmp observation one trajectory within long time interval precisely purpose work show one may obtain kernel methods class consistent estimators jump rate one may choose among class optimal way best knowledge nonparametric estimation jump rate general framework never investigated pdmp may model large variety problems methods developed many authors statistical inference presented randomness pdmp governed two characteristics transition kernel jump rate consequence two main questions arise estimation problem process namely statistical inference features one hand papers investigate nonparametric methods estimating transition function pdmp either specific model general setting process hand estimation jump rate associated density function extensively studied several authors without attempting give exhaustive survey literature topic one may refer reader references therein book author studies likelihood processes observation pdmp could lead inference methods parametric setting papers deal nonparametric estimation pdmp used modeling population observed along lineage tree articles authors rely specific form features process interest order derive asymptotic behavior estimation procedure techniques generalized introduce nonparametric method estimating jump rate specific class pdmp monotonic motion deterministic breaks say transition measure dirac mass location depending procedures developed papers obviously great interest strongly use particular framework involved investigated models thus well adapted general setting authors show famous multiplicative intensity model applies estimating jump rate modified version underlying pdmp leads statistical method approximating conditional density associated jump rate process defined bounded metric state space main difficulty throughout present paper articles related presence deterministic jumps path reaches boundary state space feature often used modeling deterministic switching quantitative variable rises certain threshold statistical point view interarrival times therefore deterministic clock depending state space leads technical difficulties would like emphasize techniques developed references take account likely presence forced jumps dynamic one may also find literature papers focus estimation various functionals family stochastic models precisely authors provide numerical methods expectations exit times pdmp addition article deals pdmp introduced modeling temporal evolution exposure food contaminant consider statistical inference procedure estimating functionals first passage times methods studied model absorbed certain threshold many aspects approach papers different complementary indeed devoted estimation functionals pdmp focus direct estimation primitive data process article introduce kernel estimator computed observation embedded markov chain pdmp composed locations travel times along path establish pointwise consistency well asymptotic normality theorem estimate consider recursive may computed sequential data may relevant many applications deduce result two first corollaries nonparametric estimation conditional density interarrival time time conditionally event see corollary survival function see corollary also investigate corollary asymptotic behavior estimator composed function obtained ratio derive uncountable class indexed states hitted reverse flow time consistent estimates jump rate words get good estimate show one may choose among class estimators minimizing asymptotic variance state procedure equivalent maximize criterion along curve denotes invariant measure locations deterministic time reach following choice criterion far obvious without precisely computing limit variance central limit theorem presented corollary indeed naive criterion maximize invariant distribution along larger larger number data around higher quality estimation nonetheless simple criterion take account estimate also depends time interest see remark question also investigated numerical point view subsection show synthetic data choice criterion better naive one bandwidths kernel methods free parameters exhibit strong influence quality estimation discuss choice bandwidth parameters classic procedure consists minimizing integrated square error computed along reverse flow introduce procedure markov setting prove convergence propositions finally would like highlight regularity conditions impose non restrictive particular neither deterministic exit time state space assumed bounded function transition kernel supposed case see assumptions addition forms transition measure deterministic flow specified sequel paper organized follows begin section precise formulation framework see subsection main assumptions need article see subsection section devoted presentation statistical procedure related results convergence precisely kernel estimator times introduced investigated subsection derive class estimators jump rate propose choose among subsection crucial choice bandwidth parameters studied subsection finally presentation whole estimation procedure provided subsection illustrated sequel section three different application scenarios various sample sizes state space dimensions involving simulated real datasets precisely focus tcp window size process used modeling data transmission internet subsection estimation bacterial motility tackled subsection acceleration fatigue crack propagation considered subsection proofs technicalities postponed appendix end paper problem formulation section devoted definition pdmp presentation main assumptions impose paper list notations paper denotes borel algebra endowed euclidean norm addition lebesgue measure denoted particular notation case ball radius center denoted closure set denoted stands boundary definition notation motion pdmp may described solution ordinary differential equation punctuated random times random jumps governed transition kernel see figure random jumps occur either deterministic motion hits boundary state space non homogeneous rate taken along curve defined differential equation figure schematic path pdmp paper devoted nonparametric estimation statedependent rate governs spontaneous generation jumps trajectory precisely pdmp defined three local characteristics deterministic flow satisfies semigroup property jump rate transition kernel define deterministic exit times flow reverse flow inf inf sequel consider pdmp evolving open subset context impose usual standard conditions addition restrict case transition kernel admits density respect lebesgue measure assumption natural one considers multivariate pdmp satisfied various problems arising biology population dynamics insurance insurance model starting initial condition motion may described follows distribution first jump time given exp else words process jumps either flow hits boundary state space time fashion rate next location time defined transition kernel test function path first jump time given else starting location one chooses next time future location similar way one obtains strong markov process sequence jump times convention times defined integer finally denotes stochastic sequence locations xtn randomness pdmp contained stochastic sequence markov chain addition locations also form markov process state space condition paper denotes distribution nth location integer denotes markov kernel stands conditional distribution given exp else light would like highlight conditional distribution absolutely continuous respect unidimensional lebesgue measure sometimes singular component conditional density may obtained deriving exp sequel stands conditional survival function associated defined highlighted process forms markov chain set defined denotes transition kernel process denotes distribution couple assumptions main assumption impose present paper condition ergodicity markov chain property often keystone statistical inference markov processes may directly imposed established primitive features data assumption exists distribution initial distribution lim stands total variation norm assumption may checked directly markov kernel existence function doeblin condition instance theorem following remark establish first property sequence limit remark since transition kernel assumed absolutely continuous respect lebesgue measure kernel given locations also admits density consequence integer distribution thus invariant measure introduced assumption admit density state space sake clarity write slight abuse notation add regularity conditions main features process show convergence estimates theorem assumptions finite conditions used proof theorem find upper bound non diagonal terms square variation process interest functions lipschitz lip lip conditions used proof theorem control diagonal terms variation process study convergence remainder terms survival function lipschitz lip condition used proof theorem investigate convergence remainder terms deterministic exit time continuous condition used find admissible initial bandwidths finally consider additional condition transition kernels flow order ensure lipschitz mixing property markov chain sufficient establish almost sure convergence remainder term adequate rate proof theorem assumptions transition kernel markov chain satisfies addition composed function belongs regularity class defined positive numbers satisfying estimation procedure inference times integer introduce functions defined defined interior let recall set given fbn gbn vid vid vid denotes kernel function bandwidths defined integer initial positive values already noted quantities great interest statistical study times pdmp indeed see ratio estimates conditional density defined ratio estimates conditional survival function defined ratio estimates composed function addition name suggests state good estimate density unique invariant distribution locations relevant estimation problem pdmp already investigated proposition sequel impose assumptions kernel functions assumptions kernel function assumed nonnegative smooth function satisfying following conditions kkp finite supp together initial bandwidths condition avoids compute kernel estimator data located boundary state space see also remark function lipschitz lip condition used show proof proposition remark particular assumptions ensure finite used find upper bound ofrthe non diagonal terms variation process proof theorem addition integral also finite needed establish almost sure convergence remainder terms proof result sequel admissible set bandwidth parameters given min part main result obtained use vector martingales stated following theorem theorem couple inf appears third item assumptions almost sure convergence asymptotic normality gbn matrix degenerate one positive term position defined proof proof stated appendix remark existence couple satisfying obvious whenever exit time continuous see assumptions condition ensures times used calculus fbn gbn obtained forced jumps process reaches boundary state space case obvious interarrival times consistency asymptotic normality therefore still accurate without condition remark theorem choice initial bandwidths locally dependent point interest may appear restrictive may avoided considering elements compact subset inf indeed always exists couple inf inf thus satisfies point remark matrix appearing asymptotic normality presented theorem degenerate component positive means rate estimators gbn faster one fbn straightforward fbn obtained smoothing empirical distribution data spatial temporal directions contrary gbn proof previous result may adapted show central limit theorem diagonal keystone state convergence behavior given hook vector martingale assume geometric ergodicity markov chain one may also obtain rate convergence variances estimates fbn gbn uniformly compact subset parameters may uniformly chosen see remark proposition let assume exists geometric ergodicity particular ensured doeblin condition see theorem set couple sup var fbn sup var gbn var let recall defined rate convergence fbn faster one given theorem whenever proof proof similar demonstrations proposition corollary relies control covariance process functionals geometrically ergodic markov chain see theorem present sequel corollaries theorem interest estimation problem times first define estimator fbn fbn fbn usual convention following result convergence corollary couple satisfying fbn fbn proof result direct application theorem slutsky lemma another feature interest times survival function one may estimate quantity convention properties convergence stated following corollary corollary couple proof almost sure convergence direct application theorem central limit theorem consequence asymptotic normality established remark slutsky lemma optimal estimation jump rate propose estimate composed function ratio defined fbn gbn convention pointwise convergence asymptotic normality consequence theorem corollary couple satisfying proof result direct application theorem slutsky lemma together sequel focus estimation rate composed function estimate defined particular fixed value introduce indexed elements curve described reverse class estimators flow propose choose estimate class optimal way proceeding define notation definition addition define unique time satisfying thus following trivial result consequence propose define class estimators estimates see corollary already defined virtue corollary using one goes infinity asymptotic variance given paper choose approximate element minimizing asymptotic variance words optimal estimator jump rate obtained arg max remark good criterion maximize invariant measure indeed large large frequency locations around may available dataset nevertheless roughly speaking quantities interest well estimated large number locations around together times around naive criterion may corrected including quality estimation time say maximizing product also refer reader simulation study presented section precisely figure criterion generally uncomputable known features pdmp thus remains estimated light theorem definition naturally propose approximate quantity consequence propose estimate jump rate statistical approximation maximizing estimated criterion precisely bnbn arg max high oscillations alternatively high smoothness kernel estimator bandwidth suggest choice parameter appearing see crucial maximization step choose bandwidth parameters part devoted choice bandwidth parameters criteria introduce part choose features defined line integrals along curve consequence need ensure kind quantity well defined setting assumption starting condition reverse flow defines change variable say diffeomorphic mapping common literature minimize integrated square error ise choose optimal bandwidths kernel estimator particular dependent data one may refer reader another classical solution investigate behavior mean integrated square error mise framework gaussian dependent data authors shown optimal bandwidths obtained minimizing ise mise close dependence short range say covariance function integrable see theorem noted geometric ergodic markov chain satisfies kind condition see theorem let recall appears first computation estimated criterion need maximize along curve indeed computed estimate implicitly depends propose choose bandwidth parameter minimizing ise associated defined isen function given gbn gbn one may remark ise unusually computed along curve interest dependency holds function consequence optimal parameter minimizing stochastic function also minimizes ise function generally computable since unknown quantity appears definition consequence propose estimate popular technique selecting bandwidth minimizes ise authors would like highlight involves two main difficulties first estimators computed dependent data identically distributed addition almost surely data set integration whenever dimension larger propose specific procedure adapted framework defining estimate need introduce quantities first denotes hyperplane orthogonal addition introduce notation furthermore denotes tube around radius finally denotes unique time particular noted small enough focus method estimating quantity estimate defined observation embedded markov chain another pdmp independent first one distributed according parameters denotes usually euler function regularity conditions necessary investigate asymptotic behavior estimate proposition assumptions finite lipschitz lip deterministic exit time reverse flow lipschitz lip flow lipschitz lip finite lipschitz lip proposition conditionally lim proof proof stated appendix virtue proposition one may obtain estimate optimal bandwidth parameter arising small enough large enough addition minimizing quantity quantity also appears calculus estimator particular choice may done computing denominator consequence remains choose optimal way bandwidth parameters arising formula numerator fbn similar way propose choose minimizing ise associated fbn computed along curve isen fbn fbn implicitly depends stands given fbn fbn previous part propose estimate observation another pdmp define estimate embedded chain fbn fbn zek zek convergence investigated proposition proposition conditionally lim proof proof similar demonstration proposition stated appendix simulation study section provide presentation estimation procedure well three application scenarios simulated real datasets estimation algorithm sequel devoted presentation estimation procedure provided paper precisely interested estimation preliminary computations preliminary computations require manipulate flow state space compute curve choose compute compute unique solution mapping defined preliminary estimates preliminary computations require choose kernel functions well two positive values couple compute compute choice bandwidth parameters observation two independent embedded chains one determines optimal bandwidth parameters appearing preceding estimates practice one trajectory underlying pdmp one may divide data two categories largest one used estimation step relies one compute arg max choose compute arg max zek zek estimation finally compute best estimate preceding computations compute arg max compute bnf gbng remark time complexity algorithm depends several parameters namely number observed jumps number observed data steps also numbers discretizing state spaces maximization procedure easy see step complex remains polynomial precisely process application focus part variant famous tcp window size process appearing modeling transmission control protocol used data transmission internet presented protocol designed adapt traffic conditions network connection maximum number packets sent given random variable called congestion window size time step packets successfully transmitted one tries transmit one packet congestion appears model presented part two dimensional sake clarity use following notation denote components consider pdmp evolving state space deterministic part model defined flow given jump rate defined transition kernel defined starting process evolves unit square always right jump appears either motion hits boundary non homogeneous rate according weibull distribution two components location independent governed way process tends high probability jump left location jump one obtains process see figure second dimension models quality network upper second component higher probability congestion figure approximately asymptotic behavior process may represented invariant distribution locations since quantity unknown propose show figure estimate defined computed observed jumps figure approximately present procedure estimating jump rate location quality network average context class estimators indexed elements noted invariant distribution quite low see figure nevertheless method expected work pretty well even unfavorable framework simulation study assume observe embedded markov chain jump procedure computed additional chain independent first one observed jump boxplots presented computed replicates begin choice bandwidth parameter appearing procedure relies minimization estimate depends positive parameter present figure quantity function different values fortunately new parameter seems little influence behavior estimation ise along figure approximately denoted sequel maximize estimated criterion along curve see figure obtain optimal point compute estimator crucial role maximization step illustrated figure figure approximately continue choice couple implicitly appearing estimator fbn optimal parameters denoted sequel obtained minimizing estimate related ise see figure figure approximately compute estimators different values optimal bandwidths related boxplots presented figure procedure makes able choose best index time corresponds estimate least bias variance proves strong interest estimation algorithm developed paper figure approximately bacterial motility present model bacterial motility motile bacteria move use flagellum several flagella bacteria moves direction flagellum flagellum behaves like rotary motor periodically flagellum changes direction results reorientation bacteria allows bacteria change direction bacteria sense nutrients move towards additionally move response temperature light etc bacteria favorable environment frequency changes direction low intelligent behavior allowed fact jump rate direction depends environment example propose models trajectory bacteria coli particular author uses pdmp describe movement bacteria influence external attractive chemical signal present variant model presented path bacteria described pdmp evolving space state unit disk sake clarity use following notation denote components case position bacteria unit disk direction flagellum bacteria moves direction flagellum words flow given cos sin bacteria changes direction according jump rate function depends environment position current angle jump rate describes interaction bacteria environment bacteria changes direction new direction chosen preferentially direction favorable environment transition kernel models change direction case suppose bacteria information priori quality environment around thus defined starting position bacteria evolves direction jump appears either bacteria hits boundary environment rate cos sin direction next randomly chosen bacteria continues path new direction possible path model presented figure figure approximately simulation study jump rate taken constant equal words interaction bacteria environment course assumption taken account estimation goal estimate jump rate different points order check depends position actually provides estimate influence likely external attractive signal bacteria estimation algorithm presented paper yields estimate jump rate threedimensional state nevertheless one take account jump rate depend angle explain procedure reduce dimension estimate jump rate location bnbn let fixed angle estimate best index jump rate location bnbn estimates reproduce preceding step obtain connected trajectory around class optimal estimators indexed elements jump rate depend direction aggregate estimators obtain following jump rate estimate bnbn investigate estimation jump rate following target points trajectory changes direction sequel aggregated estimator approximated discretization interval step target points bnbn discretization grid present figure boxplot estimates figure approximately shown figure trajectories defined similar close boundary explain fact follows target state optimal point maximizes along cos sin satisfies cos sin recall gbn estimator first hand exp small enough point maximizes hand important number data close boundary due fact bacteria jumps hits boundary maximization gbn compromise near boundary target points close together close true value finally estimates small variance good indicator constant jump rate see figure figure approximately investigate effect size dataset quality estimation compare results simulations observed data trajectories time close boundary due fact case areas state space observed locations target point optimal point obtained location see figure addition estimated curves obtained data close ones computed observed jumps figure approximately course larger number data better estimations see figures particular difficult conclude jump rate constant observed jumps variance estimates large nevertheless dataset size may sufficient conclude jump rate depend location figures approximately fatigue crack propagation fatigue crack propagation stochastic phenomenon due inherent uncertainties originating material properties cyclic mechanical loads consequence stochastic processes offer appropriate framework modeling crack propagation fatigue life may divided crack initiation two crack growth periods namely linear stable regime described paris equation acceleration unstable regime modeled forman equation crack length time measured number loading cycles stress ratio represents maximal value stress intensify factor required induce failure range stress intensity factor given cos size test specimen stress range addition two unknown material parameters context models proposed analyze fatigue crack growth data authors propose estimate transition time paris forman regimes assuming crack propagation follows trajectory piecewisedeterministic markov process estimation procedure performed virkler dataset identical aluminium alloy specimens tested constant amplitude loading mpa stress ratio number loading cycles crack tip advance predetermined increment recorded initial crack length final length crack growth histories obtained tests see figure figure approximately paper assume crack growth propagation follows paris equation random parameters switching forman equation another set parameters distributed according transition measure transition occurs random time given survival function exp deterministic flow paris equation parameters sequel computed method explicit solution differential equation propose estimate jump rate stochastic model real crack growth data noted jump times directly observed virkler dataset estimated jump rate significant importance understanding transition stable unstable regimes crack propagation never estimated framework pdmp emphasize virkler dataset composed independent experiments thus directly follow theoretical framework developed manuscript nevertheless estimation procedure expected perform pretty well favorable context independent curves instead one markov path material parameters unobserved also estimated virkler experiments literature strong linear relationship features characteristic highlighted shown figure order reduce dimension thus simplify model propose parametrize paris equation one parameter say log obtained linear relationship figure approximately focus estimation crack length parameter paris equation exists unique deterministic time paris flow reaches time starting criterion maximum estimated different values see figure particular one may observe larger target length larger optimal parameter figure approximately crack length estimate jump rate parameter maximizing estimated criterion obtain estimated function displayed figure curve increasing expected describes transition rate stable region propagation towards acceleration regime leads fracture makes able detect conditions crack growth instability could used predict critical length fatigue crack propagation given level confidence figure approximately ergodicity invariant measures section present preliminary technical results invariant distributions underlying markov chains begin properties markov chain locations pdmp interest particular assumption one may state following result proposition following statements positive aperiodic unique invariant distribution proof proof similar demonstration proposition properties make able apply law large numbers markov chain see theorem propose focus sequence also forms markov process whose transition kernel given let recall denotes distribution state space defined couple define measure conditional distribution given particular unique invariant distribution proposition following statements initial distribution lim positive aperiodic unique invariant distribution proof proof similar demonstrations lemma proposition able apply law large numbers markov chain virtue result noted measures share property absolute continuity respect lebesgue measure presented remark remark thus limit admit density interior state space expression thanks remark finally would like highlight link measures may expressed another way indeed expression transition kernel virtue proposition formula useful investigations proof theorem section use notation addition classical symbols must understood hold almost sure sense sketch proof proof theorem relies following decomposition integer sequence martingale defined studied appendix remainder term defined studied appendix appendix establish remainder term almost surely goes tends infinity first step show almost sure convergence presented theorem addition investigate rate convergence remainder term appendix additional lipschitz mixing condition stated assumption enough prove asymptotic normality given theorem rest proof deals study martingale term prove appendix process vector martingale investigate asymptotic behavior studying square variation process hmin appendix thanks results state law large numbers central limit theorem appendix finally almost sure convergence presented theorem direct application together asymptotic normality obtained remainder term part paper devoted asymptotic study remainder term sequence appearing definition remainder term one may write components defined five terms define given uvj vwj uvj uvj vwj vwj vwj four terms define given uvj uvj uvj finally three terms define given uvj almost sure convergence investigate first component since terms may treated similar arguments first obvious sequence almost surely tends addition term converges virtue ergodic theorem applied markov chain thanks proposition together functions lipschitz bounded see assumptions lip lip lip means sequences tend thus rate convergence first item assumptions markov chain thus process satisfy contraction assumption given theorem applying theorem function see second item assumptions obtain therefore couple set result obvious couples condition min finally obtain goes infinity martingale term vector martingale let process defined component defined terms vjd given uvj uvj vwj vjd uvj uvj vjd uvj claim process keystone show proof presents particular difficulty except first component provide details let recall denotes transition kernel markov chain see subsection vnd vnd let recall bandwidth sequence decreasing together third item assumptions obtain supp supp similar arguments obtain supp supp inf condition couple together expression obtain one may conclude change variable predictable square variation process asymptotic behavior martingale may investigated studying predictable square variation process denote usual hmin time hmin symmetric matrix defined hmin calculate coefficient matrix beginning diagonal terms first hmin one hmi terms defined uvj uvj vwj vjd uvj uvj vwj since functions bounded see assumptions easily obtain let introduce additional notation ten uvj uvj vwj remark kernel functions since lipschitz bounded see assumptions together supp see assumptions stochastic sequences ten limit thanks lip lip ten mean addition almost sure ergodic theorem see proposition together obtain virtue lemma obtain together goes infinity hmin second diagonal term predictable variation process may studied similar way obtain hmin uvj uvj uvj uvj goes infinity hmin third last diagonal term may also investigated way uvj hmin uvj goes infinity hmin focus non diagonal terms integer hmin uvj uvj vwj uvj hmi uvj uvj vwj uvj uvj uvj uvj vwj uvj uvj vwj hmi uvj uvj uvj uvj uvj uvj bounded see assumptions together fact integral finite see remark easily obtain hmi conclusion one may sum asymptotic behavior predictable variation process hmin following formula hmin noted coefficients positive assume statement theorem limit theorems vector martingale law large numbers propose apply law large numbers vector martingales see theorem process interest sequel denotes trace matrix hmin stands minimum eigenvalue first light trace almost surely tends infinity thus able apply third item theorem function obtain log goes infinity min consequence using whenever together law large numbers tends infinity kmn central limit theorem investigate asymptotic normality vector martingale apply corollary sequence defined first assumption result obviously satisfied sequence hmin almost surely converges positive matrix indeed hmin degenerate matrix consequence check lindeberg condition order establish central limit theorem words prove study three components uvj uvj vwj uvj uvj uvj thus obtain kkd kkd kkd together condition consequence exists integer event almost surely empty shows lindeberg condition finally obtain tends infinity proof proposition virtue ergodic theorem see proposition applied markov chain tends infinity gbn conditionally together definition expression remark obtain gbn conditionally definition gbn gbn addition one gbn gbn consequence assumptions conscientious calculus together shows gbn gbn obtain expected result remarking normalizing constant remark order prove one may split integral interest two terms integral remainder term integrated first one clearly upper bounded integral lipschitz function lip thus bounded integral lip second integral obviously bounded kgbn lip references odd olai aalen statistical inference family counting processes proquest llc ann arbor thesis california berkeley terje aven uwe jensen stochastic models reliability volume applications mathematics new york new york romain recursive nonparametric estimator transition kernel piecewisedeterministic markov process esaim probability statistics january romain dufour anne nonparametric estimation jump rate marked renewal processes ann inst probab romain dufour anne estimation conditional distribution interjumping times markov processes scandinavian journal statistics romain alexandre genadot inference absorption features model test anis ben abdessalem romain marie anne monique puiggali stochastic modelling prediction fatigue crack propagation using markov processes accepted publication journal risk reliability patrice bertail stephan jessica tressou storage model random release rate modelling exposure food contaminants mathematical bioscience engineering patrice bertail stephan jessica tressou statistical analysis dynamic model dietary contaminant exposure journal biological dynamics adrien brandejsky saporta dufour numerical method expectations markov processes camcos adrien brandejsky saporta dufour numerical methods exit time markov process adv appl djalil chafai florent malrieu katy paroux long time behavior tcp window size process stochastic processes applications april julien chiquet nikolaos limnios method compute transition function piecewise deterministic markov process application reliability statist probab julien chiquet nikolaos limnios mohamed eid piecewise deterministic markov processes applied fatigue crack growth modelling journal statistical planning inference gerda claeskens peter hall effect dependence stochastic measures accuracy density estimations annals statistics alina crudu arnaud debussche muller ovidiu radulescu convergence stochastic gene networks hybrid piecewise deterministic processes annals applied probability mark davis markov models optimization volume monographs statistics applied probability chapman hall london saporta dufour huilong zhang charles elegbede optimal stopping predictive maintenance structure subject corrosion journal risk reliability marie doumic marc hoffmann nathalie krell lydia robert statistical estimation model observed genealogical tree bernoulli appear marie doumic marc hoffmann patricia vincent rivoirard nonparametric estimation division rate population siam journal numerical analysis marie duflo random iterative models applications mathematics berlin alexandre genadot thieullen averaging fully coupled markov process infinite dimensions adv appl jeffrey hart philippe vieu bandwidth choice density estimation based dependent data annals statistics pages jianghai shankar sastry modeling subtilin production bacillus subtilis using stochastic hybrid systems alur pappas editors hybrid systems computation control number lncs berlin martin jacobsen point process theory applications marked point piecewise deterministic processes probability applications tae yoon kim asymptotically optimal bandwidth selection rules kernel density estimator dependent observations journal statistical planning inference tae yoon kim dennis cox bandwidth selection kernel smoothing time series journal time series analysis nathalie krell statistical estimation jump rates specific class piecewise deterministic markov processes march sean meyn richard tweedie markov chains stochastic stability cambridge university press cambridge second edition norris exploring optimality various bacterial motility strategies stochastic hybrid systems approach phd thesis massachusetts institute technology othmer xin xue excitation adaptation bacteria model signal transduction system controls taxis spatial pattern formation international journal molecular sciences ovidiu radulescu muller alina crudu limites pour les processus markov sauts technique science informatiques lydia robert marc hoffmann nathalie krell stephane aymerich jerome robert marie doumic division escherichia coli triggered rather timing mechanism bmc biology tindall maini porter armitage overview mathematical approaches used model bacterial chemotaxis bacterial populations bulletin mathematical biology virkler hillberry goel statistical nature fatigue crack propagation engng mater technol first dimension second dimension first dimension time secon asure dime nsion iant dim ensi invar firs second dimension figure two representations simulated path model interest jump vector field graph given left observe trajectory first component versus time right first dimension figure estimation invariant distribution locations computed first jumps model estimated ise additive constant estimated ise additive constant estimated ise additive constant parameter parameter parameter estimated criterion estimated criterion figure choice bandwidth parameter appearing criterion bnx obtained minimizing estimate related ise parameter seems small influence minimization estimated error computed left right deterministic path deterministic path figure optimal index calculated maximizing criterion bnx computed optimal parameter left side figure compare estimate bnx full line dashed line computed confirms estimation performs pretty well quantity seems admit one one absolute maximum estimated criterion deterministic path alp parameter ise figure parameter plays crucial role estimation bnx compare curves computed already presented figure full dashed lines oscillating estimate obtained dotted line parameter figure optimal bandwidth parameters implicitly appearing fbn obtained minimizing estimate computed related ise parameter seems little influence estimation error comparison estimation deterministic path computed different values index optimal figure estimator bandwidth parameters replicates optimal points time located times enhanced gray colors seems correspond estimators least bias variance particular obtain better result index around see remark maximizing estimated invariant measure figure simulated trajectory bacteria unit disk jump estimated jump rate figure target points boxplot optimal estimators estimated jump rate taken uniform discretized grid step presented black thick lines boxplots defined dashed line represents true correspond aggregated estimate jump rate figure target points curve well estimate computed observed data provided left boxplot estimates also given right presents bias small dispersion estimated jump rate estimated jump rate estimated jump rate figure target points curve well estimate computed top bottom observed data provided left boxplots estimates also given cases right figure target point indexed estimated jump rates computed different datasets estimated jump rate crack length computed datasets different sizes figure boxplots estimated jump rates time number cycles parameter log figure virkler dataset contains independent crack growth histories starting parameter figure material parameters log strongly linked linear relationship used reduce dimension underlying model log estimated criterion estimated criterion best parameter estimation parameter estimated criterion parameter parameter crack length jump rate unstable region figure estimation criterion different values target length top left top right bottom left relationship optimal parameter maximizing target crack length also presented bottom right crack length figure estimation jump rate different crack lengths stochastic model fatigue crack propagation virkler dataset
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classification major depressive disorder via weighted lasso model dajiang brandalyn neda nynke dan ian matthew danai james cynthia henrik ilya thomas lianne dick paul imaging genetics center usc stevens neuroimaging informatics institute keck school medicine university southern california usa bcn neuroimaging center department neuroscience university groningen university medical center groningen netherlands dept psychiatry mental health university cape town south africa neurosciences program department psychology stanford university usa department psychiatry behavioral sciences stanford university usa dept neuroimaging institute psychiatry psychology neuroscience king college london dept psychology school arts social science city university london department medicine imperial college london department psychological medicine king college london dept psychiatry psychotherapy berlin germany department psychiatry trinity college dublin ireland dept psychiatry psychotherapy otto von guericke university magdeburg germany dept psychiatry neuroscience campus amsterdam university medical center netherlands orygen national centre excellence youth mental health australia center youth mental health university melbourne australia abstract collaborative analysis brain imaging data psychiatry neurology offers new source statistical power discover features boost accuracy disease classification differential diagnosis outcome prediction however due data privacy regulations limited accessibility large datasets across world challenging efficiently integrate distributed information propose novel classification framework weighted lasso site performs iterative weighted lasso feature selection separately within iteration classification result selected features collected update weighting parameters feature new weight used guide lasso process next iteration features help improve classification accuracy preserved tests data five sites patients major depressive disorder mdd normal controls method boosted classification accuracy mdd average result shows potential proposed new strategy effective practical collaborative platform machine learning large scale distributed imaging biobank data keywords mdd weighted lasso introduction major depressive disorder mdd affects million people worldwide takes immense personal toll patients families placing vast economic burden society mdd involves wide spectrum symptoms varying risk factors varying response treatment unfortunately early diagnosis mdd challenging based behavioral criteria consistent structural functional brain abnormalities mdd beginning understood neuroimaging large cohorts identify characteristic correlates depression may also help detect modulatory effects interventions environmental genetic risk factors recent advances brain imaging magnetic resonance imaging mri variants allow researchers investigate brain abnormalities identify statistical factors influence relate diagnosis outcomes researchers reported brain structural functional alterations mdd using different modalities mri recently working group found adults mdd thinner cortical gray matter orbitofrontal cortices insula cingulate temporal lobes compared healthy adults without diagnosis mdd subcortical study largest date showed mdd patients tend smaller hippocampal volumes controls diffusion tensor imaging dti reveals average lower fractional anisotropy frontal lobe right occipital lobe mdd patients mdd patients may also show aberrant functional connectivity default mode network dmn functional brain networks fig overview proposed framework even classification mdd still challenging three major barriers first though significant differences found previously identified brain regions brain measures always consistent markers mdd classification second besides imaging modalities including dti functional magnetic resonance imaging fmri commonly acquired clinical setting last always easy collaborating medical centers perform integrated data analysis due data privacy regulations limit exchange individual raw data due large transfer times storage requirements thousands images biobanks grow need efficient platform integrate predictive information multiple centers available datasets increase effort increase statistical power identify predictors disease diagnosis future outcomes beyond site could identify study introduce weighted lasso model boost classification performance individual participating site integrating knowledge feature selection results classification shown fig proposed framework features following characteristics site retains data performs weighted lasso regression feature selection locally selected brain measures classification results shared sites information selected brain measures corresponding classification results integrated generate unified weight vector across features sent site weight vector applied weighted lasso next iteration new weight vector leads new set brain measures better classification performance new set brain measures sent sites otherwise discarded old one recovered methods data demographics study used data five sites across world total number participants older years old demographic information site participants summarized table sites total total total age controls mdd controls mean patients groningen stanford brcdecc berlin dublin combined age mdd mean female mdd female total table demographics five sites participating current study data preprocessing common clinical settings mri brain scans acquired site quality control analyses performed locally right cortical gray matter regions subcortical gray matter regions lateral ventricles segmented freesurfer detailed image acquisition brain segmentation quality control methods may found brain measures include cortical thickness surface area cortical regions volume subcortical regions lateral ventricles total brain measures considered study algorithm overview better illustrate algorithms define following notations selected brain measures features classification performance weight vector performing weighted lasso weight vector svm performing svm classifier using feature set algorithms two parts run site integration server first integration server initializes weight vector ones sends sites site use weight vector conduct weighted lasso section data locally selected features better classification performance send new features corresponding classification result integration server improvement classification accuracy send old ones integration server receives updates sites generates new weight vector section according different feature sets classification performance detailed strategy discussed section algorithm integration server initialize features weighted one send sites least one site improvement update section send sites end send null sites table main steps algorithm algorithm received null section svm send integration server else send integration server end end end table main steps algorithm ordinary lasso weighted lasso lasso shrinkage method linear regression ordinary lasso defined lasso arg min observations predictors known sparsity parameter minimizes sum squared errors penalizing sum absolute values coefficients lasso regression force many coefficients zero widely used variable selection however classical lasso shrinkage procedure might biased estimating large coefficients alleviate risk adaptive lasso developed tends assign predictor different penalty parameters thus avoid larger coefficients penalized heavily small coefficients similarly motivation weighted lasso penalize different predictors brain measures assigning different weights according classification performance across sites generating weights brain measure feature model discussed section generation weight algorithm integration server receives information selected features brain measures corresponding classification performance site generates new weight feature new weight feature feature selected site number sites classification accuracy proportion participants relative total number participants sites penalizes features survived small number sites contrary specific feature selected sites meaning sites agree feature important tends larger weight consider classification performance proportion samples site achieved high classification accuracy relatively small sample size compared sites features selected conservatively recommended sites general feature selected sites resulted higher classification accuracy larger weights weight lasso section define weighted lasso model arg min represents mri measures controlling effects age sex intracranial volume icv managed within different sites label indicating mdd patient control brain measures features study model feature larger weights implies higher classification performance recognition multiple sites hence penalized less greater chance selected sites consider feature previous iteration results classification improvements model study applied algorithm algorithm data five sites across world first iteration integration server initialized weight vector ones sent sites therefore five sites conducted regular lasso regression first round small set features selected using similar strategy within site performed classification locally using support vector machine svm shared best classification accuracy integration server well set selected features integration server generated new weight according sent back sites second iteration site performed none improvement classification result total five sites ran six iterations classification performance round summarized fig fig applying data coming five sites subfigure shows classification accuracy acc specificity spe sensitivity sen iteration shows improvement classification accuracy site performing though stanford berlin sites show improvements second iteration classification performance brcdecc site dublin continued improving sixth iteration hence terminated sixth round fig shows improvements classification accuracy five sites average improvement sparsity level lasso set means features tend selected lasso process section shows reproducibility results different sparsity levels conducing svm classification kernel rbf used performed grid search possible parameters best classification results adopted analysis features process new set features resulting improvements classification accepted otherwise prior set features preserved new features also recommended sites increasing ing weights new features fig displays changes involved features six iterations top features selected majority sites fig number involved features six iterations top five consistently selected features across sites within subfigure top showed locations corresponding features bottom indicated many sites selected feature process cortical thickness surface area measures first iteration features selected five sites number decreases iterations features preserved six iterations average classification accuracy increased moreover feature originally selected majority sites tends continually selected multiple iterations fig promising features accepted fewer sites first might incorporated sites iteration increased fig reproducibility selected improvement selected improvement features acc spe sen features acc spe sen table reproducibility results different sparsity levels column selected features represents percentage features preserved lasso procedure average improvement accuracy sensitivity specificity sparsity problems solution selection sparsity level highly data dependent validate model repeated algorithm algorithm different sparsity levels leads preservation different proportions features reproducibility performance proposed summarized table conclusion discussion proposed novel weighted lasso model heuristically improve classification performance multiple sites sharing knowledge features might help improve classification accuracy sites site multiple opportunities reconsider set selected features strive increase accuracy iteration study average improvement classification accuracy five sites offer proof concept distributed machine learning may scaled disorders modalities feature sets references world health organization world health organization depression fact sheet available http fried depression sum score parts individual dsm symptoms different risk factors psych med schmaal cortical abnormalities adults adolescents major depression based brain scans cohorts worldwide enigma major depressive disorder working group mol psych doi schmaal subcortical brain alterations major depressive disorder findings enigma major depressive disorder working group mol psych liao depression disconnection syndrome diffusion tensor imaging studies patients mdd psych neurosci sambataro revisiting default mode network function major depression evidence disrupted subsystem connectivity psychl med significant variables automatically good predictors pnas https zhu classification major depressive disorder via distributed lasso proc spie tibshirani regression shrinkage selection via journal royal statistical society qingyang collaborative imaging genetics studies risk genetic factors alzheimer disease across multiple institutions miccai zou adaptive lasso oracle amer statist assoc koutsouleris individualized differential diagnosis schizophrenia mood disorders using neuroanatomical biomarkers brain supported part nih grant see ref additional support coauthors cohort recruitment
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criterion completeness dec peter schenzel bstract let denote ideal commutative noetherian ring let completion defined lim called complete whenever natural homomorphism isomorphism let main result paper shown complete flat test module system elements rad rad result extends several known statements starting jensen result see proposition finitely generated local ring complete flat ntroduction let denote commutative noetherian ring ideal consider completion defined lim case local ring finitely generated shown see following conditions equivalent complete flat iii elements generate ideal denotes localization respect extension local case result jensen see proposition proved equivalence first two conditions main result present paper extension case completion precisely prove following result theorem let ideal commutative noetherian ring let denote arbitrary rmodule following conditions equivalent iii complete extir flat elements system elements rad rad note paper see proposition jensen proved following let denote semi local ring finitely generated complete flat resp countably generated flat different proof vanishing extir flat complete follows results buchweitz flenner see theorem mathematics subject classification primary secondary key words phrases adic completion flat module ext peter schenzel moreover another criterion completeness shown frankild satherwagstaff let ideal contained jacobson radical let finitely generated paper see proved complete extir dimr denotes completion whence main result present paper construction simple flat test module completeness topology terms vanishing generalizes case maximal ideal proved instead matlis duality used available case topology use homological techniques reliminary esults following denotes commutative noetherian ring ideal consider completion defined lim say induced complete whenever natural homomorphism natural surjections isomorphism implies begin preparatory results needed later proposition let denote ideal let denote suppr proof assumption implies words first let finitely generated certain therefore necessarily finitely generated lim direct system finitely generated submodules ordered inclusions whence suppr lim required let denote ideal left derived functors defined lim denotes flat resolution functors first systematically studied greenlees may see simon see recently note natural surjective homomorphism since noetherian follows see section whence completion complete following discuss assumption proposition provides certain kind nakayama lemma proof known see also lemma add arguments proposition let denote ideal commutative noetherian ring let denote arbitrary proof lim versely suppose required clearly let denote free presentation onto therefore onto required criterion completeness following let denote system elements rad rad consider complex system see section note complex introduced direct limit koszul complexes natural morphism context importance complex following result allows expression right derived functors different terms theorem let denote ideal commutative noetherian ring let denote natural isomorphisms homr denotes injective resolution proof statement one main results section formulation use technique derived functors derived categories advanced exposition based derived categories derived functors interested reader might also consult let summarize basic results completions used sequel following results proposition let denote ideals commutative noetherian ring let denote system elements let denote suppose complete also complete suppose rad rad assume complete topology complete proof proof see let denote completion topology topology defined equivalent follows lim therefore element form since complete lim maps complete proves following result helpful order compute projective limits lim denotes first right derived functor projective limit see let denote consider following projective system transition map multiplication lemma previous notation follows lim extir extir lim homr proof let denote direct system transition map multiplication lim well known therefore homr lemma see also lemma short exact sequences lim extir lim extir whence results first part follow vanishing homr easy see peter schenzel previous result slight modification lemma lemma remark morphism two complexes called homology isomorphism whenever induced homomorphisms cohomology modules isomorphisms definitions homr refer details homological algebra complexes modules refer adic ompletions let denote ideal commutative noetherian ring first main result shall prove vanishing result certain complete theorem let denote arbitrary let denote flat satisfying extir proof proof use techniques summarized theorem end fix notation let denote system elements rad rad let denote complex respect defined section bounded complex flat natural morphism complexes let denote injective resolution applying homr induces morphism complexes homr next investigate complex homr left bounded complex injective rmodules moreover virtue theorem follows homr therefore homr homr since complete see therefore morphism homr consider complexes homr homr homr claim homologically trivial complexes reason enough show homologically trivial since bounded complex injective since flat yields vanishing apply proposition suppr shown homr left bounded complexes injective applying functor homr induces homr homr homr complex right cohomologically trivial cohomology complex left extir therefore vanish required following result prove another behaviour certain respect completion theorem let denote ideal let arbitrary natural map suppose suppr extir natural homomorphism extir extir isomorphism criterion completeness proof let denote elements rad rad let denote complex respect short exact sequence complexes global complex dxi dxi let denote injective resolution applying functor homr short exact sequence complexes provides short exact sequence homr homr left bounded complexes injective complex middle injective resolution follows fact complete see therefore complex right considered complex concentrated homological degree zero order prove statement let denote projective resolution extir homr homr see enough show last modules vanish end consider isomorphism complexes homr homr homr bounded complex flat therefore enough show complex homologically trivial true since dxi suppr proves statement view investigations homr second complex injective resolution proves statement continue result vanishing certain step towards main result shall prove partial result order characterize completion theorem let denote commutative noetherian ring let denote arbitrary let element following conditions equivalent complete extir proof let follows definition statements may replace without loss generality may assume commutative diagram exact rows passing inverse limit provides exact sequence lim lim view lemma yields lim extir furthermore short exact sequence induces isomorphism extir extir follows long exact cohomology sequence extir peter schenzel vanishing note multiplication acts isomorphism zero map homomorphism isomorphism extir proves proof theorem implication consequence theorem moreover implications iii trivial iii easy see order prove first note theorem implies radically complete since generates radical follows complete proposition since separated completes proof alternative proof heorem section alternative proof theorem based results inverse limits results following two lemmas might also independent interest statements particular cases certain spectral sequences inverse limits see give elementary proof based description lim discussed lemma let denote commutative ring let direct system let denote arbitrary short exact sequence lim extir lim lim extir proof result proved lemma noetherian ring arguments work general case following need certain dual statement lemma certain sense key argument second proof theorem lemma let denote commutative ring let denote arbitrary let inverse system lim short exact sequence lim extr extir lim lim extir proof lim short exact sequence lim third homomorphism transition map see induces long exact cohomology sequence extir lim extr extr end recall ext transforms direct products direct products second variable cohomology commutes direct products see known see coker lim ker lim extir completes proof criterion completeness remark assumption lim fulfilled whenever projective system satisfies condition instance transition map surjective note proof following theorem motivated arguments done buchweitz flenner see theorem let denote commutative ring let denote let denote flat satisfying extir proof definition lim lemma short exact sequence lim extir lim extir order show vanishing extr enough show vanishing exti claim extir order show isomorphisms let projective resolution flat torir projective resolution adjunction isomorphisms complexes homr taking cohomology proves claim vanishing follows consequence assumption remark fact theorem slight sharpening theorem end recall ideal completion necessarily complete used proof theorem explicit example see bartijn thesis page note example ring noetherian grows iii exerc see also yekutieli see example cknowledgements author thanks simon reviewer careful reading manuscript suggesting comments eferences lonso tarr eremias ipman local homology cohomology schemes ann scient norm avramov oxby homological dimensions unbounded complexes pure appl algebra artijn flatness completions regular sequences trois thesis utrecht ourbaki commutative hermann paris uchweitz lenner power series rings projectivity manuscripta math nochs enda relative homological algebra revised walter gruyter berlin rankild ather detecting completeness proc amer math soc reenlees derived functors completion local homology algebra ensen vanishing lim algebra peter schenzel ensen les foncteurs lim leurs applications des modules lecture notes math springer chenzel proregular sequences local cohomology completion math scand chenzel structure endomorphism ring certain local cohomology module algebra chenzel criterion completeness appear imon homological properties complete modules math proc cambr phil soc imon adic completion dual homological results publ matematiques univ autonoma barcelona trooker homological questions local algebra london math lect note ser cambridge univ press eibel introduction homological algebra cambridge univ press ekutieli flatness completion infinitely generated modules noetherian rings comm algebra artin uther niversit alle ittenberg nstitut aale ermany address nformatik alle
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localization tracking multiple animals autonomous uav feb hoa van nguyen michael chesser fei chen hamid rezatofighi damith ranasinghe school computer science university adelaide australia email aerial robots provide new possibilities study habitats behaviors endangered species efficient gathering location information temporal spatial granularities possible traditional manual survey methods present novel autonomous aerial vehicle system track localize multiple animals simplicity measuring received signal strength indicator rssi values high frequency vhf commonly used field exploited realize low cost lightweight tracking platform suitable integration unmanned aerial vehicles uavs due uncertainty nonlinearity system based rssi measurements tracking planning approaches integrate particle filter tracking localizing partially observable markov decision process pomdp dynamic path planning approach allows autonomous navigation uav direction maximum information gain locate multiple mobile animals reduce exploration time consequently conserve battery power also employ concept search termination criteria maximize number located animals within power constraints aerial system validated online approach extensive simulations field experiments two mobile vhf ntroduction understanding basic questions ecology animals use habitat movements activities necessary addressing numerous environmental challenges ranging invasive species diseases spread animals saving endangered species extinction conservation biologists ecologists well natural resource management agencies around world rely numerous methods monitor animals predominant tool suitable wide range animal sizes high frequency vhf radio collar tagging subsequent radio telemetry measurements locate spatiotemporal information concerned species however traditional method radio tracking typically requires researchers trek long distances field armed cumbersome vhf radio receivers handheld antennas battery packs manually home radio signals emitted collared animals consequently precious spatial data acquired radio tracking comes significant cost researchers terms manpower time funding problem often compounded challenges low animal recapture rates equipment failures inability track animals move inaccessible terrain furthermore many fig overview uav tracking platform sensor system endangered species also happen difficult track due small size inconspicuousness location remote habitats automated tracking location wildlife autonomous unmanned aerial vehicles uavs provide new possibilities better understand ecology native wildlife safeguard biodiversity manage natural resources present approach capable realization lightweight payload transforming existing commodity drone platforms autonomous aerial vehicle systems shown fig empower conservation biologists track localize multiple animals main contribution work new autonomous aerial vehicle system simultaneously tracking localizing multiple mobile animals using vhf radiocollars commonly used field particular system realized payload suitable multitude versatile easy operate uavs without remote pilot license lightweight less system achieved new sensor design exploits simplicity software defined radio architecture capturing received signal strength indicator rssi value multiple vhf radio tags compact lightweight vhf antenna geometry due noisy complex nonlinear characteristics rssi data integrate sequential monte carlo implementation bayesian filter also known particle filter tracking localization jointly partially observable markov decision process pomdp divergence prior posterior estimates animal locations autonomy dynamic online path planning minimize flight time maximizing number located animals formulation considers location accuracy resource constraints uav maneuverability power constraints develop practical solution iii validate method extensive simulations field experiments mobile vhf elated ork since application related locating animals focus progress made towards autonomous localization tracking multiple animals estimations radio beacon data logged uav demonstrated pioneering achievements autonomous wildlife tracking made simulation studies experimentally demonstrated systems recent years particular first demonstration uav proved recent approaches localization static target assuming stationary wildlife used wireless signal characteristics captured receiver estimate location particular aoa radio beacon determined using array antennas information related receiver location estimations although approach conveniently manage topological variations terrain aoa systems require large bulky receiver system multiple antenna elements well long observation times seconds per observation reported moreover antenna systems mounted top uav likely lead difficulty tracking terrestrial animals although suitable locating avian species dwelling trees see investigations studied problem locating animals using autonomous robots although system based recently evaluated locate stationary animal development lightweight autonomous system capable flights localization multiple mobile animals still remains especially significant realization system widely accessible conservation biologists field small less flown without formal pilot license fewer restrictions given exclusion category uavs regulatory regimes present alternative approach exploiting rssi based range measurements ability use simpler sensing system board commodity uavs realize lower cost longer flight range uavs tracking localizing multiple animals together strongly principled approach joint tracking planning lightweight autonomous aerial robot platform provides method wildlife conservation management iii racking lanning roblem ormulation tracking requires online estimator dynamic planning method section presents tracking localizing formulation principled framework bayesian filter tracking pomdp planning strategy tracking localizing tracking use bayesian filter online estimation technique deals problem inferring knowledge unobserved state dynamic problem changes time sequence noisy measurements suppose respectively system kinematic state vector state space measurement observation vector observation space problem estimating state measurement calculating marginal posterior distribution sequentially prediction update steps case nonlinear system noise general solution bayesian recursion generally form therefore problem use particle filter implementation approximate solution bayesian filtering problem due highly nonlinear measurement model particle filter particle filter uses sampling approach represent form posterior density samples distribution represented set particles particle weight assigned represent probability particle sampled probability density function particles representing nonparametric form propagated time simplest version particle filter known bootstrap filter first introduced gordon samples directly generated transitional dynamic model reduce particle degeneracy resampling injection techniques implemented detailed algorithm found measurement model update process requires derivation likelihood measurements problem based estimating target radio tag receiver require realistic signal propagation model obtain likelihood receiving given measurement employ two vhf signal propagation models suitable describing rssi measurements outdoor environments denoting rssi measurement function target observer uav state log distance path loss model logpath received power line sight power component transmitted transmitter subjected signal attenuation absorption propagation loss target position observer uav position cartesian coordinates uav state includes heading angle euclidean distance target position uav position uav receiver antenna gain depends heading position target position received power distance exponent characterizes signal losses absorption propagation losses parameter depends environment typical values range log distance path loss model fading multipath received power composed line sight power component transmitted transmitter power component reflected ground plane subjected signal attenuation absorption propagation loss addition terms contact angle reflected path ground sin sin cos ground reflection coefficient relative permittivity ground phase difference two waves wave length ptx pux pty puy ptz puz environments received power usually corrupted environmental noise assumption noise white total received power gaussian white noise covariance notably even rssi noise completely characterized white noise model still model gaussian noise using higher covariance account unknown uncertainty use data captured experiments using sensor system validate physical sensor characteristics see sec defined environmental characteristics well estimate propagation model reference power parameter noise see sec measurement likelihood based gaussian noise likelihood measurement given target sensor position respectively time normal distribution mean covariance path planning uav planning problem similar problem agent computing optimal actions partially observable markov decision process pomdp maximize reward kaelbling shown pomdp framework implements efficient optimal approach based previous actions observations determine true world states pomdp conjunction particle filter provides principled approach evaluating planning decision realize autonomous system tracking general pomdp described set uav target states set uav actions function reward function set observations observation function current state next state respectively taken action measurement goal pomdp find optimal policy maximize total expected reward lookahead horizon steps discount factor serves value difference current reward versus future reward state action time step expectation operator reward function calculated using different methods strategies uncertainty high information gain approach preferable reduce target location uncertainty hence used method calculate reward function several approaches evaluate information gain robotic path planning shannon entropy kullbackleibler divergence divergence adapted approach implement divergence reward function since fits naturally montecarlo sampling method divergence calculates distance prior posterior densities log prior density calculated propagating current posterior particles sampled time using prediction step posterior density future measurement set observed action taken calculated applying prediction update steps time however using bayes update procedure computationally expensive prohibitive setting instead implement computationally efficient approach using black box simulator proposed along monte carlo sampling approach hence problem transforms find optimal action maximize total expected reward arg max number future measurements tracking particle filter proposed sec extended tracking mtt however mtt normally deals complex data association problem difficult determine measurement belongs target contrast system target estimated measurement based signal frequency tracked independently thus need solve data association problem notably targets detected due example uav movements measurement range limits imposed propagations losses receiver sensitivity therefore target detected solver update estimated position besides maximizing number targets localized tracked formulated termination condition target conserve uav battery power target considered localized location determined particle sufficiently small found targets forgotten aid solver prioritize computing resources targets high uncertainty ystem mplementation implemented experimental aerial robot system based tracking planning formulation overview complete system described fig experimental system used uav platform new sensor system built compact directional vhf antenna design software defined signal processing module capable simultaneously processing signals multiple targets remotely communication ground control system tracking planning system ardupilotmega apm uav transmits back global positioning system gps location telemetry host tool developed group communicate apm module using mavlink protocol mhz full duplex radio channel sensor system together antenna sdr receiver embedded compute delivers targets rssi data ghz radio channel ground control system gps targets rssi data delivered tracking planning telemetry host using restful web service solver estimates target locations calculates new control actions per pomdp cycle command uav mavlink fly new location order ensure safety meet university regulatory requirements also employ popular flight control mission planning monitor abort autonomous navigation detail sensor system signal processing module contrast previous work propose using software defined radio sdr receiver implement signal processing components key advantages choice ability reduce weight receiver rapidly scan large frequency spectrum track multiple animals beaconing different vhf frequency channels iii signal processing chain defined software ability reconfigure system fly work use hackrf one open source platform developed ossmann capable directly converting radio frequency signals digital signals using converter adc intel edison board embedded compute module implemented discrete fourier transform dft filter isolate multiple signals unique vhf frequency channel associated animal radio collar measure signal strength received signal antenna lightweight folded yagi antenna specially designed sensor system design achieves low profile antenna capable within form factor commodity uavs suitable easy operation field similar standard yagi antenna folded design one reflector one driven element shown fig antenna operates frequency range mhz typical range wildlife radio tags center frequency mhz length driven reflector elements respectively inductive loading ring diameter wavelength antenna gain model calculated design shown fig planning implementation system implementing planning algorithms systems always challenging high computational demand thus section present approaches minimize planning computational time scarifying overall localization performance notably rssi data uncertainty estimation target location reduced maximum gain directional antenna mounted uav points toward target position hence increase localization accuracy uav heading angle must controlled path planning process although uav maneuvered without changing heading adopted set discrete uav rotation angles control actions helps reduce computational complexity pomdp planning process limiting number possible actions evaluate solver performs planning every observation cycles instead every observation approach helps ensure solver prioritizes limited computational resource tracking targets instead performing planning steps iii coarse planning interval planning procedure implemented minimize computational time reducing number steps still horizon example want estimate target state second horizon use normal interval estimate target state times use coarser interval perform estimation twice latter approach computationally less expensive instead selecting best action possible action space domain knowledge receiver antenna gain used select subset actions gives highest received gain using alg fig full communication channels uav ground control system main softwares protocols folded yagi antenna used sensor system observations algorithm calculate control action subset input number preferred actions antenna gain target position output ask get calculate glr end ask glr top following implementation approach uav motion includes two modes changing heading angle hovering moving forward direct location one planning procedure cycles uav needs cycles rotate spends remaining cycles move forward without changing heading floor absolute operator respectively uav maximum rotation angle one cycle sign decides rotation direction clockwise imulation xperiments implementing real system difficult hence want validate systems firstly several simulation experiments verify tracking planning algorithms investigate planning parameters different values divergence number discrete actions created alg contribute overall algorithm performance iii compare proposed divergence based planning technique methods impact horizon parameters computational time localization accuracy simulation experiments processed intel core cpu ram tracking planning simulation simulation implemented validate approach experiment uav attempted search localize moving targets randomly located area following list parameters used simulation sampling time step second since tag emits pulse signals every second solver performed planning procedure every horizon parameters number horizon planning interval uav started home location moved constant velocity maximum heading rotation angle number particles target capped future sample measurement divergence parameter number actions addition target considered localized location uncertainty determined particles covariance small chosen limit logpath measurement model used verify proposed algorithm demonstrate algorithm able localize mobile targets animal usually wanders around area considered hence random walk model used describe behavior single target transitional density identity matrix diag fig shows localization results mobile targets estimation details annotated next target position two indicators rms flight sec definitions summary scenario took uav seconds localize ten moving targets maximum error distance less except outlier target rms flight time finishing localizing last target target uav sent command fly back original station case total uav travel distance results demonstrate algorithm search accurately localize multiple numbers targets real time minutes travel distance commercial shelf drones monte carlo simulations experiment monte carlo setup parameters kept sec except investigated ones addition ensure results random conducted experiments performed monte carlo runs tracking algorithm evaluated based following criterion estimation error absolute distance ground truth estimated target location drms ntg drms drms xtruth xest ytruth yest flight time uav localize targets includes hovering time uav waits commands solver take action uav travel distance distance find target positions rms flight time rms flight time rms flight time rms rms flight time flight time rms flight time rms flight time rms flight time fig simulation results moving targets localized using single uav particle filter pomdp table ocalization performance different alpha values rms flight time uav travel distance computational cost calculated scenarios execution time solver perform tracking algorithm called time execution time solver select best action planning step well conduct tracking called planning time firstly search localization algorithms evaluated using different values reward function table presents monte carlo results general values significant impact overall performance however applying provides best localization results terms estimation error search duration applying proposed results worst performance increases flight time travel distance necessary complete localization task using considered using divergence popular information gain helps save uav travel distance sacrificing location accuracy one explanation scenario noisy measurement causes predicted posterior less informative due high uncertainty therefore reward function pace emphasis current posterior instead using small value setting completely ignore future posterior also explains selecting equally weighting current future posterior result worst localization performance equally important conducted experiments understand action space set created alg affects tracking performance term planning time localization error table shows monte carlo results presents interesting result applying provides best localization performance number actions rms flight time uav travel distance planning time rms flight time travel distance number targets travel distance rms flight time flight time rms table ocalization performance different number actions real position target estimated position target uav trajectory rms flight time fig localization performance different number targets ntg increase terms estimation error flight duration travel distance desirable result realizing real time planning limited computational resources another aspect want examine proposed algorithm performs maximum number targets ntg change depicted fig algorithm estimation error stable invariant number targets moreover reasonable flight time travel distance increased linearly target numbers took time power track targets lastly examined divergence performs different horizons compared shannon entropy naive approach moves uav closest estimated target location iii uniform search predefined path used table iii shows monte carlo comparison results among various planning algorithms parameters reused sec except updated based previous experimental results result demonstrated divergence reward function superior planning strategies term localization accuracy including shannon entropy horizon settings reward function large look ahead horizon number helps improve localization accuracy however requires higher computational power planning causes uav travel using provides best computational time accuracy summary according simulation results select planning parameters field experiment since parameters provides lowest computational cost best performance term location estimation error travel distance flight time fig waterfall plot rotor noise experiment four motors full rotation speed normalized antenna gain red line gain modeled pattern black line normalized measured gain pattern measurements collected rotating uav heading intervals plot measured rssi data points estimated models distance intervals table iii ocalization performance different planning algorithm rms flight time uav travel distance plan time non plan time uniform closest target shannon ield xperiments describe extensive experiments regime validate approach evaluate performance aerial robot system field aim investigate possibility signal interference spinning motors uav rssi measurements estimate model parameters sensor model validate proposed model iii conduct field trials demonstrate evaluate system capabilities rotors noise investigated rotor noise confirm system affected electromagnetic interference uav motors also helps clear concern raised rotor noise may affect rssi measurements four motors shown used experiment rssi data radio collar measured across mhz mhz frequency spectrum four motors operating maximum speed rounds per minute fig shows frequency spectrum received signal see difference frequency characteristics rotors states result confirms rotors spin fast enough generate interference impact rssi measurements sensor model validation parameter estimation antenna gain antenna gain pattern measured verify directivity compared antenna gain model physical design discussed sec fig shows measured modeled radiation patterns angle uav heading direction position target position plane result shows ratio smaller expected artifact folding reflector design signal propagation model parameter validation collected rssi data points range uav vhf radio tag tag uav kept height ground experiment tag stationary times uav directed move away straight line tag intervals whilst hovering location allow collection approximately measurements uav heading maintained ensure consistent antenna gain experiment since operated open terrain grassland selected path loss exponent suitable modeling free space path loss fig shows measured rssi propagation models obtained using nonlinear regression algorithm estimate model parameters following results reference power reference distance measurement noise variance logpath model multipath model results show models expected derived similar reference power whilst providing reasonable fit measurement data affirms validity propagation model although logpath model reasonable multipath model accurate yields smaller measurement noise variance results confirms impact ground reflections especially close signal source field trials present two sets field experiments validate two measurement models conducted total autonomous flights demonstrate system capabilities experiments designed around anonymous university regulations governing conduct experimental uav research given need operate autonomous mode flight zone well scope experiment restricted university owned property designated uav flight tests therefore uav task set search localize two mobile targets search area instead wildlife relied two people wearing vhf radio tag forearm mobile phone based gps data logger table omparison ocalization performance model logpath multipath cliff target type mobile mobile stationary trials rms total flight time travel distance hands obtain ground truth two extra personnel stationed maintain constant sight uav people field abort autonomous mode transfer control manual operations pilot volunteers radio tags asked walk randomly instructions given fig shows tracking localization results along uav trajectories based two different measurement models table presents summary comparison results location estimates two measurement models smaller rms root mean square estimation error values suggest higher accuracy shorter flight times travel distance localize targets highly desirable practicable system given power constrained nature commodity uavs result confirm multipath model superior standard logpath model since able account ground reflections uav required approach target closely using logpath model reduce measurement uncertainty discussion also summarise results cliff comparison table table presents complete comparison proposed system cliff system notably search area smaller compared cliff due test flight zone restrictions however set initial distance uav home position farthest target position target case equivalent distance stationary target approximately results table demonstrate proposed method localize two mobile targets shorter flight time flight time multipath model cliff better accuracy moreover search locate two mobile targets contrast cliff method implemented locate single stationary target general shown table system compact lighter payload consequently capable longer flight times given uav although reliance sdr without preamplifier resulted shorter detection range total system mass significant since enables ecologists operate system without remote pilot licenses repl moreover ability instantly collect measurements also helps reduce flight time significantly compared method requiring full rotations information regarding total flight reported cliff however shown fig cliff one observation took one trial needed observations hence total flight time fig field experiment results search track localize two mobile tags two different measurement models standard logpath multipath table omparison system liff system quadcopters smaller drone software defined radio rapidly scan multiple frequencies support multiple frequencies cliff octocopters relatively larger drone detection range measurement model exploiting simplicity measurement system payload total mass drone type receiver architecture filtering method planning nature targets particle filter operations per iteration divergence multiple mobile target tracking analog filtering circuit difficult new frequency antenna array uav rotation grid points phase difference measurement system filter operations per iteration shannon entropy single stationary target localization uav observation point shown table furthermore discussed computational cost methods used increases dramatically number cells whilst grid must dense enough achieve accurate estimations filter cells conducts operations per iteration similar particle filter particles requires operations hence filter method works case stationary targets expensive computational step prediction step skipped moreover shown table iii planning algorithm based divergence superior shannon entropy approach important metrics accuracy flight time vii onclusion developed demonstrated autonomous aerial vehicle system range tracking localization vhf animals rssi based measurement uncertainty mobility targets discovery field joint particle filter pomdp divergence based reward function provided accurate method explore track locate multiple animals considering resource constraints underlying uav platform addition realized uav system ensure practicability accessibility technology conservation biologists demonstrated successful system formulated approach tracking problem ideally suitable tracking endangered species largely flat terrains grasslands consequently current approach suitable tackle tracking wildlife hills mountainous areas would require uav capability maintain fixed relative altitude ground formulating tracking problem extend method topographical conditions leave latter future work viii acknowledgments work jointly supported western australia parks wildlife australian research council defense science technology group university adelaide unmanned research aircraft facility eferences sanjeev arulampalam simon maskell neil gordon tim clapp tutorial particle filters online bayesian tracking ieee transactions signal processing michael anthony beard ngu sanjeev arulampalam void probabilities divergence generalized labeled multibernoulli models ieee transactions signal processing casa remotely piloted aircraft systems online accessed oliver cliff robert fitch salah sukkarieh debbie saunders robert heinsohn online localization wildlife autonomous aerial robot system robotics science systems neil gordon david salmond adrian smith novel approach bayesian state estimation iee proceedings radar signal processing alfred hero christopher kreucher doron blatt information theoretic approaches sensor management springer david hsu wee sun lee nan rong pomdp planner target tracking proc ieee icra pages austin jensen david geller yangquan chen monte carlo simulation analysis tagged fish radio tracking performance swarming unmanned aerial vehicles fractional order potential fields journal intelligent robotic systems leslie pack kaelbling michael littman anthony cassandra planning acting partially observable stochastic domains artificial intelligence roland kays sameer tilak margaret crofoot tony fountain daniel obando alejandro ortega franz kuemmeth jamie mandel george swenson thomas lambert tracking animal location activity automated radio telemetry system tropical rainforest computer journal pages fabian raphael speck ali haydar salah sukkarieh autonomous airborne wildlife tracking using radio signal strength proc iros pages sophocles orfanidis electromagnetic waves antennas rutgers university new brunswick michael ossmann software defined radio hackrf posch salah sukkarieh uav based search radio tagged animal using particle filters australasian conference robotics automation acra sydney australia dec pages branko ristic particle filters random set models new york branko ristic sensor control multiobject estimation using random finite sets automatica branko ristic sanjeev arulampalam neil chercheur gordon beyond kalman filter particle filters tracking applications artech house branko ristic mark morelande ajith gunatilaka information driven search point sources gamma radiation signal processing david silver joel veness planning large pomdps advances neural information processing systems pages bindi thomas john holland edward minot wildlife tracking technology options cost considerations wildlife research pratap tokekar deepak bhadauria andrew studenski volkan isler robotic system monitoring carp minnesota lakes journal field robotics joshua vander hook pratap tokekar volkan isler cautious greedy strategy active localization analysis field experiments journal field robotics neeti wagle eric frew characterization airborne radio frequency environments ieee globecom workshops pages jakes microwave mobile communications wiley new york
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moonshine distilling cheap convolutions nov elliot crowley gavin gray amos storkey school informatics university edinburgh abstract model distillation compresses trained machine learning model neural network smaller alternative could easily deployed resource limited setting unfortunately requires engineering two architectures student architecture smaller first teacher architecture trained emulate paper present distillation strategy produces student architecture simple transformation teacher architecture recent model distillation methods allow preserve performance trained model replacing convolutional blocks cheap alternative addition distillation attention transfer provides student network performance better training student architecture directly data introduction despite advances deep learning variety tasks lecun deployment deep learning embedded devices mobile phones digital cameras vehicle navigation systems relatively slow due resource constraints devices operate big neural networks fit devices networks big expensive dominant memory cost neural networks number parameters need stored networks substantially fewer parameters without commensurate loss performance possible take large teacher network use outputs aid training smaller student network caruana distillation process student network powerful trained solely training data closer performance larger teacher student network typically architecture shallow thinner mean filters less channels romero teacher possible arbitrarily approximate network another urban limit neural network performance least part due training algorithm rather representational power paper take alternative approach designing student networks instead making networks thinner shallow take standard convolutional block networks possess replace cheaper convolution block keeping original architecture example resnet standard block pair sequential convolutions show comparable number parameters student networks retain architecture teacher cheaper convolutional blocks outperform student networks original blocks smaller architectures model compression strategy effective time transformation easy implement deep learning framework replacing convolutional blocks simple substitution existing architecture furthermore optimisation scheme used teacher network repeated student making another round hyperparameter optimisation unnecessary cheap convolutional blocks suggest described section well overview methods employ distillation section train number student networks task image classification krizhevsky datasets demonstrate cheap convolutions perform better traditional student networks given parameter cost level parameter reduction competitive much complicated methods literature howard methods may complementary han though possible train resulting architectures directly demonstrably less effective distilling larger teacher model related work parameters deep networks great deal redundancy shown many predicted subset parameters denil however challenge remains find good ways exploit redundancy without losing model accuracy observation along desire efficiency improvements driven development smaller less convolutions one prominent examples depthwise separable convolution sifre applies separate convolutional kernel channel followed pointwise convolution lin channels depthwise separable convolutions used several architectures ioffe szegedy chollet xie explicitly adapted mobile devices howard however separating spatial elements way simplify convolution jin authors propose breaking general convolution set pointwise convolutions along different axes authors wang start separable convolutions add topological subdivisioning way treat sections tensors separately bottleneck spatial dimensions methods demonstrate models several times smaller original model maintaining accuracy separable convolution expensive part pointwise convolution proposed operation could also grouped sets channels however maintain connections channels helpful add operation mixing channels together zhang simply squared reduction achieved applying bottleneck channels spatial convolution xie iandola paper examine potency separable bottleneck structure work discussed thus far section involves learning compressed network scratch clear alternatives retraining reducing number parameters han interested learning smaller network student distillation caruana conjunction large teacher network small student complex function large deep teacher network theoretically approximated network single hidden layer enough units cybenko difficulty practice learning function knowledge distillation caruana hinton proposes use information logits learnt network train smaller student network early experiments shown effective networks much smaller original could trained small increases error however modern deep architectures prove harder compress example deep convolutional network trivially replaced feedforward architecture urban two methods proposed deal first romero authors use linear map activations intermediate points produce extra loss function second attention transfer zagoruyko komodakis authors choose instead match activations taking mean channels context paper found attention transfer effective experiments described section model compression cheap convolutions given large deep network performs well given task interested compressing network uses fewer parameters flexible widely applicable way reduce number parameters model replace convolutional layers cheaper alternative replacement invariably impairs performance reduced network trained directly data fortunately able demonstrate modern distillation methods enable cheaper model performance closer original large network distillation paper utilise compare two different distillation methods learning smaller student network large teacher network knowledge distillation hinton caruana attention transfer zagoruyko komodakis briefly explain methods knowledge distillation let denote cross entropy two probability vectors lce log assume dataset elements one element denoted element corresponding class label denote vector corresponding given trained teacher network teacher outputs corresponding logits denoted likewise student network outputs logits student perform knowledge distillation train student network minimise following loss function averaged across data items lkd lce softmax function temperature parameter parameter controlling ratio two terms first term standard cross entropy loss penalising student network incorrect classifications second term minimised student network produces outputs similar teacher network idea outputs teacher network contain additional beneficial information beyond class prediction attention transfer consider choice layers teacher network corresponding layers student network chosen layer teacher network collect spatial map activations channel vector atij let ati collect atij likewise student network correspondingly collect asij asi given choice mapping maps collection form vector attention transfer involves learning student network minimising lat lce ati asi asi hyperparameter zagoruyko komodakis recommended using pna nai number channels layer words loss targeted difference spatial map average squared activations spatial map normalised overall activation norm let examine loss first term standard cross entropy loss second term however ensures spatial distribution student teacher activations similar selected layers network explanation networks paying attention things layers cheap convolutions large layers longer commonplace convolutions make almost parameters modern therefore desirable make smaller present several convolutional blocks may introduced place standard block network substantially reduce parameter cost first let consider standard two dimensional convolutional layer contains nout filters size nin assuming square convolutions nout number channels layer output nin number channels input kernel size convolution modern neural networks almost always case nin nout let max nin nout parameter cost layer nin nout bounded typical residual network block contains two convolutions refer standard block outlined table one alternative full convolutions parameters scale approximately break convolution groups shown figure restricting convolutions mix channels within group groups obtain substantial reduction number parameters grouped computation example nin nout cost changes standard layer groups parameter convolutions hence reducing parameter cost factor provide mixing following grouped convolution pointwise convolution parameter cost nin nout change channel size occurs across pointwise convolution refer substitution operator grouped convolution groups illustrate figure original resnet paper authors introduce bottleneck block parameterised denoted table input first channels decreased factor via pointwise convolution full convolution carried finally another pointwise convolution brings representation back desired nout reduce parameter cost block even replacing full convolution grouped one bottleneck grouped pointwise block referred illustrated figure substitute blocks compared table computational costs simplicity take case nin nout given practice varying bottleneck size number groups network parameter numbers may vary two orders magnitude enumerated examples given tables parameters introduced batch normalisation negligible compared convolutions however included completeness table figure grouped convolutions operate passing independent filters tensor separated groups channel dimension consider grouped convolution input output channels filters needs operate channels reduces parameter cost convolution factor standard figure grouped pointwise block substitutes full convolutions standard block grouped convolution followed pointwise convolution reduce parameters pointwise bottleneck used grouped pointwise convolution using grouped convolutions bottlenecks common methods parameter reduction designing network architecture easy implement deep learning framework sparsity inducing methods han approximate layers yang may also provide advantages complementary approaches structured reductions grouped convolutions bottlenecks advantageous sparsity methods sparsity structure need stored claim paper structured parameter reductions sufficient achieve model compression results line state art using effective model distillation block structure conv conv gconv gconv conv gconv conv params params table convolutional blocks used paper standard block grouped pointwise block bottleneck block bottleneck grouped pointwise block blocks use conv refers convolution gconv grouped convolution pointwise convolution refers batchnorm layer followed relu activation assume input output block channels channel size change particular convolution unless written explicitly applicable number groups grouped convolution bottleneck contraction give parameter cost convolutions block terms parameters parameter cost assuming running kept normalisation also given markedly smaller experiments section train evaluate number student networks distilled large teacher network distil knowledge distillation attention transfer also train networks without form distillation observe whether distillation process necessary obtain good performance way demonstrate high performance comes distillation achieved directly training student networks using data comparison also study student networks smaller architectures fewer teacher enables test block transformations propose key simply matter distilling networks smaller numbers parameters compare smaller student architectures student architectures implementing cheap substitute convolutional blocks architecture teacher different convolutional blocks summarised table student networks described detail section experiments conducted datasets results group output size structure pool classes block block block classes fully connected table summary wide resnet structures used experiments matching zagoruyko komodakis bulk parameters consist blocks channel width controlled explore effect substituting blocks cheaper alternatives classes refers number object classes perhaps unsurprisingly given table figure results found table figure results discussed detail section network descriptions experiments utilise competitive wide residual network wrn architecture zagoruyko komodakis briefly summarised table bulk network lies groups network depth determines number convolutional blocks groups network width denoted affects channel size filters blocks note employ attention transfer student teacher outputs groups used second term equation teacher network use wrn depth width standard blocks kernels used convolutions student teacher networks unless stated otherwise student networks use blocks student networks thinner shallow teacher represent typical student networks works employ blocks kernels replaced dilated kernels described koltun allows see possible naively reduce parameters effectively zeroing elements kernel using bottleneck block channel contraction using grouped pointwise block group sizes number channels block allows explore spectrum full convolutions fully separable convolutions bottlenecked grouped pointwise block use groups sizes number channels bottleneck use notation represents fully separable convolutions easily denote divisions thereof also used observe effect extreme compression implementation details experiments conducted pytorch paszke training used minibatches size minibatch images padded zeros random crop taken image flipped probability half training conducted epochs using sgd standard momentum fixed initial learning rate learning rate reduced factor start epochs knowledge distillation set used temperature attention transfer set analysis observations figure compares parameter cost student network log scale test error obtained attention transfer plot ideal network would lie corner parameters low error fascinating almost every network architecture teacher cheap convolutional blocks blue green cyan lines performs better given parameter budget reduced architecture networks standard blocks red line outperforms despite considerably fewer parameters several networks blocks significantly outperform use less parameters figure test error parameters student networks learnt attention transfer note logarithmically scaled points red curve correspond networks convolutional blocks reduced architectures networks architecture teacher cheap convolutional blocks green blue cyan blocks described detail table notice student networks cheap blocks outperform smaller architectures standard convolutions given parameter budget encouraging significant compression possible small losses several networks perform almost well teacher considerably less parameters blue error close teacher fifth parameters less tenth parameters teacher cost increase error similar change error compression rates exceed found contemporary papers cohen welling even simply switching convolutions smaller dilated equivalents allows one use half parameters similar performance important lesson learnt regarding grouped pointwise convolutions often used chollet form number groups equal total number channels convolution however networks half quarter number groups perform substantially better modest increase parameters parameters compared error lower make number groups smaller performance gets close teacher network structure getting closer closer original convolutions teacher neglecting pointwise component number groups easy parameter tune trade performance smaller network grouped pointwise convolutions also work well conjunction bottleneck size although large bottlenecks error increases rather significantly seen despite still comparable performance half parameters observe similar trends figure figure test error parameters student networks learnt attention transfer points red curve correspond networks convolutional blocks reduced architectures networks architecture teacher cheap convolutional blocks green blue cyan also observe attention transfer teacher network substantially better knowledge distillation training network structure directly data consider table shows attention transfer errors figure error column alongside networks trained knowledge distillation error distillation error cases student network trained attention transfer better student network trained distillation process appears necessary performances particularly impressive blocks error higher teacher despite network half many parameters also noticeable knowledge distillation gives similar even worse performance student network trained conclusion mirrored training table romero note performance issues knowledge distillation occur networks depth layers zagoruyko komodakis also observe experiments training cnn imagenet would converge using knowledge distillation conclusion training large deep model may prohibitively time consuming design model compression strategy order deploy many problems may also difficult achieve desired performance smaller model demonstrated model compression strategy fast apply require additional engineering furthermore optimisation algorithm larger model sufficient train cheaper student model cheap convolutions used paper chosen ease implementation future work could investigate complicated approximate operations described moczulski could make difference convolutions final layers network one could also make use custom blocks generated large scale black box optimisation zoph equally many methods low rank approximations could applicable jaderberg garipov sainath hope work encourages others consider cheapening convolutions compression strategy acknowledgements project received funding european union horizon research innovation programme grant agreement bonseyes work supported swiss state secretariat research innovation seri contract number opinions expressed arguments employed herein necessarily reflect official views funding bodies block teacher params error error error table student network test error network wide resnet given first column block type second standard convolutional block standard block dilated kernels grouped pointwise block groups bottleneck block contraction bottleneck block contraction grouped convolution groups refers channel width block refers channel width bottleneck applicable total parameter cost network given third column errors reported learning distillation error knowledge distillation teacher error attention transfer teacher error teacher used training given first row table shows attention transfer possible cut number parameters network retain high performance similar number parameters students cheap convolutional blocks outperform expensive convolutions smaller architectures block teacher params error error error table student network test error network wide resnet given first column block type second blocks described detail section total parameter cost network given third column errors reported learning distillation error knowledge distillation teacher error attention transfer teacher error teacher used training given first row references lei jimmy rich caruana deep nets really need deep advances neural information processing systems cristian rich caruana alexandru model compression acm sigkdd international conference knowledge discovery data mining chollet xception deep learning depthwise separable convolutions corr url http taco cohen max welling group equivariant convolutional networks maria florina balcan kilian weinberger editors proceedings international conference machine learning volume proceedings machine learning research pages new york new york usa jun pmlr url http george cybenko approximation superpositions sigmoidal function mathematics control signals systems mcss misha denil babak shakibi laurent dinh marc aurelio ranzato nando freitas predicting parameters deep learning advances neural information processing systems timur garipov dmitry podoprikhin alexander novikov dmitry vetrov ultimate tensorization compressing convolutional layers alike corr url http song han huizi mao william dally deep compression compressing deep neural networks pruning trained quantization huffman coding corr url http kaiming xiangyu zhang shaoqing ren jian sun deep residual learning image recognition proceedings ieee conference computer vision pattern recognition kaiming xiangyu zhang shaoqing ren jian sun identity mappings deep residual networks european conference computer vision geoffrey hinton oriol vinyals jeff dean distilling knowledge neural network corr url http andrew howard menglong zhu chen dmitry kalenichenko weijun wang tobias weyand marco andreetto hartwig adam mobilenets efficient convolutional neural networks mobile vision applications corr url http forrest iandola matthew moskewicz khalid ashraf song han william dally kurt keutzer squeezenet accuracy fewer parameters model size corr url http sergey ioffe christian szegedy batch normalization accelerating deep network training reducing internal covariate shift international conference machine learning pages max jaderberg andrea vedaldi andrew zisserman speeding convolutional neural networks low rank expansions british machine vision conference jonghoon jin aysegul dundar eugenio culurciello flattened convolutional neural networks feedforward acceleration international conference learning representations alex krizhevsky learning multiple layers features tiny images master thesis university toronto yann lecun yoshua bengio geoffrey hinton deep learning nature zhe xiaoyu wang xutao tianbao yang small effective pattern networks corr url http min lin qiang chen shuicheng yan network network international conference learning representations marcin moczulski misha denil jeremy appleyard nando freitas acdc structured efficient linear layer corr url http adam paszke sam gross soumith chintala gregory chanan pytorch tensors dynamic neural networks python strong gpu acceleration https accessed october adriana romero nicolas ballas samira ebrahimi kahou antoine chassang carlo gatta yoshua bengio fitnets hints thin deep nets corr url http tara sainath brian kingsbury vikas sindhwani ebru arisoy bhuvana ramabhadran matrix factorization deep neural network training output targets ieee international conference acoustics speech signal processing laurent sifre scattering image classification phd thesis polytechnique gregor urban krzysztof geras samira ebrahimi kahou ozlem aslan shengjie wang rich caruana abdelrahman mohamed matthai philipose matt richardson deep convolutional nets really need deep convolutional international conference learning representations min wang baoyuan liu hassan foroosh factorized convolutional neural networks corr url http saining xie ross girshick piotr zhuowen kaiming aggregated residual transformations deep neural networks proceedings ieee conference computer vision pattern recognition zichao yang marcin moczulski misha denil nando freitas alex smola song ziyu wang deep fried convnets proceedings ieee international conference computer vision fisher vladlen koltun context aggregation dilated convolutions international conference learning representations sergey zagoruyko nikos komodakis wide residual networks british machine vision conference sergey zagoruyko nikos komodakis paying attention attention improving performance convolutional neural networks via attention transfer international conference learning representations zhang zhou lin sun shufflenet extremely efficient convolutional neural network mobile devices corr url http barret zoph vijay vasudevan jonathon shlens quoc learning transferable architectures scalable image recognition corr url http
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serra jul stochastic model catalytic reaction networks protocells roberto alessandro marco alex chiara tommaso dept physics informatics mathematics modena reggio emilia university european centre living technology eclt university foscari venice italy dept informatics systems communication university milan bicocca italy sysbio centre systems biology piazza della scienza milano italy university turin department economics statistics cognetti martiis torino february abstract protocells supposed played key role processes leading emergence life existing models either describe protocell architecture dynamics given existence sets collectively molecules granted describe emergence aforementioned sets ensemble random molecules simple experimental setting closed system flow reactor properly describe protocell paper present model goes beyond limitations describing dynamics sets replicating molecules within lipid vesicle adopt simplest possible protocell architecture considering membrane selects molecular types allowed enter exit protocell assuming reactions take place aqueous phase internal compartment first approximation ignore protocell growth division dynamics behavior catalytic reaction networks simulated means stochastic model accounts creation extinction species reactions yet exhaustive protocell model already provides clues regarding processes relevant understanding conditions enable population protocells undergo evolution selection serra keywords autocatalytic sets molecules catalytic reaction sets origin life stochastic simulations protocell introduction widely believed origin life required formation sets molecules able collectively well compartments able undergo fission proliferate particular order observe lifelike behavior necessary chemical reactions coupled rate proliferation compartments several protocell architectures proposed identifying compartment lipid vesicle may spontaneously fission suitable circumstances hand many distinct models proposed describe sets reactions involving randomly generated molecules many cases although principle required assumed catalyzed reactions take place significant rate therefore sets also termed catalytic reaction sets briefly crss worth noting appearance new molecules implies appearance new reactions involving new molecules set molecular types set reactions change time hence possible certain time set molecules able catalyze formation emerges refer autocatalytic set acs noticed crs contain one acss none even though models protocell actually describe coupling reaction networks dynamics lipid container consider fixed set molecular species reactions hence providing incomplete representation complex interplay conversely several studies collectively self replicating sets molecules continuously stirred tank reactor cstr including provide limited information behavior protocell therefore order develop framework may unify crss protocell modeling approaches necessary analyze behavior crss vesicle investigate coupling evolving chemical population growth lipid container fission paper propose step towards first goal deferring second one work particular analyze behavior dynamical model crss simplified model vesicle best knowledge novel approach important remarks let first observe cstr good model protocell least two reasons general protocells constant inflow protocells semipermeable membranes allow molecular types serra contrary open flow reactors contained inflow enters reactor dissolved reactor washed another important limit cstr concerns evolvability argued presence different asymptotic dynamical states ability shift may essential achieve viable evolution first forms life recent works found models catalytic reaction networks cstrs generally one states found apart fluctuations furthermore order accomplish goal work need better specify model catalytic reactions sets protocell far former concerned studied dynamics random sets molecules revisiting model kauffman proposed interesting way build new molecular species existing ones see section description original version model relied purely arguments important fail appreciate effects dynamics including noise fluctuations effects dynamics later introduced farmer described kinetics using ordinary differential equations however formalism account chance species become extinct finite amount time may instead well happen reaction graph may grow never shrinks order overcome limitations bagley proposed empirical correction setting zero concentration values happen fall certain threshold works rather use beginning stochastic approach analyze dynamics gillespie algorithm order deal rigorous way low concentrations fluctuations note kauffman model largely relies upon randomness particular every polymer system fixed probability may vanish catalyze possible reaction therefore different simulations species catalyze different reactions leading formation different chemistries thus exactly language choose set tuples species catalysis reaction species catalyzes reaction called chemistry describes possible artificial simulate different chemistries look generic properties set chemistries different series experiments also keep chemistry fixed simulate various time histories principle may differ since discovery given catalyst early phase finite system might channel following evolution way another since number molecules species may small principle legitimate limitations outflow modeled chemostat supposing molecules larger certain size precipitate washed away worthwhile notice presence catalysis within tuple allows possibility species catalyze one reaction reaction catalyzed one species serra ignore aspect stochastic model particularly well suited analyze shown section course conditions simulations given chemistry converge asymptotically chemical mixture moving protocell model note usually based lipid vesicles approximately spherical structures aqueous interior membrane composed lipid bilayer spontaneously form lipids mixed water certain conditions even though different protocell architectures proposed consider simplest model namely key reactions take place aqueous phase inside protocell would indeed straightforward model coupling molecules growth protocell following approach similar previous studies yet main objective present work studying dynamics crss embedded vesicle simplify treatment ignoring growth dynamics protocell keeping volume fixed implies study limited time intervals short respect describing growth whole protocell selective character membranes key ingredient model suppose simplicity molecules shorter certain length cross membrane transmembrane motion permeable species supposed driven difference concentrations internal aqueous volume protocell external aqueous environment assume transmembrane diffusion extremely fast always equilibrium concentrations species cross membrane adiabatic hypothesis could easily relaxed future furthermore assume protocells turgid approximation implies also neglect issues related osmotic pressure another related aspect model since assumed permeable species equilibrium ones never cross barriers infinite concentration growth possible obviously nonphysical behavior model validity limited time simplifications removed subsequent studies also justified fact main goal studying dynamics crss affected embedded vesicle model used order investigate behavior system different conditions address important questions first perhaps important one reason compartments seem necessary life indeed first studies molecules interested aspect crss supposed exist pond beaker yet life seems require compartments ubiquitous important understand whether major differences may happen protocell happens bulk phase serra would unconvincing postulate priori internal external environments different indeed likely assume vesicles form aqueous environment average internal milieu essentially external membrane surrounds portion fluid happen makes difference let first observe protocells small typical linear dimensions ranging imagine population protocells exists overcrowed total internal volume typically much smaller total external volume fortiori true isolated one moreover every point interior protocell allowed far away surface protocell contains observations imply effect surfaces much larger within protocells outside suppose example membrane hosts catalytic activities important molecules synthesized close boundaries inside outside diffuse freely membrane width much smaller protocell radius internal external surface areas close external volume much larger external one therefore internal concentrations much higher external environment case system behavior interior significantly different external one note also effect may different different molecules formation might catalyzed membrane others might unaffected even relative concentrations different chemicals may differ two cases indeed important protocell models based active catalytic role membrane cases easy understand role protocell since provides essential catalysts way keep products closer protocells might able give rise internal environment different bulk even catalytic activity absent reason seemingly counterintuitive behavior smallness protocells note considering case new molecules formed already interior protocell plus cross semipermeable membrane concentrations high likely total numbers newly formed molecules quite low different protocells might host different groups molecules might even happen molecular type present protocells others order get feeling possibility let provide realistic estimates number molecules different types present protocell let consider typical vesicles linear dimension small ones typical concentrations macromolecules may observed superconcentration phenomena take place particular circumstances neglect serra typical small table excepted number molecules given species given protocell rows refer protocell volumes columns concentrations millimolar nanomolar range excepted numbers molecules single protocell therefore given table let recall numbers refer excepted values fluctuations may relevant small numbers involved example case concentration small vesicles average molecule every cells apparent different protocells widely different initial compositions therefore come conclusion creation small compartments allow formation population different individuals fluctuations environment looks macroscopically homogeneous yet sufficient definitely necessary condition darwinian evolution take place obviously supposing compartments divide division rate depends upon composition moreover small stochastic systems may also happen different trajectories stemming initial conditions due order new molecules synthesized aspects protocell dynamics important model spite current limitations well suited explore related phenomena indeed possible analyze different possible stochastic effects include path dependency induced random order new molecules generated particular regard low concentration effects catalyst discovered advance respect another system evolution may different studied comparing different simulations referring chemistry starting initial conditions differences induced different initial conditions randomly generated distribution iii different behaviors distinct chemistries real world rules chemistry given kind analysis performed also interesting understand different chemistries may affect behavior system diversity population protocells example role raf sets overall dynamics section present results simulations model concluding section discuss main findings paper propose analysis refinement models serra main features model entities interactions model describes open system simple molecules interact elementary catalyzed reactions basic entities system monomers polymers identified ordered strings letters oriented left right taken finite alphabet refer letters also bricks term monomer reserved molecular types composed single brick every species composing entire set species characterized specific amount either quantity concentration denoted number copies specific species defines number molecules two basic reactions cleavage cutting species composed one brick two shorter species abba condensation concatenation two species longer one bba aab bbaaab key assumption model reaction occurs catalyzed one specific catalysts hence exclude presence spontaneous reactions assuming sufficiently high activation energy reaction particular condensation requires intermediate reaction temporary complex catalyst substrate formed restriction regarding catalysis imposes species composed least minimum number bricks catalysts additionally neglect presence backward reactions exception made dissociation reaction intermediate complexes hypothesizing gibbs energy reaction sufficiently large schematic representation various reaction types given fig reaction network provided catalyst reaction exists species system could condensate species system split cutting point smaller species cardinality set conceivable reactions system given time therefore given cardinality length number bricks specific species first term refers conceivable cleavages second one conceivable condensations hence set reactions actually allowed chemistry system determined catalysts involved various reactions following kauffman define chemistry system random regard species finite fixed probability chosen catalyst serra condensation comp decomp cond cleavage cleav figure schematic representation two possible reaction types namely condensations cleavages represent substrates reaction species catalytic function transient complex formed first step condensation stand products reaction kcleav kcomp kdiss kcond respectively stand kinetic rates cleavage complex formation complex dissociation final condensation randomly chosen reaction among belonging worth stressing although reaction network built probabilistically remains invariant time words species chosen catalyst given reaction species always catalyst reaction details concerning structures model found dynamics system dynamics simulated means extension gillespie algorithm stochastic simulation chemical reaction systems allow appearance novel species reactions present system initial conditions fact cleavage condensation elements either present within protocell entering external environment generate new species turn involved new reactions catalysts substrates container introduction protocell classical formulation catalytic reaction network modeled within continuous reactor cstr ingoing outgoing fluxes could adjusted according experimental needs work introduce major modification model introducing membrane separates catalytic reaction network external environment see fig membrane modeled allowing species enter leave protocell namely shorter serra figure schematic representation membrane conceived model protocell membrane represented lipid bilayer shapes spherical vesicle example species shorter letters cross membrane entering leaving internal compartment dynamics actually simulated model catalytic reaction networks opposite longer species confined either inside outside vesicle trary length lperm species longer lperm either remain entrapped within protocell never enter outside another important assumption concentration permeable molecules homogeneous inside outside protocell take value assume infinitely fast diffusion bulk phases across membrane chemical potentials permeable species species cross membrane also defined buffered species work consider volume protocell constant planned introduce protocell size duplication dynamics developments model also remark model transport processes vesicle treated way albeit simplified parallels actual dynamics small flow reactors also proposed protocell models indeed well suited since require constant inflow physical analogue vesicle usually also allow outflow solutes serra algorithm formal definition model system let alphabet symbols denoting monomers infinite set polymers complexes let denote resp polymer resp complex obtained concatenating polymers let state system multiset counting occurrences consider following reaction schema variables notation cleavage rcl catalyst polymer breaking complex formation rcf complex dissociation rcd final condensation substrates catalyst substratecatalyst complex polymer composed variable polymers take values whereas complexes reactions generation instantiation map evaluated via standard pattern matching transform schema possible rewriting rules rcl rcf rcd multiset map assigns values required maximal univocal must yield possible rewriting rules applicable instantiated consistently via deterministic association evaluation yield markov chain firing reaction via classical gillespie algorithm new state system generated particular implementation model polymers shorter fed instantiated catalysts see appendix simulation proceeds new state previous step results simulations preliminary experiments protocell model performed keeping fixed key structural parameters see appendix details serra creating different random chemistries particular present details two specific chemistries specifically selected among many total simulated chemistries exhibit specific dynamical properties regard first important introduce concept raf set fundamental description systems following hordijk given entire chemistry subset reactions reflexively autocatalytic every reaction catalyzed least one reaction belonging every substrate produced food set means reactions belonging iii reflexively autocatalytic raf previous conditions satisfied chemistries two presented protocells differ raf set raf short particular first protocell rafs present whereas second protocell raf formed autocatalysis consuming molecules food set present order assess behavior system method measure similarity two different states system proposed kaneko based comparison vectors describe chemical compositions let define vectors whose components concentrations species present time systems respectively positions vectors refer species hence species present system present concentration equal viceversa similarity two systems computed means normalized inner product cos angle two vectors measured time throughout paper angles measured degrees low concentration effects section investigate low concentration effects protocells end selected chemistries tested respect different initial uniform concentrations chemical species present inside vesicle detail initial concentration species equal conversely amount buffered species always fixed notice general protocell model presented lead stationary chemical distributions indeed situations possible particular species indefinitely increases concentration example raf set particular species catalyses formation using two buffered species substrates case exponential growth achieved another simple case could linear growth species produced directly buffered species means catalyst involved reactions serra conc molarity molecules per species average mean max mean raf max mean max table table average maximum values relative distinct simulations different initial conditions rows table shown measures reported different chemistries one without rafs one rafs also computed without taking account species belonging raf column rafs conditions differ average magnitude initial concentrations initial set molecular species belonging buffered flux food set realization initial concentration drawn random poisson distribution according given parameters maintained invariant across different runs present constant concentration course complex situations possible even directly originating food set yet many simulations show transient chemicals distribution rapidly change followed long changes limited small adjustments applying parameters adopted present work state already achieved seconds except case low concentrations diversity table statistics varies according different cases shown particular possible appreciate increase dissimilarity among protocells correspondence decrease initial concentration indicated average maximum value runs specific chemistry specific initial condition computed time since runs characterized identical chemistries identical initial conditions angle reported indeed result dynamical evolutions differ simulation random seed worth remarking lower concentration profile higher distance among final angle reaches maximum exceptionally case computed second least reactions occurred within simulation reason low concentrations involve slow dynamics seconds enough order observe significant chemical changes regard considering value excluding species belonging raf set last columns table since molecules belonging raf set reach concentration much greater respect molecules considering angle computation would misrepresent distance among simulations serra value regard low initial concentrations molecule per species average interesting notice displayed low concentration effects respect overall similarity system found chemistries hinting generic property systems independently possible presence raf sets influence raf set dynamics asynchronous framework implies one reaction occurs time given rafs general involve one reaction order detect presence analyze system time interval sufficiently large let significant number reactions occur yet embracing whole simulation order avoid presence rare reactions worth stressing since analysis made post respect simulation system affect simulated dynamics way representing fig show presence absence raf every seconds using time window respect aforementioned different average concentrations note initial conditions autocatalytic species concentration higher zero thus corresponding raf always able achieve growth viability fig possible observe strong correlation average concentration species within protocell presence rafs long concentration decreases probability detecting raf yet long run simulations rafs eventually emerge sensitivity initial conditions analyze effects variations initial concentrations single species present within protocells maintaining average concentration fixed simulate different variations single species concentrations case average concentration equal condition average concentration condition fig display variation similarity time couple simulations respect conditions order provide picture sensitivity overall diversity initial conditions note low concentrations condition average one molecule species imply certain protocell realizations species missing particular simulation starts different set species composed average species possible may explain high values respect condition fig besides one see condition shows evident bifurcation fig given simulations low concentration effects set sampling frequency time threshold windows taking advantage several initial model threads essential comprehension article serra figure top bottom four traces accounting conditions chemistry one containing raf set depicting presence raf simulation time flows left right row represents different run condition black zones stands absence raf set gray zones denote presence raf important remark rafs searched dynamic network created reactions occurred mobile time window seconds autocatalytic species raf present absent initial condition system indeed reach different regions state space leading deeply different histories opposite condition shows aforementioned regulatory activity always active raf figure angles couple different simulations time condition left panel condition right panel finally important remark simulation many species disappear many appear compact way follow process monitoring run variation angle initial current chemical setting fig graph indicates approaches value close degrees means quasiorthogonality system respect initial condition serra dynamics system term molecular concentrations leading divergence beginning shown fig reasonable hypothesize clear inversions trajectories concentrations certain species may responsible phenomenon figure left panel average angle standard deviation measured chemical distribution within single run conditions average molarity average molarity shown worthwhile remark stands complete orthogonality chemical distributions order appreciate convergence towards right panel concentration species time particular simulation shown simulation condition run exponential growth concentration autocatalytic species shown limit conclusion developments paper introduced simplified model protocell investigated behavior stochastic model catalytic reaction networks environment best knowledge novel approach crucial importance small size protocell stressed effects fact chemicals present low numbers investigated broader analysis ongoing shown possible reach different compositions chemical species particular case species present bulk low concentrations also shown two different possibly overlapping reasons diversity random sequence molecular events involving species random differences initial concentrations also stressed importance raf sets influencing overall dynamics several ways work might seed research obvious relaxing physical limitations considered infinitely fast diffusion yet except may change major conclusions summarized serra obviously interesting direction considering protocell able grow divide processes involved protocell growth replication indeed complex particular necessary condition existence replication coupling rates molecules replication cell growth shown elsewhere existence coupling suffices guarantee general conditions long run rate cell division duplication replicating molecules converge value thereby allowing sustainable growth population however results achieved supposing fixed set genetic memory molecules possible extinction could sound extend approach case evolving chemical reaction sets verify whether synchronization occurs important aspect addressed case growing vesicles also effect volume growth concentrations various chemicals preliminary investigation found besides explicitly considered possibly catalytic role membranes discussed section might major cause difference intracellular environment bulk fixed volume model effect lumped effective reaction rates consider growing protocell take account differences surfaces volumes might also lead interesting phenomena analyzed future developments conclude different protocells may host different mixtures molecular species even share chemistry inhabit world might extremely interesting model behavior populations different protocells kind may show different growth rates may also undergo phenomena like coalescence exchange material etc thus investigations indeed necessary assess different generations protocell populations possible evolution pathways last least interesting extend studies protocell architectures like los alamos bug gard models others acknowledgements stuart kauffman norman packard wim hordijk kindly shared deep understanding autocatalytic sets several useful discussions useful discussions ruedi davide lucrezia timoteo carletti andrea roli giulio caravagna also gratefully acknowledged authors also grateful giulia begal kindly drawing image fig wishes acknowledge project sysbionet imp cup financial support work final publication available springer via http property proved earlier munteanu los alamos bug mode serra appendix simulation environment parameter settings simulations performed carness developed research group following baseline setting system used simulations reported parameters variated different experiments please refer text alphabet volume average catalysis probability catalyzed reaction species maximum length species lmax lperm monomers dimers catalyze kcleav kcomp kdiss kcond references bagley farmer spontaneous emergence metabolism artificial life santa institute studies sciences complexity cans meghan christine keating positioning lipid membrane domains giant vesicles aqueous cytoplasm mimic chem soc carletti serra villani poli filisetti sufficient conditions emergent synchronization protocell models theor biol timoteo carletti alessandro filisetti stochastic evolution protocell gillespie algorithm dynamically varying volume computational mathematical methods medicine article xiaofeng dai olli andre ribeiro determining noisy attractors delayed stochastic gene regulatory networks multiple data sources bioinformatics oxford england september tereza pereira souza pasquale stano frank steiniger erica aguanno emiliano altamura alfred fahr pier luigi luisi encapsulation ferritin ribosomes complexes inside liposomes insights origin metabolism origins life evolution biosphere journal international society study origin life october lisa dominak christine keating polymer encapsulation within giant lipid vesicles langmuir freeman dyson origins life cambridge cambridge university press eigen schuster hypercycle principle natural part emergence hypercycle die naturwissenschaften november farmer kauffman autocatalytic replication polymers physica nonlinear phenomena https serra filisetti graudenzi serra villani lucrezia poli role energy stochastic model emergence autocatalytic sets lenaerts giacobini bersini bourgine dorigo doursat editors advances artificial life ecal proceedings eleventh european conference synthesis simulation living systems pages mit press cambridge filisetti serra carletti villani poli protocell models synchronization chaos european physical journal june alessandro filisetti alex graudenzi roberto serra marco villani davide lucrezia rudolf fuchslin stuart kauffman norman packard irene poli stochastic model emergence autocatalytic cycles journal systems chemistry alessandro filisetti alex graudenzi roberto serra marco villani rudolf norman packard stuart kauffman irene poli stochastic model autocatalytic reaction networks theory biosciences theorie den biowissenschaften pages october ganti chemoton theory vol theory fluyd machineries vol theory livin system new york kluwer daniel gillespie exact stochastic simulation coupled chemical reactions journal physical chemistry daniel gillespie stochastic simulation chemical kinetics annual review physical chemistry martin hanczyc shelly fujikawa jack szostak experimental models primitive cellular compartments encapsulation growth division science new york october wim hordijk jose fontanari catalytic reaction sets decay preservation information wim hordijk jotun hein mike steel autocatalytic sets origin life entropy june wim hordijk mike steel detecting autocatalytic sets chemical reaction systems journal theoretical biology april wim hordijk mike steel stuart kauffman structure autocatalytic sets evolvability enablement emergence page may jain krishna model emergence cooperation interdependence structure evolving networks proc natl acad sci sanjai jain sandeep krishna autocatalytic set growth complexity evolutionary model phys rev lett kunihiko kaneko life introduction complex systems biology understanding complex systems new york secaucus usa kauffman autocatalytic sets proteins theor biol luisi ferri stano approaches minimal cells review naturwissenschaften sheref mansy model protocells lipids international journal molecular sciences serra sheref mansy jason schrum mathangi krishnamurthy sylvia douglas treco jack szostak synthesis genetic polymer model protocell nature harold morowitz bettina heinz david deamer chemical logic minimum protocell origins life evolution biosphere september simon conway morris life solution inevitable humans lonely universe university cambridge mouritsen life matter fat emerging science lipidomics springer berlin first edition munteanu attolini steen rasmussen ziock generic darwinian selection protocell assemblies doi sfi working papers santa institute andreea munteanu ricard phenotypic diversity chaos minimal cell model theor biol alla polozova xingong tong shangguan paul meers daniel schuette nozomi ando sol gruner walter perkins formation homogeneous unilamellar liposomes interdigitated matrix biochimica biophysica acta bba biomembranes steen rasmussen liaohai chen david deamer david norman packard peter stadler mark bedau transitions nonliving living matter science steen rasmussen liaohai chen martin nilsson shigeaki abe bridging nonliving living matter artificial life january tristan rocheleau steen rasmussen peter nielsen martin jacobi hans ziock emergence protocellular growth laws philos trans soc lond biol sci lancet composing life embo reports september roberto serra timoteo carletti irene poli syncronization phenomena surfacereaction models protocells artificial life roberto serra timoteo carletti irene poli marco villani alessandro filisetti conditions emergent synchronization protocell jost helbing eds proceedings european conference complex systems paper roberto serra alessandro filisetti marco villani chiara damiani alex graudenzi tommaso panini stochastic model catalytic reaction networks protocells submitted artificial life roberto serra marco villani mechanism formation density gradients semipermeable membranes physical review june ricard andreea munteanu carlos javier synthetic protocell biology reproduction computation philos trans soc lond biol sci stadler dynamics autocatalytic reaction networks inhomogeneous replicator networks bio systems january serra stadler schuster dynamics small autocatalytic reaction bifurcations permanence exclusion bulletin mathematical biology january peter stadler wolfgang schnabl christian forst peter schuster dynamics small autocatalytic reaction networks replication mutation catalysis pasquale stano pier luigi luisi achievements open questions selfreproduction vesicles synthetic minimal cells chemical communications cambridge england june szostak bartel luisi synthesizing life nature vera vasas chrisantha fernando mauro santos stuart kauffman eors szathmary evolution genes biology direct january wesson beyond natural selection mit press cambridge ting zhu jack szostak coupled growth division model protocell membranes journal american chemical society
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adaptive delivery caching networks seyed ali saberali hamidreza ebrahimzadeh saffar lutz lampe ian blake jul abstract problem content delivery caching networks investigated scenarios multiple users request identical files redundant user demands likely file popularity distribution highly user demands positively correlated adaptive method proposed delivery redundant demands caching networks based redundancy pattern current demand vector proposed method decides transmission uncoded messages coded messages delivery moreover lower bound delivery rate redundant requests derived based cutset bound argument performance adaptive method investigated numerical examples delivery rate several specific demand vectors well average delivery rate caching network correlated requests adaptive method shown considerably reduce gap delivery rate lower bound specific cases using adaptive method gap shrinks almost average rate index terms adaptive delivery algorithm average delivery rate coded caching correlated requests placement optimization redundant demands ntroduction local content caching promising technique meet unprecedented traffic demands next generation communication networks caching networks take advantage users contextual information predict future user demands enables network store popular content storage nodes also known caches close end users satisfy user requests locally caching networks operate two phases commonly referred placement delivery phases placement phase caches fill memories parts popular files storage capacity phase takes place network traffic low contrast delivery phase performed network congested phase upon users requests cache provides users parts files available remaining parts files conventionally delivered users separate unicast transmissions performed central server channel shared users recent caching approach known coded caching central server uses simultaneous deliver requested content users reduce network congestion formulation coded caching developed authors defined delivery rate total traffic shared communication link due server messages users requests satisfied moreover proposed centralized scheme reduce delivery rate later work decentralized caching scheme proposed require coordination caches operate decentralized nature method made building block several caching schemes designed later complicated scenarios used peak delivery rate figure merit caching network peak rate results demand vector users request distinct files however average delivery rate also significant performance metric caching network average delivery rate depends statistics user requests thus statistical patterns user demands significantly affect design caching scheme one statistical property user demands popularity distribution files caching schemes used popularity distribution uniform hand proposed different caching schemes account popularities particular caching schemes designed based grouping files several popularity groups files group relatively close popularity levels provide storage resources files popular groups use decentralized caching scheme within group separately also groups library files two groups popular unpopular files requests popular files delivered delivery algorithm requests unpopular files delivered uncoded messages problem investigated assuming zipf popularity distribution independent identically distributed user requests placement based partitioning file equal length packets randomly distributing packets bits caches unlike schemes delivery based delivery algorithm chromatic number index coding contrast scheme restrict coding opportunities requests within popularity group however implementation complicated requires vertex coloring conflict graph statistics users requests affect design caching networks increasing chance multiple identical requests scenario one might able modify delivery algorithm benefit redundancies user demands reduce average delivery rate redundant demands likely made files significantly different popularity levels positive correlations among requests different users case file popularities schemes take effect identical requests account delivery phase delivery schemes based delivery designed demand vectors distinct requests addition popularity levels correlated user requests likely many practical scenarios considerable amount multimedia requests made social networks like facebook twitter instagram movie providing websites like netflix scenarios users overlapping circles friends ones follow people pages live geographical area common personal social professional interests likely get suggestions content media feeds therefore request files paper investigate delivery redundant demands caching networks study model placement fixed yet requests changing time delivery adapts requests propose adaptive delivery scheme based message selection minimize delivery traffic specifically upon receiving demand vector users server exploits redundancy pattern user demands decide whether use uncoded messages coded messages deliver part files requested assume placement phase accomplished placement schemes ensures peak delivery rate exceed delivery rates link capacity constraints satisfied file popularities relatively uniform little prior knowledge popularity distribution available placement time natural accomplish placement delivery phase however users reveal demands server server use knowledge side information adapt choice coded uncoded messages accordingly benefit possible redundancies requests best authors knowledge paper first work literature consider scenario specifically design scheme delivery redundant requests although use placement schemes proposed delivery method based optimization formulation content placement problem namely use modified version problem optimize choice coded uncoded messages proposed delivery scheme side result placement optimization problem generalization centralized placement arbitrary cache sizes particular derive parameters centralized caching analytically cases total cache capacity integer multiple total size files library show superiority adaptive method numerical examples several specific demand vectors derive lower bound delivery rate redundant requests based cutset bound argument compare rate proposed delivery method lower bound moreover study dynamics caching system correlated user demands apply gibbs sampling generate sample demand vectors based stochastic modeling dependencies among user requests shown proposed method superior conventional method terms average delivery rate specific cases adaptive method decreases gap average rate scheme lower bound almost remainder paper organized follows sec present network model review caching schemes formulate rate minimization problem sec iii sec propose adaptive delivery scheme derive lower bound delivery rate sec presents numerical examples simulation results finally conclude paper sec roblem odel eview section explain problem model briefly review caching schemes assume network central server caches server able communicate caches broadcast link see fig denote set caches network library popular files given file bits long assume files available central server cache memory capacity bits represents ratio cache size library size placement phase placement phase caches fill memories parts popular files based placement algorithm assume placement takes place remains unchanged delivery phase resulting distribution bits caches described follows given file given subset caches denote vsn subset bits file exclusively server cache cache cache cache fig network caches central server stored caches note resulting subsets bits partition set bits every file partitions define portion bits file exclusively stored subset caches cardinality assumed depends particular neither depends particular choice caches long cardinality holds symmetry assume uniform distribution file popularities placement phase performed either centralized scheme decentralized scheme centralized caching scheme used integer centralized placement split file subfiles length assign one subfiles subset caches manner store bits belonging subfile caches corresponding results xcen decentralized placement cache stores bits file uniformly random shown large xdecen high probability delivery phase delivery phase network serves one user every cache time denote requests users caches respectively refer vector demand vector note demand vector evolves time delivery phase represent number distinct files demand vector call demand vector redundant addition denote number requests requested file current demand vector thus call redundancy pattern demand vector demand vector define delivery rate traffic shared broadcast link due server messages caches successfully recover files requested express rate terms equivalent total number files must transferred shared link rate files equivalent bits construct file cache needs receive vsdk server delivers bits caches coded delivery messages given algorithm proposed notice delivery method centralized caching special case algorithm note file requested multiple users including user algorithm embeds several messages user side information decode one messages result server needs send messages even though algorithm delivery algorithm require vsn placement phase procedure delivery server sends end end demand vector redundant case messages cases uncoded messages deliver bits stored cache system users request file decode needs sent result traffic due uncoded messages instead thus total delivery rate note substitution gives peak rate centralized caching scheme peak rate decentralized caching scheme one notes redundant demand vectors actual rate algorithm smaller centralized decentralized caching schemes respectively observation basis analysis sec iii ptimality entralized lacement delivery based lgorithm formerly discussed use either methods placement phase caching scheme section show centralized placement scheme optimal placement minimize peak rate delivery algorithm also generalize application centralized placement cases integer optimal placement characterized optimal parameters lead smallest peak delivery rate algorithm based peak rate minimization problem formulated minimize subject first constraint ensures resulting subsets partition bits file also guarantees second constraint represents storage capacity constraint objective function delivery rate algorithm present analytical solution proposition proposition optimal placement demand let solution otherwise table optimal file placement parameters integer dte btc btc btc dte dte otherwise integer btc dte denote largest integer smaller smallest integer larger respectively proof see appendix proposition shows centralized placement optimal algorithm integer generalizes centralized placement scheme caching systems table shows optimal placement parameters system caches library files various storage capacities note two values daptive aching cheme design adaptive delivery method benefits redundancies user requests without changing cache content derive lower bound delivery rate redundant demand vectors adaptive delivery method adaptive method introduce extra step delivery phase takes place receiving request vector transmission server messages users step server decides whether send part requested files corresponding coded message algorithm uncoded message use uncoded messages instead coded messages deliver file equivalent transferring bits vsn notice transfer cache ignores parts content change actual placement files let represent subset bits file exclusively cached transfer done ysn delivery method server first optimizes ysn arbitrarily picks ysn bits vsn form adds rest bits finally uses algorithm delivery based resulting subsets instead vsn find optimal lengths updated partition sets minimize sum lengths messages subsets assume caches requested distinct files current demand vector denote set distinct files requested current demand vector note evolve time fixed demand vector current demand vector rate minimization problem given minimize ysk subject max ysdk ysdk known placement phase given length decentralized centralized placements respectively thus objective function rate algorithm operating based message adjusted subsets similar equality constraint partition constraint also constraints ranges parameters let server use uncoded messages instead coded messages vice versa problem posed linear programming problem standard technique defining ancillary variables max adding extra constraints sec resulting linear programming problem solved numerically algorithm shows adaptive delivery scheme simplified adaptive delivery simplified version message selection step formulated taking number distinct requests account ignoring redundancy pattern demand vector symmetry set ysn leads minimize subject simplified message selection problem algorithm original adaptive delivery algorithm require vsn placement phase procedure adaptivedelivery message selection step unique set distinct files requested solution problem initialization first bits vsdk last bits vsdk end end message construction step server sends end end optimal parameters simplified message selection proposition let problem given proof transfer bits subsets vsn resulting change rate transfer bits difference negative case results parameters algorithm shows simplified adaptive delivery scheme algorithm simplified adaptive delivery algorithm require vsn placement phase procedure simplifiedadaptivedelivery message selection step size unique number distinct requests vsdk corresponds first rule corresponds second rule else vsdk corresponds third rule end end end message construction step server sends end end lower bound let denote smallest rate achievable every possible demand vector distinct requests proposition gives lower bound based cutset bound argument proposition cutset bound assume caches request distinct files must satisfy max proof see appendix umerical xamples imulation esults section investigate performance proposed adaptive delivery method numerical examples computer simulations numerical examples specific demand vectors first consider performance adaptive methods specific instances demand vector fig shows delivery rates delivery scheme algorithm simplified original adaptive schemes lower bound proposition network caches placement cases identical accomplished centralized scheme parameters also calculate rate scheme example considered four redundancy patterns demand vector distinct file requests shown fig rate scheme simplified adaptive scheme lower bound depend specific redundancy pattern contrast rate original adaptive method depends redundancy pattern led different rates different patterns fig observe considerable improvement delivery rate adaptive methods used table shows reduction gap nonadaptive delivery rate lower bound adaptive schemes used storage simplified adaptive adaptive adaptive adaptive adaptive lower bound rate files fig comparison rate different delivery schemes system caches cases centralized placement used users request distinct files shows number users requesting file delivery redundancy method pattern simplified adaptive adaptive adaptive adaptive adaptive table improvement performance gap lower bound fig capacities observe reduction gap redundancy patterns respectively also notice symmetric redundancy pattern adaptive methods led delivery rate redundancy pattern gets asymmetric gap rate original simplified adaptive methods increases observe unlike adaptive schemes delivery rate method increases rate files simplified adaptive adaptive lower bound centralized placement rate files simplified adaptive adaptive lower bound decentralized placement fig comparison rate different delivery schemes system caches rate files simplified adaptive adaptive average lower bound centralized placement rate files simplified adaptive adaptive average lower bound decentralized placement fig effect number distinct files requested delivery rate storage capacity small shows inefficiency algorithm deliver redundant requests fig compares performance delivery methods two different redundancy levels results shown figs cases centralized decentralized placement schemes used respectively rate improved original adaptive method general simplified method requires higher redundancy levels smaller compared original adaptive method able improve rate fact shown fig delivery rates plotted versus original adaptive method delivery rate averaged redundancy patterns distinct requests one notices reduction delivery rate method considerable smaller small large number bits subsets need delivered uncoded messages based algorithm number uncoded messages decreases decreasing reduction rate larger small simulation network dynamics investigate average rates different delivery methods stochastic modeling dynamics caching network consider graph representation network vertices represent caches undirected edge two vertices shows requests corresponding caches correlated model correlation requests assume cache requests file either based neighbours previous requests probability independently probability former case chooses file set last files requested neighbours uniformly random however choosing independently cache picks file library files based popularity distribution files simulations mainly use uniform popularity distribution focus paper also consider scenario file popularities assumed uniform placement phase actual demands delivery phase follow distribution use zipf distribution parameter model file popularities gives maximum average average table iii empirical correlation coefficients resulting number distinct files per demand vector simulations fig correlation coefficient requests caches larger popularity distribution typical values corresponds uniform distribution model described completely determines conditional probabilities users requests chance requesting file cache written otherwise set last files requested neighbour caches use gibbs sampling sec sec generate sample vectors joint distribution user demands based network graph simulations set assume complete graph network vertex degree use control dependency level users requests also control popularity distribution use gibbs sampling need give underlying markov chain enough time reach stationary distribution use estimated potential scale reduction espr convergence criterion sec chains determine time required ignoring first sample vectors samples suffices get shows stationary distribution reached use sample vectors time evaluate average rate different delivery schemes table iii presents details correlation coefficients redundancy levels obtained empirically simulation average rate files simplified adaptive adaptive lower bound fig performance different delivery schemes terms avergae delivery rates central placement used fig shows resulting average delivery rates also shows lower bound average rate calculated taking average lower bounds sample demand vectors used observe requests become correlated larger file popularities get larger adaptive method makes larger improvements rate also adaptive schemes effective decreasing average delivery rate improvement performance gap lower bound shown table onclusion proposed new delivery scheme caching networks exploits redundancies users demand vector reduce delivery traffic proposed scheme allows server decide use coded messages uncoded messages delivery part files requested choice made based redundancy pattern requests current demand vector server decision making process formulated linear programming problem must solved numerically facilitate decision delivery method adaptive simplified adaptive delivery method adaptive simplified adaptive table improvement performance gap lower bound fig making process simplified decision rule also derived analytically derived lower bound delivery rate redundant demands based cutset bound argument proposed adaptive schemes shown significantly improve delivery rate several numerical examples decreased performance gap method lower bound highly redundant demand vectors also investigated dynamics caching network markov chain simulations reported average delivery rate adaptive schemes adaptive methods considerably outperform methods terms resulting average delivery rates also generalized application centralized placement scheme caching networks ppendix roof roposition proof kkt conditions sec optimization problem get lagrange multiplier inequality constraint lagrange multipliers capacity inequality constraint partition equality constraint respectively kkt conditions require result requires two indices provide two degrees freedom set coefficient matrix given linear equations rank result either one two values greater zero consider case separately first assume equality constraint capacity constraint require respectively integer optimal solution achieved given otherwise storage capacity used solution optimal optimal solution one second case let exactly two values namely storage partition constraints get since requires given objective function simplifies function decreasing increasing region specified therefore minimize objective function must take largest value must take smallest value substitution values gives optimal parameters completes proof proposition ppendix roof roposition proof modify cutset bound argument sec bound minimum delivery rate demand vectors distinct requests let subset caches two caches identical user requests assume caches requested files library files let denote server input shared link determines files similarly assume users request files server input determines files requested let consider cut separating caches corresponding users see fig since assume coded caching scheme works files perfectly decoded total information available users cut equal total information requested words since accept value results eferences niesen decentralized coded caching attains tradeoff trans networking vol niesen fundamental limits caching ieee trans inf theory vol may bastug bennis debbah living edge role proactive caching wireless networks ieee commun vol shanmugam golrezaei dimakis molisch caire femtocaching wireless content delivery distributed caching helpers ieee trans inf theory vol tulino llorca caire average performance caching coded multicasting random demands proc international symposium wireless communications systems iswcs hachem karamchandani diggavi content caching delivery heterogeneous wireless networks ieee conference computer communications infocom apr server cache cache cache cache cache fig example cutset separating caches server users caches solid dashed lines represent information flow users selected selected cutset server messages users color identical requests notice two users request picked niesen coded caching nonuniform demands ieee conference computer communications workshops infocom wkshps apr zhang lin wang coded caching arbitrary popularity distributions proc information theory applications workshop ita pedarsani niesen online coded caching proc ieee int conf communications june hachem karamchandani diggavi effect number users coded caching proc ieee int symp information theory june karamchandani niesen diggavi hierarchical coded caching proc ieee int symp information theory june murphy machine learning probabilistic perspective mit press fischer igel introduction restricted boltzmann machines progress pattern recognition image analysis computer vision applications springer boyd vandenberghe convex optimization new york usa cambridge university press breslau cao fan phillips shenker web caching distributions evidence implications ieee conference computer communications infocom mar
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nonlinear observer limited gain power oct daniele astolfi lorenzo marconi note deal new observer nonlinear systems dimension canonical observability form follow standard paradigm instead observer dimension gain grows power design observer dimension gain grows power estimating state addressed means nonlinear observer form index nonlinear observers observers diag ntroduction note consider problem state observation nonlinear systems form state measured output sufficiently smooth functions bounded disturbance measurement noise among different techniques observer design available literature see particularly interested methods shown effective many control scenarios respect assume pair fulfils uniform observability assumption see definition implies existence diffeomorphism dynamic new state variable described canonical observability form see theorem locally lipschitz function inverse namely triplet prime form dimension system defined set class systems fact problem asymptotically case astolfi university bologna italy mines paristech psl research university cas paris france marconi university bologna italy work supported european project sherpa ict design parameter taken sufficiently large chosen matrix hurwitz eigenvalues lefthalf complex plane appropriate saturated version matter fact proved uniformly lipschitz namely exists chosen bounded agree observation error originating exponentially converges origin exponential decay rate form exp positive constants possible initial condition long particular note exponential decay rate may arbitrarily assigned value polynomial peaking order worth noting uniform lipschitz condition automatically fulfilled compact set case identically zero long bounded observer guarantees bound estimation error depends bound value particular following asymptotic bounds proved lim sup max lim sup lim sup positive constant diag previous asymptotic bound holds possible long note high value leads arbitrarily small asymptotic gain component disturbance hand boundedness automatically guaranteed compact turn typical case observers used semiglobal output feedback stabilisation problems large value general detrimental sensitivity asymptotic estimate sensor noise first disturbance components observers form routinely used many observation control problems instance feature exponential decay rate asymptotic bound last component arbitrarily imposed value main reason observer plays fundamental role output feedback stabilisation setting semiglobal nonlinear separation principles case set arbitrarily large compact set made invariant design state feedback stabilisation law observer observe although asymptotic gain respect increase observer anyway able guarantee iss respect sensor noise main drawback observers form though related increasing power order parameter makes practical numerical implementation hard task large motivated considerations note propose new observer class systems preserves features substantially overtakes implementation problems due powered order specifically present highgain observer structure gain grows power instead price observer state dimension instead esult start presenting technical lemma instrumental proof main result presented proposition let matrices defined positive coefficients let matrix defined turns eigenvalues arbitrarily assigned appropriately choosing coefficients claimed next lemma lemma let arbitrary hurwitz polynomial exists choice characteristic polynomial coincides proof lemma deferred appendix constructive procedure designing given presented structure proposed observer following form triplet prime form dimension diag blkdiag times col appropriate saturated version variable represents asymptotic estimate state obtained extracting components state according matrix defined clarified next redundancy observer used extract extra state estimation blkdiag times following proposition shows observer recovers asymptotic properties two estimates standard observer statement proposition let col col proposition consider system observer coefficients fixed matrix defined hurwitz see lemma let bounded function agrees assume bounded exist following bound holds max exp proof consider change variables col system transforms element vector col rescale variables follows letting col easy calculation shows zero column vector dimension last position col diag uniformly lipschitz bounded exists rest proof follows standard lyapunov arguments sake completeness briefly recalled let consider lyapunov function taking derivative along solutions using previous bounds letting one obtains exist positive constants max kdn symmetric definite positive turns respectively smallest highest eigenvalue using bounds previous implication leads conclude long max kdn exp using bound terms following estimate easily obtained max exp kdn using fact previous bound leads following estimate max exp kdn claim proposition immediately follows bearing mind definition noting iii bout sensitivity observer high frequency noise linear case speed state estimation sensitivity measurement noise fact observer theory respect observers tuned obtain fast estimation dynamics necessarily sensitive noise bounds estimation error presence measurement noise standard observers studied instance different techniques developed order improve rejection mainly based gain adaptation see among others section compare properties standard observer proposed observer respect measurement noise specialising analysis linear systems particular consider systems form linear function form row vector dimension moreover contest consider sin constants shown ratio asymptotic estimation error state variable provided new observer one provided standard observer strictly decreasing polynomial function noise frequency regard new observer better asymptotic properties respect noise far state estimation variables concerned except first one formalised next proposition proposition let let fixed error dynamics observers hurwitz moreover denoting position index first non zero coefficient vector zero vector let constants defined min exist lim sup lim sup proof consider system standard observer letting read defined linear system hurwitz choices similarly consider system new observer defined proof proposition obtain system compactly rewritten hurwitz system choices consider systems given dynamics input outputs denote harmonic transfer functions systems simple computations show systems relative degree defined statement proposition definition harmonic transfer function fact systems hurwitz turns furthermore fact respectively relative degrees turns exist positive result immediately follows xample bserver uncertain van scillator let consider uncertain van der pol oscillator see measured output uncertain constant parameters let assume compact set containing origin state belongs compact invariant set limit cycle van der pol oscillator observe depends following system extended immersed system canonical observability form matter hence letting letting immediately seen system restricted immersed system triplet prime form dimension following prescriptions section implemented proposed observer lim sup lim sup denote harmonic transfer functions systems simple computation shows systems relative degree similarly consider systems given dynamics input outputs fact let col vector time derivatives let compact set simple computations show locally lipschitz bounded function agrees coefficients roots notation simulations fixed gain initial conditions figures show error state estimate proposed observer first two components namely estimation state sensor noise following section iii compared observer standard observer presence sensor noise numerically taken sin observer implemented eigenvalues table shows normalized asymptotic error magnitudes proposed observer standard observer normalized asymptotic error estimate defined lim sup although result section iii given linear systems numerical results shown table show remarkable improvement sensitivity measurement noise new observer respect standard observer standard high gain observer modified observer modified observer table normalized asymptotic errors presence noise numerically simulated previous section complete characterisation sensitivity sensor noise new observer interesting research topic investigation peaking phenomenon due wrong initial conditions fast convergence typical observers prevented proposed structure however techniques deal peaking saturations timevarying gains gradients techniques others available adopted improve proposed observer structure work consider multioutput case specific class systems diffeomorphic block triangular form block associated output triangular dependence states subsystem see proposed structure simply applied obtain observer apart case complete extension case immediate investigation acknowledgment wish thank laurent praly suggesting design procedure presented appendix ppendix procedure assign eigenvalues consider matrices recursively defined fig error state estimate col defined definition note letting note depend depends let pmi characteristic polynomials use notation col mij col characteristic polynomial pmi computed pmi fig error state estimate red line blue line onclusions presented new observer design based techniques tunable convergence speed respect standard observers state dimension larger instead clear benefit observer implementation due power highgain moreover specialised linear systems showed benefit proposed observer respect standard observer terms noise rejection benefits clearly confirmed also nonlinear pol example hence simple although lengthy computations show coefficients pmi related follow zero vector first position zero matrix identity matrix lower left block note invertible hence first one obtains embedded second third yield relations zero vector last position observe polynomial odd order consequence always exists least one real fulfilling previous results used set basic assignment algorithm used iteratively solve eigenvalues assignment matrix basic assignment algorithm let arbitrary polynomial exist real polynomial pmi matter fact letting coefficients possible take real solution take coefficients polynomial previous algorithm hand design assign arbitrary characteristic polynomial immediately done following steps desired characteristic polynomial compute running basic assignment algorithm compute iteratively running basic assignment algorithm compute eferences ahrens khalil observer presence measurement noise approach automatica vol astolfi praly output feedback stabilization siso nonlinear systems observer original coordinates ieee conference decision control december astolfi praly marconi note observability canonical forms nonlinear systems ifac symposium nonlinear control systems september atassi khalil separation principle stabilization class nonlinear systems ieee transactions automatic control ball khalil observer tracking performance presence measurement noise american control conference nonlinear observers applications springer bornard hammouri high gain observer class uniformly observable systems ieee conference decision control december deza busvelle gauthier rakotopara high gain estimation nonlinear systems systems control letters vol april forte isidori marconi robust design internal models nonlinear regression ifac symposium nonlinear control systems september lyapunov matrix equation system stability control academic press gauthier bornard observability class nonlinear systems ieee transactions automatic control vol august gauthier kupka deterministic observation theory applications cambridge university press hammouri bornard busawon high gain observer structured nonlinear systems ieee transaction automatic control vol april khalil praly observers nonlinear feedback control robust nonlinear control maggiore passino separation principle class non uniformly completely observable systems ieee transactions automatic control vol july sanfelice praly performance observers gain adaptation measurement noise automatica vol teel praly global stabilizability observability imply stabilizability output feedback systems control letters vasiljevic khalil differentiation observers presence measurement noise ieee conference decision control december
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nov probabilistic characterization random malicious communication failures networked control cetinkaya ishii hayakawa abstract control problem linear dynamical system network explored network assumed subject packet drops malicious nonmalicious nodes well random malicious data corruption issues utilize asymptotic bounds transmission failure ratios characterize links paths network well network probabilistic characterization allows take account multiple failures depend coordinated malicious attacks network obtain sufficient condition stability networked control system utilizing probabilistic approach demonstrate efficacy results different scenarios concerning transmission failures network introduction network control systems incorporate communication networks facilitate exchange measurement control data plant controller recently networks utilized networked control operations network wireless hoc network consists number nodes connected number communication links ensuring orderly operation networked control system challenging packets may sometimes fail transmitted different parts network due various reasons one reasons transmission failures network channel noise communication links may corrupt contents state control input data packets occurrence data corruption communication network may modeled using random processes addition channel noise network congestion may also cause packet transmission failures routers may drop packets mitigate congestion furthermore become apparent malicious attacks may also hamper transmissions networked control system instance jamming attacks cetinkaya ishii department computer science tokyo institute technology yokohama japan hayakawa department systems control engineering tokyo institute technology tokyo japan addresses ahmet ishii hayakawa key words phrases networked control networks malicious attacks random failures work supported part jst crest grant jsps scientific research grant characterization failures networked control adversary may interfere communication links effectively prevent transmission packets issue investigated several works viewpoints wireless communications well control game theory transmissions state control input information plant controller may also fail due malicious activities routers malicious routers may intentionally drop packets coming headed certain nodes network detection routing attacks challenging especially malicious nodes act normal certain periods time see grayhole attacks furthermore networks may face malicious routing random packet losses due link errors understanding effects malicious attacks networks important viewpoint cyber security networked control systems goal paper explore effects random malicious transmission failures general communication network develop network characterization used analysis networked control systems key problem characterize failures overall network nonconservative way still taking account packet failures coordinated malicious attacks network literature researchers proposed different characterizations packet failures networked control system specifically explored control network multiple links introduce random packet drops results obtained utilize packet drop probabilities edges constitute network graph approach also utilized network characterization furthermore discussed almost sure networked control system stability studied networked state estimation problem recently investigated mean square stability robustness delays packet losses markov jump linear system framework studies markov processes certain transition probabilities used modeling packet losses network paper consider network random failures also malicious activities nodes links result failures may always modeled markov processes packet losses may always probabilities researchers explored network model malicious nodes fault detection isolation problem explored case nodes induce delay transmissions consider different setup different modeling approach particular investigate detection problem model effect delays paper would like characterize overall packet exchange failures network plant controller using properties paths network communication links paths also note network characterization provides high level model tailored utilized stability analysis networked control instead specifying underlying physical channel models routing protocols take probabilistic approach characterize transmission failure ratios links paths network well network approach modeling overall packet failures network built upon bounds processes describe occurrences failures network specifically link network described asymptotic bound transmission failure characterization failures networked control ratio link approach different typical random packet loss modeling approach assigning probabilities failures approach capture failures occur due malicious nonmalicious reasons fact utilized approach study combined effects following version attack model random packet losses modeled markov chains differently show paper bounds links available obtain bounds describing overall failures individual paths network bounds used deriving bounds describing overall failures network using proposed characterizations obtain probabilistic average number packet exchange failures plant controller use almost sure stability analysis linear networked control system setting location failures whether multiple failures depend critically affect quality communication hence stabilization performance controller especially situation becomes serious network targeted number adversaries launch coordinated attacks different locations communication network bound approach handle situations unified manner addition investigation situations also explore case one known associated random failures corresponding indicator processes mutually independent cases show tighter results obtained done exploiting certain properties hidden markov models characterize random failures organization rest paper follows section explain networked control problem network present multihop network characterization sections specifically section give characterization overall transmission failures network based failures individual paths network furthermore section investigate failures paths occur due packetdropping issues nodes communication links demonstrate utility results section conclude paper section note conference version paper appeared paper provide detailed discussions throughout paper furthermore present new results sections new examples section paper use denote nonnegative positive integers respectively notation denotes probability probability space filtration binary numbers notation represents oroperation moreover represents control networks section investigate control linear plant network network plant controller exchange state measurement control input packets transmissions network subject delay however may failures packet exchange attempts plant controller characterization failures networked control describe linear dynamics plant respectively denote state input matrices furthermore state control input vectors respectively plant controller attempt exchange state measurement control input packets time instant denote success failure states packet exchange attempts using process case successful packet exchange plant transmits state measurement controller controller uses received state information together linear static feedback control law compute control input sent back plant transmitted control input applied plant side certain time instants either state packet control input packet delivered due network issues packet drops jamming attacks communication errors cases packet exchange attempts fail control input plant side set one common approaches literature see references therein characterization control input applied plant side given represents feedback gain although consider static control setup techniques develop paper also used conjunction control approaches particular predictive control approach control approach earlier work studied within context networked control using network characterizations develop introduce class processes useful describing packet failure indicators paper definition given scalar define class processes definition furthermore binaryvalued process satisfies explored problem similar one discuss paper considered single direct communication channel plant controller describe packet losses channel proposed probabilistic characterization based following assumption assumption packet exchange failure indicator assumption allows characterize range scenarios unified manner scalar instance case packet exchange attempts fail described setting moreover characterization failures networked control case packet exchange attempts successful addition two extreme cases illustrate throughout paper assumption also used describe random failures malicious attacks packet exchange failures networked control system satisfy assumption scalar also represents bound asymptotic packet exchange failure ratio specifically follows lemma almost surely inequality holds lim small means packet exchanges fail statistically rarely showed plant stabilized network sufficiently small stability analysis method developed allows obtain following result presents sufficient stability conditions networked control system theorem consider dynamical system suppose assumption holds scalar exist matrix scalars zero solution system asymptotically stable almost surely theorem conditions characterize stability instability dynamics scalars scalars also appear condition hence packet exchange failure ratio sufficiently small satisfied almost sure asymptotic stability implies sections present key methods obtaining networked control system facilitate stability analysis theorem specifically consider setting state measurement control input packets attempted transmitted networks instead single direct channel considered use assumptions similar assumption characterize packet transmission failures paths plant controller well individual links paths show packet exchange failures represented overall networked control system satisfy assumption scalar depends network structure well failure models links facilitate analysis following sections define two classes processes distinct goal characterize specific models random malicious failures earlier work utilized markov chains characterizing random failures single communication channel consider network composed number channels face random transmission failures required introduce different characterization overall failures network depends failures individual channel hence overall failures always described markov characterization failures networked control chain order describe random failures networks utilize timeinhomogeneous hidden markov models see process hidden markov model binary valued function set finite number elements moreover timeinhomogeneous markov chain initial distributions transition probability functions plq satisfying plq process also called output process hidden markov model notice depends markov chain function necessarily markov chain specifically certain cases may shows markov property see hold example case represents failures channel see also note hidden markov models naturally arise description networks instance may failure indicator path multiple links even failures individual links may satisfy markov property overall failure indicator satisfy due dependence states individual links cases follows hidden markov model markov chain represents combined states individual links given output process associated hidden markov model let given next definition introduce class output processes associated hidden markov models definition given scalars define class output processes hidden markov models plq following result establishes relation classes proposition consider output process hidden markov model give proof proposition first present technical result provides upper bounds tail probabilities sums involving binaryvalued output process associated hidden markov model lemma let markov chain transition probabilities let given function characterization failures networked control furthermore let process independent assume moreover lemma essential tool dealing different failure scenarios specific networks generalizes lemma previous work particular case lemma recovers lemma proof lemma given appendix ready prove proposition proof proposition notice since case show employing lemma specifically let define processes since conditions satisfied follows lemma completes proof next introduce class processes employ characterizing timing malicious attacks definition given scalars define class processes characterization class based version malicious attack model used model occurrences malicious attacks described process represents number initial attacks represents bound average attack rate lemma shows note malicious attack characterization class require process follow particular distribution time key difference class class represents random failures characterization failures networked control several ways attacker strategize cause transmission failures instance methods used attacker decide timing attacks important property class characterizes attacks maximum average attack rate specific timing strategy follow thus using capture uncertainty generation attacks may follow deterministic strategy may involve randomness interestingly attacker also make use system dynamics well state information decide timing attacks cause damage system attacks illustrated section explored characterization discussed process belongs either classes also belongs class suitable value observation suggests random failures malicious attacks characterized utilizing class make clear following sections class useful properties instance two processes belong classes processes defined setting belong classes depend properties enable model failures links paths network using processes belong class certain values random malicious packet failures networks section present framework modeling random malicious packet transmission failures networks used exchanging state control input packets plant controller network model follow approach represent networks plant controller using directed acyclic graphs model network state packets transmitted plant controller consider directed acyclic graph denotes set nodes denotes set edges nodes edges correspond respectively communication devices links represent nodes plant controller respectively path node another node identified sequence nonrepeating edges write denote number edges path similarly network used transmission control input packets controller plant represented graph plant controller nodes practice physical network may used transmission state control input packets cases nodes would correspond physical devices assume exists least one directed path node node least one directed path node node example show example fig nodes corresponding plant controller directly connected state control input packets still transmitted help intermediate nodes characterization failures networked control figure networked control system intermediate nodes networks forward data packets receive incoming edges outgoing edges depending communication protocol forwarding method may differ instance broadcast method intermediate nodes forward data packets receive incoming edges nodes connected outgoing edges hand may also case intermediate nodes follow specific routing scheme packet coming certain incoming edge forwarded certain outgoing edge assume information packets propagate delay networks represented although may transmission failures due packet drops malicious nodes prevent communication nonmalicious nodes avoid congestion data corruption links random channel errors malicious jamming attacks hence packet exchange plant controller may fail state control input packets dropped get corrupted note corrupted data packets allowed transmitted intermediate nodes detected discarded nodes codes used purpose note also controller receives corrupted versions state packet control input computed following sections present key results analysis packet failures network directly applicable analyzing particular characterize failures terms failures different paths plant controller present set results relate data corruption packet dropping issues nodes links failures individual path results enable obtain assumption essential analyzing system fig theorem emphasize central problem find overall network fig nonconservative way still taking account packet failures coordinated attacks network packet transmission failures networks use process indicate transmission failures specifically means state packet sent plant node successfully characterization failures networked control received controller node hand indicates failure controller receive state let denote number paths graph node node let denote paths example network fig paths note different paths may include link hence packet transmission attempted multiple paths link shared paths may used multiple times instance hence may utilized twice forward packets coming hand framework describe also allows modeling case one packets dropped node transmitted furthermore packet drop random malicious use lpi indicate whether state packet successfully transmitted controller path specifically lpi represents successful transmission hand lpi may indicate path utilized transmission due particular routing scheme may indicate failure failures occur packets get dropped path get corrupted thus network transmission state packet node node results failure lpi paths therefore given lpc following result presents probabilistic asymptotic bound packet transmission failure ratio function bounds individual paths proposition assume path lpi proof first let arg follows hence lpi therefore result follows since scalars proposition represent bounds asymptotic packet failure ratios different paths network proposition indicates minimum scalars also bound packet failure ratio whole network observe path characterization failures networked control means state securely reliably transmitted controller time instants transmission path never fails hand paths indicating packet transmission attempts fail almost surely note proposition assume lpi processes allows deal scenarios transmission failures different paths may depend particular consider coordinated packet dropout attacks several malicious routers different paths instance two malicious routers fig may skip forwarding packets time transmissions paths given would fail similarly proposition also useful links different paths attacked time coordinated jamming attackers another scenario explored proposition related packet drops nonmalicious routers prevent congestion example nonmalicious router fig may choose forward one packets coming would dependent processes particular failures network remark utilizing additional properties indicator processes lpi paths obtain better asymptotic failure bound one provided proposition particular one paths known associated random failures corresponding indicator processes mutually independent obtain tighter results proposition end first present following result properties process obtained using operation output processes two hidden markov models theorem consider output processes hidden markov models suppose markov chains associated processes mutually independent let min process defined output process hidden markov model moreover proof let define bivariate process follows markov chain ini tial distribution transition probabilities plq plq respectively denote initial distribution characterization failures networked control transition probability function markov chain associated output process furthermore follows let given follows holds replaced thus output process hidden markov model next goal prove showing first show observe let follows hence obtain plq plq plq plq plq plq plq pql characterization failures networked control since plq plq plq furthermore plq since summation possible states using inequalities obtain since holds min furthermore noting obtain using inequality follows holds next show notice noting use obtain plq plq plq plq implies since hold theorem shows two hidden markov output processes combined operation resulting process also hidden markov output process furthermore theorem provides values result applied obtain instance consider case output processes hidden markov models follows theorem suppose notice since direct application proposition gives min however applying proposition obtain smaller value fact proposition characterization failures networked control obtain notice case theorem applied repeatedly instance use theorem first consider case graph possesses paths indicator processes mutually independent associated random failures even case obtain results tighter proposition end first provide following result derive properties process obtained using operation hidden markov output process binary valued process theorem consider processes satisfy process defined satisfies proof first show min let note follows defined satisfies furthermore defined satisfies follows lemma min min since follows holds implies min since consequently min min theorem concerned operation applied process hidden markov model class another process class shown operation results process satisfies note application proposition situation would allow show min notice proposition case conservative since min hand note proposition allows deal processes mutually independent packet transmission failures paths network far previous section looked packet failures paths network affect overall packet transmission rate section goal explore effect failures individual nodes links path characterization failures networked control end first consider scenario packet transmission failures occur due data corruption explore case data corruption packet drops may occur path characterization data corrupting paths let denote jth edge path use process lpii denote data corruption indicator link example fig consider second edge path state indicates time packet flowing path faces data corruption link may due jamming attack link due channel noise moreover may also case node maliciously corrupts packet notation data corruption indicator allows distinguish data corruption issues consider communication link different paths instance communication link may case node corrupts packets transmitted along none packets transmitted along situation described setting state packet transmitted path subject data corruption data corruption one edges path hence lpi lpi next result shows asymptotic transmission failure rate path given sum failure rate bounds links path proposition consider lpi given assume lpii satisfy lpi proof lpi lpi hence lpi lpii obtain last inequality used lpii result follows since lpii lpii proposition used characterize overall failures path note proposition indicators links necessarily mutuallyindependent processes allows model failures different links characterization failures networked control depend particular explore effect interference links well coordinated jamming attacks targeting several links time note certain cases result provided proposition improved terms conservativeness particular one links path known associated random failures corresponding indicator processes mutually independent obtain less conservative results comparison proposition following result counterpart theorem operation concerned output processes two hidden markov models theorem consider output processes hidden markov models suppose markov chains associated processes mutually independent let min process defined output process hidden markov model moreover proof let define bivariate process follows markov chain ini tial distribution transition probabilities plq plq respectively denote initial distribution transition probability function markov chain associated output process furthermore follows let given follows holds replaced thus output process hidden markov model show apply theorem end first note output process complementing process given output process hidden markov model furthermore since hlc characterization failures networked control consequence following show proving given satisfies first observe since finally noting obtained using operation processes use theorem obtain implies theorem shows two hidden markov output processes combined operation resulting process also hidden markov output process furthermore provides values theorem applied obtaining given path consider example path links assume failure indicator processes associated links mutually independent follows theorem lpi min furthermore min proposition lpi min hand direct application proposition provides conservative result apply proposition notice first hence proposition obtain value implies lpi inequality min allows conclude theorem provides less conservative range compared proposition certain scenarios path may composed communication links indicator processes associated random failures scenarios possible obtain results less conservative proposition specifically following result obtain properties process obtained using operation hidden markov output process binary valued process theorem consider processes satisfy process defined satisfies proof notice since process next show first implies characterization failures networked control follows let define let min furthermore let following show first note max max max holds since use proposition replaced replaced obtain next use lemma show obtain result first observe since moreover min hence let defined since furthermore since conditions lemma hold together processes defined setting hence lemma implies finally arrive shows theorem concerned operation applied process hidden markov model class another process class characterization failures networked control shown operation results process satisfies notice application proposition situation would allow show proposition case conservative since remark advantage proposition may allows deal processes mutually independent general characterization paths data corruption packet drops previous section provided characterization paths links follows extend characterization consider effects data corruption packet dropouts without loss generality assume links path either note original network link subject issues artificially add node edges graph consider link link link packets available always transmitted content may externally manipulated damaged transmission link hand link packets available may may transmitted never get corrupted link packet dropout may occur malicious router intentionally skips forwarding packets see blackhole grayhole attacks nonmalicious router may also drop packets avoid congestion addition two issues packet may also dropped header part packet includes information destination packet corrupted furthermore scenarios implemented intermediate nodes corruption data part packet detected result corrupted packet need transmitted scenario also studied within packet drop framework consider link path let denote indices packets node possesses receives previous nodes path observe links first packet index receives may different may state later time may dropped reaching node use lpi indicate status transmission packet index node possesses node goal obtain relation asymptotic packet failure ratio path failure ratios links path end use recursive characterization describing packet failures paths specifically consider path links consider associated process state indicates packet first node possesses successfully transmitted last node whereas indicates failure consider case let respectively denote first link characterization failures networked control subpath composed rest links illustrate left side fig next show transmission failures path characterized transmission failures link subpath let denote indicators transmission failures link subpath link dataf corrupting link would relation hold link index represent different packets introduce new process link define setting hand link definition denotes number packets successfully transmitted node node among first packets node possesses hence scalar represents index packet received moreover indicates whether packet successfully transmitted observe set drops packet possesses hand transmits packet state indicates transmission packet subpath failed whereas indicates success result hence holds characterization recursive sense recursively applied describe failures means failures first link subpath example fig see path first link subpath together sample trajectories note link drops first two packets consequence furthermore link successfully transmits packet value represents whether packet successfully transmitted subpath since packet first packet receives transmission state represented result characterization failures networked control figure left paths link right trajectories indicating successful transmission indicates packet possessed first node successfully transmitted last node following result essential characterizing failures path first link subpath obtain result first establishing key inequality applying arguments employed proving proposition lemma consider path given assume scalars follows proof first show link consider therefore situation link case since hence follows also characterization failures networked control finally since holds hence follows together completes proof given path repeated application lemma allows obtain following result theorem assume lpii satisfy lpi note theorem allows consider packetdropouts links hence generalizes proposition remark utilizing theorem together proposition obtain average number packet exchange failures networks note controller either receive state packet receives corrupted versions discarded hence control input packet attempted transmitted setting similar situation discussed packet dropping links consider whole networked system path node node corresponds indicator first packet dropping link corresponds packet transmission failure indicator rest path hence using arguments similar ones used show value utilized stability analysis theorem illustrative numerical examples section present illustrative examples demonstrate utility results characterization communication failures networks also investigate effect failures stability networked control system consider networked control system together networks fig system explored previously single channel network model differently consider characterization failures networked control networks incorporate multiple paths multiple links packet transmissions follows investigate various scenarios demonstrate utility results sections characterizing overall network failures different scenario goal find level transmission failures tolerated communication links stability system guaranteed data dropout issues networks consider scenario links networks subject data corruption packet dropout issues explore general situation failures may depend general setup use proposition theorem characterization overall transmission failures plant controller explained remark overall packet exchange failure process satisfies assumption asymptotic failure ratios networks find values use proposition theorem particular proposition theorem obtain similarly number paths denoted controller node plant node network consequently assumption holds min min effect asymptotic packet failure ratios stability networked system analyzed using theorem first identify values asymptotic packet exchange failure ratio assumption stability conditions hold numerical example exist matrix scalars satisfy less hence theorem guarantees zero solution system asymptotically stable almost surely asymptotic packet transmission failure ratios satisfy min min system operator guarantee ensuring least one path network sufficiently secure reliable particular exist path network path network holds stability guaranteed regardless paths another approach guarantee ensure certain level links example links sufficiently secure reliable holds stability guaranteed see first note network characterization failures networked control figure paths plant node controller node network figure paths controller node plant node network contains paths number links paths given furthermore network contains paths contains links follows min min min min min min implies holds hence stability guaranteed notice example made particular assumption independence randomness failures links fact links may subject failures caused actions coordinated adversaries case occurrence failures may nonrandom moreover processes characterize failures different links would depend example case attacks scenario would failures paths synchronized packet transmissions necessarily fail parallel paths paths graph guarantees time notice condition failures happen sufficiently rarely average long run thus networked stabilization achieved successful exchanges measurement control data following examples illustrate results sections used scenarios information properties communication links available characterization failures networked control jamming attacks random transmission failures multiple links consider network paths shown fig assume paths subject malicious attacks particular node assumed controlled malicious agent packets arriving node dropped hence implies moreover first second links path assumed subject jamming attacks cause data corruption scenario happens attackers coordinate jam one links would maximize number packet losses within total energy constraint attackers results take account particular suppose attacked links satisfy provides bound average failures link related energy available attacker fact implies follows proposition hence notice holds even scenario mentioned provides average number overall failures path scenario assume link path subject random data corruption associated failure indicator process satp isfies consider ideal communication links path thus using obtain hence also characterize overall failures network use obtain assuming failures paths follows theorem noting see next consider network suppose secure attacks unreliable subject random transmission failures characterize overall transmission failures network utilize theorems apply results first describe routing scheme network specifically network assumed router forwards incoming packets node node incoming packets node node among paths shown fig packets may transmitted never transmitted due routing scheme hence assume links path links face random data corruption issues furthermore failure indicator processes assumed mutually independent cesses belong hidden markov model class see characterization failures networked control definition links connected directly plant controller nodes considered ideal communication links words observe failure indicators paths satisfy applying theorem obtain min min next since follows theorem min min min min min min min min min min min min finally case consequence proposition obtain note overall packet exchange failure indicator satisfies since follows discussed section stability networked control system ensured follows holds implying almost sure asymptotic stability networked control system observe utilizing results sections able derive sufficient stability condition terms attack rate associated links path attacked jamming attackers well random failure parameters attacks malicious node scenarios discussed sections strategies attackers specified however certain cases attacker may access state control input information able directly cause transmission failures plant controller scenarios goal attacker might increase state norm small amount attacks section goal illustrate attack strategy particular consider case plant node compromised attacker attacker assumed access state information characterization failures networked control consider attack strategy based optimization problem particular attacker node decides whether transmit state information links solving optimization problem maximizing norm state future time represent attackers actions process indicates attacker transmits state packet indicates transmission attacker decides values process according solution optimization problem maximize subject strategy attacker decides attack time based optimization problem goal maximize sum squared state norms interval keeping attack rate certain value horizon optimization problem large attacker sufficient computational resources notice attack strategy hence observe scenario since consequence notice implies since attacker completely prevent packet exchange plant controller observe also sources transmission failures network may times even result attacker may able correctly predict state time may may failure network prevents control input reach plant however optimization problem solved time step updated state information used decision notice case links secure reliable since hence suppose network also faces failures particular consider setup section since noting stability networked control system ensured impose sufficient condition attack rate specifically scenario psfrag replacements characterization failures networked control time figure comparison state trajectories different values section holds thus networked control system almost surely asymptotically stable illustrate effect different parameters attack strategy conduct simulations first generate sample trajectories process setting failure processes outputs hidden markov models markov chain initial distributions transition probability functions satisfying notice next sample trajectory simulate networked control system attack strategy fig show state trajectories attack parameters selected top part fig horizon parameter bottom part observe larger horizon variable attacker following strategy utilize resources efficiently state norm set larger values longer durations even though cases attack belongs class notice inequality holds therefore zero solution system almost surely asymptotically stable theorem psfrag replacements characterization failures networked control number packet transmission attempts figure average number failures networks together average number overall packet exchange failures show fig average number failures networks well average number total packet exchange failures plant controller averages upper bounded long run certain scalars particular process satisfies lim see lemma result lim lim lim next consider case corresponds case attacker initial resources case consider two scenarios obtain sample state trajectories shown fig first notice case top plot fig state grows larger values longer durations comparison case top plot fig cases stability conditions hold therefore state eventually converges zero guaranteed theorem note set larger values attacker attack higher rate example set observe high attack rate sample state trajectories diverge see bottom plot fig conclusion paper explored state feedback control linear plant unreliable network may also face malicious attacks developed probabilistic psfrag replacements characterization failures networked control time figure comparison state trajectories different values approach characterize failures network terms failures different paths plant controller showed failures path described combination failures communication links particular path obtained sufficient conditions almost sure stability overall networked control system allow check stability using probabilistic obtained average number packet exchange failures plant controller appendix proof lemma section prove lemma proof lemma proving lemma use markov inequality follow approaches used obtaining tail distribution inequalities sums random variables see appendix section proof lemma following result also plays key role lemma let markov chain transition probabilities furthermore let given function denote indices proof use induction show first consider case case characterization failures networked control next consider case observe thus random variable consequently last equality follows fact measurable function hence since markov chain therefore obtain thus using arrive hence satisfied assume holds prove holds employing arguments similar used obtaining get finally obtain next utilizing lemma prove lemma proof lemma first define next let moreover characterization failures networked control utilizing definitions obtain since therefore follows follows goal find upper bounds two summation terms start second term since obtain next obtain upper bound first term observe follows moreover obtain obtain upper bound term utilize markov inequality lemma first let characterization failures networked control denote indices nonzero entries consequently htj observe since using markov inequality obtain follows lemma inequality together imply notice obtain last inequality employed fact summation term far side satisfies therefore obtain side inequality zero therefore holds also using fact together arrive characterization failures networked control next show first prove summation terms side convergent first therefore geometric series converges next show observe moreover result since obtain implies thus follows finally since obtain references hespanha naghshtabrizi survey recent results networked control systems proc ieee vol gupta dana hespanha murray hassibi data transmission networks estimation control ieee trans autom control vol alur innocenzo johansson pappas weiss compositional modeling analysis control networks ieee trans autom control vol smarra innocenzo benedetto optimal control scheduling routing control networks proc ieee conf dec innocenzo benedetto serra fault tolerant control control networks ieee trans autom control vol innocenzo smarra benedetto resilient stabilization control networks subject malicious attacks automatica vol characterization failures networked control smarra benedetto innocenzo method routing redundancy design lossy networks proc ifac congress khayam radha modeling wireless local area networks proc acm mswim floyd jacobson random early detection gateways congestion avoidance trans networking vol trappe zhang wood feasibility launching detecting jamming attacks wireless networks proc acm int symp mobile hoc network pelechrinis iliofotou krishnamurty denial service attacks wireless networks case jammers ieee commun surveys vol amin sastry safe secure networked control systems attacks proc hscc persis tesi stabilizing control ieee trans autom control vol liu liu saddik stochastic game approach security issue networked control systems jamming attacks franklin vol shi cheng chen quevedo jamming attacks remote state estimation systems approach ieee trans autom control vol persis tesi networked control nonlinear systems syst control vol awerbuch curtmola holmer rubens odsbr ondemand secure byzantine resilient routing protocol wireless hoc networks acm trans inf system security vol mahmoud shen security wireless networks springer jhaveri patel jinwala dos attacks mobile hoc networks survey proc actt mizrak savage marzullo detecting malicious packet losses ieee trans parallel distrib vol shu krunz truthful detection packet dropping attacks wireless hoc networks ieee trans mobile computing vol amin sastry research challenges security control systems proc conf hot topics security sandberg amin johansson special issue cyberphysical security networked control systems ieee control syst vol dana gowaikar palanki hassibi effros capacity wireless erasure networks ieee trans inf theory vol quevedo johansson state estimation sensor networks correlated wireless fading channels ieee trans autom control vol cetinkaya ishii hayakawa networked control random malicious packet losses ieee trans autom control vol cetinkaya ishii hayakawa random malicious packet transmission failures channels networked control systems proc ifac necsys quevedo nesic stability packetized predictive control unreliable networks affected ieee trans autom control vol cetinkaya ishii hayakawa output feedback control resilient jamming attacks random packet losses proc ifac necsys anderson wiener hidden markov models ieee control syst vol characterization failures networked control moulines inference hidden markov models springer vidyasagar hidden markov processes theory applications biology princeton university press billingsley probability measure wiley sadeghi kennedy rapajic shams markov modeling fading channels ieee signal process vol ellis pezaros kypraios perkins markov model packet loss video applications targeting residential users comput vol medhi ramasamy network routing algorithms protocols architectures morgan kaufmann madhow fundamentals digital communication cambridge university press
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two provably consistent divide conquer clustering algorithms large networks aug soumendu sundar purnamrita peter department statistics university california berkeley department statistics data sciences university texas austin august abstract article advance strategies solving community detection problem networks propose two algorithms perform clustering number small subgraphs finally patches results single clustering main advantage algorithms bring significantly computational cost traditional algorithms including spectral clustering programs modularity based methods likelihood based methods without losing accuracy even improving accuracy times algorithms also nature parallelizable thus exploiting facts traditional algorithms accurate corresponding optimization problems much simpler small problems methods provide omnibus recipe scaling traditional algorithms large networks prove consistency algorithms various subgraph selection procedures perform extensive simulations analysis understand advantages approach various settings contents glossary notation introduction two divide conquer algorithms pace averaging algorithm gale sequential algorithm remarks sampling schemes beyond community detection main results results pace results gale simulations analysis simulations comparison traditional algorithms real data analysis discussion proofs results pace results gale acknowledgments details dgcluster auxiliary results analysis pace stochastic block model community sizes random community sizes ego neighborhoods analysis adjacency spectral clustering analysis sdp proofs results gale glossary notation convenience reader collect frequently used notations used paper provide summarizing phrase well page number notation first appears nij yab number subgraphs used randomly chosen subgraph weights used number times appear subgraphs threshold nij tuning parameter pace estimated clustering matrix returned pace randomly projected version used recovering estimate confusion matrix two clusterings used match cluster membership matrix subgraph alignment cluster membership matrix subgraph alignment number nodes cluster proportion nodes cluster mink similarly normalized mismatch two clusterings normalized mismatch two clusterings normalized two clustering matrices otherwise vector number nodes cluster subgraph introduction community detection also known community extraction network clustering central problem network inference wide variety real world problems ranging finding protein protein complexes gene networks guruharsha studying consequence social cliques adolescent behavioral patterns hollingshead depend detecting analyzing communities networks problems one observes interactions pairs entities pairwise edges possibly additional node attributes goal infer hidden community memberships many real world networks massive hence crucial develop analyze scalable algorithms community detection first talk methodology uses network connections inference divided roughly two types first type consists methods derived independently model assumption typically involve formulation global optimization problem examples include normalized cuts shi malik spectral clustering etc end statisticians often devise techniques model assumptions simplest statistical model networks communities stochastic blockmodel sbm holland key idea sbm enforce stochastic equivalence two nodes latent community identical probabilities connection nodes network many extensions sbm degree corrected stochastic blockmodel karrer newman allow one model varied degrees community whereas standard sbm mixed membership blockmodels airoldi allow node belong multiple communities whereas sbm node belong exactly one cluster sbm generating network nodes communities one hidden membership matrix zik node community given memberships link formation probabilities given aij zik symmetric parameter matrix probabilities elements may decay zero grows infinity model sparse networks typically goal estimate latent memberships consistently method outputting estimate called strongly consistent permutation matrix weaker notion consistency fraction misclustered nodes goes zero goes infinity typically consistency results derived average degree network grows faster logarithm often called regime average degree bounded sparse regime plethora algorithms community detection include methods amini modularity based methods snijders nowicki newman girvan bickel chen spectral methods rohe programming sdp based approaches cai etc among spectral methods scalable since main bottleneck computing top eigenvectors large often sparse matrix theoretical guarantees spectral clustering typically proven regime mcsherry rohe lei rinaldo regularized version shown perform better random predictor sparse networks profile likelihood methods bickel chen involve greedy search possible membership matrices makes computationally expensive semidefinite programs robust outliers cai shown strongly consistent dense regime amini levina yield small error sparse regime vershynin however semi definite programs slow typically scale thousands nodes millions nodes methods like spectral clustering geodesic distances bhattacharyya bickel provably consistent case give small error sparse cases however requires computing pairs shortest paths nodes pose serious problems computation storage large graphs monte carlo methods snijders nowicki nowicki snijders popular tools bayesian frameworks typically scalable scalable alternatives variational methods gopalan blei provable guarantees consistency often suffer bad local minima far discussed community detection methods look network connections node attributes often also available may possess useful information community structure see newman clauset extensions methods mentioned earlier accommodate node attributes modularity based zhang spectral binkiewicz sdp based yan sarkar etc methods come theoretical guarantees good performance moderately sized networks existing bayesian methods mcauley leskovec newman clauset amenable incorporating covariates inference procedure often computationally expensive lack rigorous theoretical guarantees mentioned array algorithms diverse unique aspects order scale large datasets one apply different computational tools tailored different algorithmic settings stochastic variational updates may suitable scale bayesian methods pseudo likelihood methods better optimized using row sums edges inside different node blocks article propose divide conquer approach community detection idea apply community detection method small subgraphs large graph somehow stitch results together could achieve would able scale community detection method may involve covariates well network structure computationally feasible small graphs difficult execute large networks would especially useful computationally expensive community detection methods sdps modularity based methods bayesian methods another possible advantage concerns variational likelihood methods large number depending local parameters typically optimization landscape riddled local minima smaller graphs less parameters fit optimization problem often becomes easier clearly principal difficulty matching possibly conflicting label assignments different subgraphs see figure example immediately rules averaging estimates cluster membership matrices various subgraphs viable stitching method regard propose two different stitching algorithms first called piecewise averaged community estimation pace focus estimating clustering matrix labeling invariant since element matrix one simply means nodes belong cluster whereas value zero means belongs two different clusters thus first compute estimates various subgraphs average matrices obtain estimate finally apply computationally cheap clustering algorithm like approximate spectral clustering etc recover estimate also propose another algorithm called global alignment local estimates gale first take sequence subgraphs two consecutive subgraphs sequence large intersection traverse sequence aligning clustering based subgraph greedy algorithm propose article see appendix figure conflicting label assignments two subgraphs one labels denoted onion neighborhood yellow vertices solid edges constitute ego network root vertex colored red green yellow vertices together solid edges constitute onion neighborhood root averaged clustering union predecessor subgraphs sequence already aligned alignment done via algorithm called match identifies right permutation align two clusterings two subgraphs computing confusion matrix two clusterings restricted intersection two subgraphs whereas naive approach would entail searching permutations match finds right permutation log time alignment step complete get averaged clustering union subgraphs covers vertices design gale works estimates cluster membership matrices directly output estimate thus unlike pace avoids extra overhead recovering estimate rest paper organized follows section describe algorithms section state main results applications section contains simulations real data experiments section provide proofs main results relegating details appendix section finally section conclude discussions directions future work two divide conquer algorithms discussed introduction main issue divide conquer algorithms clustering one somehow match various potentially conflicting label assignments vein propose two algorithms accomplish task algorithms first compute clustering small patches network patches induced subgraph random subsample nodes neighborhoods however stitching procedures different pace averaging algorithm suppose adjacency matrix network true cluster membership nodes given matrix clusters set clustering matrix whose entry indicator whether nodes belong cluster given perform local clustering algorithm obtain estimate estimate cluster memberships may reconstructed algorithm pace piecewise averaged community estimation subgraph selection fix positive integer threshold minimum required subgraph size fix another positive integer number subgraphs sample given choose subsets nodes procedure select nodes random pick neighborhoods vertices vertices denote adjacency matrix network induced clustering subgraphs perform standard clustering algorithms like profile likelihood mean field likelihood mfl spectral clustering programming sdp etc subgraphs size least obtain estimated clustering matrices rest subgraphs set extend matrix setting least one selected let denote resulting matrix patching let yij denote indicator event selected set nij yij define combined estimator nij yij nij nij nij integer tuning parameter call piecewise averaged community estimator also abbreviated pace parameter pace seems reduce variance estimation quality discards information less credible sources pair nodes appeared subgraphs trust patching say setting nij seems work well practice choice also justified theory slight variant algorithm allow subgraph specific weights computation final estimate nij yij nij nij nij nij equals yij may call estimator standing weights equal becomes equivalent ordinary pace natural nontrivial choices including place weight estimates based large subgraphs degs degs degs denotes degree node subgraph put weight pairs high degree first prescription intimately related following sampling scheme ordinary pace pick subgraphs probability proportional sizes instance section analyze political blog data adamic glance neighborhood subgraphs chosen selecting roots probability proportional degree real world applications might make sense choose weights based domain knowledge instance may certain subnetworks known important another minor advantage weights nij becomes estimator based full graph example true typically much smaller however ordinary pace lacks property unless fact estimate returned pace identically anyway follows stick ordinary pace simplicity discuss reconstruct estimate let note may obtain binary matrix thresholding level example thresholding change consistency properties see lemma looking plot matrix gives good visual representation community structure follows work unthresholded reconstruction actually reconstruct key note members community identical rows thanks pace gotten hold consistent estimate thus may use clustering algorithm rows recover community memberships another option would run spectral clustering matrix however rows clustering algorithms typically running time best indeed main computational bottleneck distance based clustering algorithm high dimensional situation like present one computing dij takes bit operations however since gotten good estimate project rows onto lower dimensions without distorting distances much famous lemma random projections says projecting onto log dimensions one keep probability least distances projected vectors within factor true distances choosing inverse polylog need project onto polylog dimensions would readily bring computational cost distance based algorithm polylog following discussion paragraph first random projection rows onto polylog dimensions apply clustering algorithm algorithm recovering random projection followed distance based clustering select dimension random projection let cluster clustering algorithm operates rows first argument outputs clusters standard gaussian matrix dimensions cluster word asymptotic notation addition standard notations probabilistic counterparts use following less standard alternatives clean presentation places mean mean also sometimes use tilde standard notations hide polylogarithmic factors cluster may use approximate distance based clustering algorithm dgcluster greedy algorithm presented appendix algorithm gale sequential algorithm first introduce simple algorithm computing best permutation align labels one clustering another set nodes fixed ordering set idea first compute confusion matrix two clusterings note two labelings low error respect unknown true labeling confusion matrix close diagonal matrix permutations following algorithm essentially finds best permutation align one clustering another algorithm match algorithm aligning two clusterings set nodes input compute confusion matrix set left find mij tie broken arbitrarily set replace row columns return permutation matrix remark one also compute optimal permutation searching permutations labels picking one gives smallest mismatch two match brings dependence exponential quadratic note one clusterings poor match may retrieve optimal permutation however goal cluster many subgraphs using algorithm good accuracy high probability may computationally intensive profile likelihood semidefinite programming use intersections subgraphs align one another shall show later long enough members community simple algorithm sketched suffices find optimal permutation present sequential algorithm aligns labelings across different subgraphs idea first fix indexing nodes cluster subgraphs possibly parallel implementation using algorithm align clusterings along sequence subgraphs make things precise make following definition definition let vertex set edge set denote subgraphs subgraph node two nodes connectedlif corresponding subgraphs substantial overlap random threshold define traversal spanning tree sequence covering vertices along sequence adjacent traversal walk length passing vertex least construct traversal travel traversal step align current subgraph labels using match algorithm algorithm intersection union previously aligned subgraphs end subgraph labellings aligned labeling starting subgraph simply take average majority vote algorithm gale global alignment local estimates input adjacency matrix parameters base algorithm mfl sdp etc subgraph selection given choose subsets nodes procedure select random select random nodes pick neighborhoods denote adjacency matrix network induced clustering subgraphs perform algorithm subgraphs obtain estimated cluster membership matrices extend matrix setting traversal subgraphs construct traversal sxj subgraphs initial estimate also set sequential label aligning subgraph sxi traversal visited yet compute overlap current subgraph subgraphs previously visited let sxi compute best permutation match clustering set compute match permute labels nodes sxi get aligned cluster membership matrix update threshold mark sxi visited implementation details constructing traversal subgraphs done using depth first search subgraphs implementation start large enough subgraph parent pick another subgraph large overlap child align note subgraph visited recursively find another unvisited child current subgraph possible particular path cover vertices hence ideal estimate clusterings multiple traversals different starting subgraphs align clusterings take average real networks also note step find poorly clustered subgraph give bad permutation may deteriorate quality aligning subsequent subgraphs path order avoid use self validation routine let intersection current subgraph union previously visited subgraphs aligning current subgraph clustering compute classification accuracy current labeling previous labeling accuracy large enough use subgraph move next subgraph path implementation use threshold computational time storage main computational bottleneck gale building traversal random subgraphs let time computing clusterings subgraphs parallel naive implementation would require computing intersections pairs subsets show theoretical analysis take mink size cluster taking computation intersections takes time naive comparison computing subsets similar close given one would require log time subset leading log computation however building traversal one needs access subsets large overlap given subset classic example nearest neighbor search computer science one method widely used theoretically analyzed technique locality sensitive hashing lsh hash function maps data object arbitrary size another object fixed size case map characteristic vector subset number idea lsh compute hash functions two subsets two functions high probability close fact amount overlap normalized simply cosine similarity characteristic vectors two subsets efficient hashing schemes using random projections exist charikar arccos lsh schemes one needs build hash tables governs approximation quality hash table bucket corresponding index stores subsets hashed index query point one evaluates hash functions examines subsets hashed buckets respective hash tables subsets distance computed exactly preprocessing time storage total query time brings running time added algorithm specific time nearly linear time thus nearly linear time clustering algorithms gale may lead computational savings however algorithms like profile likelihood sdp well known computationally intensive gale lead significant computational saving without requiring lot storage remarks sampling schemes pace mainly used random neighborhoods onion neighborhoods many subgraph sampling schemes possible instance choosing roots hop neighborhoods probability proportional degree sampling roots high degree nodes done analysis political blog data section discussed earlier weighted sampling scheme related natural question regarding neighborhoods many hops use theory yet small world phenomenon expect need many hops typically moderately sparse networks hops enough although adaptive procedure type choosing would welcome also since neighborhood size increases exponentially hop size alternative choosing full choose smaller hopneighborhood add randomly chosen neighbors already chosen vertices possibilities include sampling certain proportion edges random consider subgraph induced participating nodes leave possibilities future work analyzed gale random sampling scheme scheme one understand behavior intersection two samples neighborhoods example one takes hhop neighborhoods sparse graphs neighborhood predominantly nodes mainly one cluster hence gale often suffers scheme show empirically section gale accuracy much improved random sampling scheme beyond community detection ideas behind pace gale restricted community detection modified application interesting problems including general clustering problems problems rohe mixed membership models among others discussed upcoming article fact mackey took similar divide conquer approach matrix completion main results section state discuss main results pace gale along applications results pace let two clusterings objects clusters usually discrepancy measured inf inf permutation group corresponding binary matrices related measure discrepancy two clusterings inf perm kzq easy see elaborate let permutation matrix corresponding permutation qij purposes however useful measure discrepancy would normalized frobenius squared distance corresponding clustering matrices compare two notions discrepancies proposition incidentally cluster sizes equal one show although lower bound terms lemma tang gives exists orthogonal matrix kkck kkc knkc kzo nmin used fact kck nmax caveat matrix need permutation matrix prove consistency pace assume clustering algorithm use consistency properties example suffice assume randomly chosen subgraph subgraph selection procedure small following main result expected misclustering rate pace theorem expected misclustering rate pace let randomly chosen subgraph according sampling scheme let maxk max nij expectation taken randomness graph randomness sampling mechanism first term essentially measures performance clustering algorithm use randomly chosen subgraph second term measures well covered full graph chosen subgraphs depend subgraph selection procedure effect algorithm use felt first term specialize theorem various subgraph selection schemes first consider randomly chosen easy corollary corollary subgraphs induced randomly chosen nodes let notice constant made close one desires means bound essentially optimal full neighborhood subgraphs much harder analyze pursued however ego networks neighborhoods minus root node see figure easy deal one also extend analysis onion neighborhoods recursively defined follows ego network vertex general shell operation denotes superposition networks ease exposition choose work ego networks onion neighborhoods corollary ego neighborhoods stochastic block model let max bab min bab let exp exp exp exp use existing consistency results several clustering algorithms conjunction bounds see conditions conditions model parameters etc required pace consistent first consider adjacency spectral clustering asp lei rinaldo use stochastic block model generative model simplicity assume link probability matrix following simple form quote slightly modified version corollary lei rinaldo model lemma lei rinaldo let consider adjacency matrix generated simple block model log output adjacency spectral clustering algorithm applied exists absolute constant probability least min actually allow resulting bound involves complicated constants depending add anything extra nature bounds chosen work particular ease exposition general constant multiplier first term instead corollary adjacency spectral clustering random assume setting lemma let min quantity log proof corollary follows corollary estimate given max obtained using lemma order first term zero need min thus balanced block sizes need qualitatively large small small separation blocks large natural expect particular fixed shows need subgraphs size many achieve consistency average degree let dnn let see computational gain get spectral clustering full graph complexity complexity pace spectral clustering mrn complexity would essentially log gain small note however parallel implementation source processing subgraphs get significant boost running time least terms constants running time would mndn corollary adjacency spectral clustering ego subgraphs assume setting lemma let min exp exp exp exp quantity log proof corollary follows corollary estimate given obtained using lemma right hand side zero assuming fixed balanced block sizes need min terms average degree means need ego neighborhoods log surprising since ego networks rather sparse case one needs use larger neighborhoods anyway writing nrn complexity adjacency spectral clustering case becomes processing units gets nmdn although analysis clear pace spectral clustering work well sparse settings numerical simulations found various regimes pace regularized spectral clustering vastly outperforms ordinary regularized spectral clustering see table seems reason pace works well sparse settings lies weights nij neighborhoods chosen subgraphs puv geodesic distance nij known spectral clustering matrix geodesic distances works well sparse settings bhattacharyya bickel pace seems inherit property although presently rigorous proof conclude section illustration pace random using sdp algorithm shall use setting theorem vershynin illustration stated slightly different notation let denote following sdp vershynin sdp maximize subject diag lemma vershynin theorem consider sbm mink bkk also let expected variance edges denotes number pairs vertices one community community fix accuracy solution satisfies corollary sdp random consider setting lemma set fix accuracy assume simple two parameter blockmodel equal community sizes average degree nodes note assumptions corollary satisfied ndn max exactly similar saw spectral clustering take particular average degree need pace succeed however bounded degree regime advantage negligible potentially smaller constant need numerical results expect subgraphs pace perform much better results gale denote unnormalized miscustering error estimated labels true labels set nodes minq perm note since binary discussed earlier number misclustered nodes half number main idea algorithm simple every approximately accurate clustering set nodes accurate permutation never recovered without true labels however align labeling permuted version truth permutation estimated another labeling set vertices done calculating confusion matrix two clusterings call two clusterings aligned cluster one clustering large overlap cluster clustering labels aligned clusterings agree confusion matrix matrix large diagonal entries idea used match algorithm estimate permutation matrix align one clustering another present main result prove consistency slightly modified weaker version algorithm algorithm every step traversal apply match algorithm intersection current subgraph union subgraphs previously aligned estimate permutation yet unaligned current subgraph however theorem presented use intersection unaligned current subgraph last aligned subgraph empirically better use scheme presented algorithm since increases size intersection requires weaker conditions clustering accuracy individual subgraph leave future work formally define estimator gale let let denote aligned clustering subgraph let define gale entries gale fractions show lemma rounding binary matrix change consistency properties note gale depends spanning tree use particular traversal spanning tree let spanningtreesg set spanning trees graph spanningtreesg let traversalst set traversals let outcome gale traversal spanningtreessm theorem misclustering rate gale let let log log consider algorithm labels random error probability least probability least gale max max constant taken close one desires thus bound also essentially optimal illustrate theorem several algorithms begin result approximate adjacency spectral clustering corollary adjacency spectral clustering gale assume setting log lemma let let let log probability least min min constant corollary max max see first term exactly first term corollary balanced graphs imposes condition particular dense well separated regime need log log need log cases need log thus regime average degree like log still computational advantage large networks also factoring parallelizability however moderately sized networks gale may lead much computational advantage present exact recovery result sdp base algorithm shall use yan sdp call let also let denote vector cluster proportions lemma theorem yan let diag optimal solution probability least max max min bkk max max assuming subsequent clustering exactly recovered scaled clustering matrix diag gives exact clustering back example distance based naive algorithm following corollary corollary sdp gale exact recovery assume setting lemma let log let let log long holds replacing probability least separation condition min max max note bound taken greater means high probability proportion misclustered nodes less hence zero leading exact recovery computational complexity note separation condition replaced restricts small consider simple sbm balanced block sizes concreteness case separation condition essentially dictates case spectral clustering thus remarks made earlier large chosen apply well using lemma shows solution small norm difference ideal clustering matrix relate directly misclustering error detailed appendix discussed earlier section even naive implementation gale result running time addition time required cluster random whereas careful implementation add time nearly linear since sdps notoriously time intensive solve gives big saving simulations analysis table present qualitative comparison pace gale four representative global community detection methods profile likelihood mean field likelihood mfl spectral clustering semi definite programming sdp computationally easy theoretical complexity scalability parallelizability hard mfl sdp pace gale table qualitative comparison various methods simulations comparison traditional algorithms simulations use following simple block model dimensional identity matrix matrix ones degree density measure relative separation within block block connection probabilities blocks prior probabilities average degree model given particular model balanced order understand emphasize role pace gale reducing computational time maintaining good clustering accuracy use different settings sparsity different methods recovering pace used random projection plus abbreviated rpkmeans spectral clustering also want point sparse unbalanced networks gale may return clusters typically small fraction nodes visited however possible unvisited nodes important information network structure example subgraphs may chosen larger clusters thereby leaving smallest cluster depend details implementation numerical accuracy sought algorithms unvisited take care computing smallest error permutations gale clustering ground truth essentially treats smallest cluster returned gale misclustered real simulated networks almost never seen gale return large number unvisited nodes sdp admm interior point methods sdps fast practice solved sdps using admm based implementation yan sarkar table see pace gale significantly reduces running time sdp without losing accuracy much fact use spectral clustering estimate last step pace get zero misclustering error mean field likelihood table see implementation mean field full graph converge acceptable solution even five half hours pace gale return much better solutions two minutes fact spectral clustering last step pace misclustering error quite good begs question improvement due spectral clustering show next simulation certain settings even spectral clustering used base algorithm pace gale lead significant improvements terms accuracy running time algorithm sdp sdp pace sdp pace rpkmeans sdp gale algorithm mfl mfl pace mfl pace rpkmeans mfl gale time taken time taken table pace gale mean field likelihood mfl base method simulation settings average degree neighborhood equal sized clusters parallel implementation matlab workers table pace gale sdp base method simulation settings average degree equal sized clusters parallel implementation matlab workers regularized spectral clustering sparse unbalanced settings regularized spectral clustering pace gale performs significantly better regularized spectral clustering full graph fact spectral clustering used last step pace hit error quite remarkable see table section see pace gale also add stability spectral clustering terms clustering degree vertices profile likelihood tabu search optimizing profile likelihood likelihood modularity bickel chen community detection combinatorial problem hard scale even ignore problem local minima table compare running time profile likelihood optimized using tabu search divide conquer versions see local methods significantly cut running time without losing accuracy much also applied profile likelihood node graphs workers although pace gale finished minutes global method finish days present results node networks instead remark seen results presented section recovering pace spectral clustering outperforms random projection based algorithms rpkmeans smaller random neighborhood onion algorithm time taken time taken time taken rsc rsc pace rsc pace rpkmeans rsc gale table pace gale regularized spectral clustering rsc base method simulation settings average degree unequal sized clusters relative sizes parallel implementation matlab workers random algorithm time taken pace rpkmeans gale neighborhood time taken table pace gale profile likelihood base method simulation settings average degree unequal sized clusters relative sizes parallel implementation matlab workers sampled neighborhoods selecting roots uniformly random nodes degree greater lower quantile degree distribution average neighborhood size ordinary pace scheme may thought discussed section networks issue spectral clustering dense matrix context table took seconds however networks much larger scale say several million nodes last step would costly spectral clustering used designing better algorithms recovering something working currently real data analysis political blog data directed network see figure hyperlinks blogs either liberal conservative adamic glance ground truth labels available comparison liberal conservative convert undirected network putting edge blogs least one directed edge resulting network lots isolated nodes isolated edges degree distribution also quite heterogeneous model would appropriate focus largest connected component use laplacian spectral clustering row normalized correct degree heterogeneity pace tables show pace gale add stability possibly eigenvector computation spectral clustering indeed pace gale able cluster leaf vertices vertices degree significantly accuracy figure network political blogs reds conservative blues liberal picture courtesy adamic glance largest comp leaves nodes without leaves nodes rsc rsc pace pace table misclustering rate political blog data pace used neighborhoods roots chosen random high degree nodes largest comp leaves nodes without leaves nodes rsc rsc pace rsc gale pace gale table misclustering rate political blog data gale pace used random subgraphs discussion summarize proposed two type algorithms community detection pace gale lead significant computational advantages without sacrificing accuracy main idea behind methods compute clustering individual subgraph stitch together produce global clustering entire network main challenge stitching procedure comes fundamental problem unidentifiability label assignments two subgraphs overlap clustering assignment pair nodes overlap may inconsistent two subgraphs pace addresses problem estimating clustering matrix subgraph estimating global clustering matrix averaging subgraphs gale takes different approach using overlaps two subgraphs calculate best alignment cluster memberships nodes subgraphs prove addition computationally much efficient base methods typcally run time methods accuracy least good base algorithm typical accuracy type subgraphs used high probability experimentally show something interesting identify parameter regimes local implementation base algorithm based pace gale fact outperforms corresponding global algorithm one example meanfield algorithm typically suffers bad local optima large networks empirically seen smaller subgraph reasonable number restarts finds local optima often highly correlated ground truth pace gale take advantage phenomenon improve time significantly another example regularized spectral clustering sparse unbalanced networks intend theoretically investigate future work finally working many subgraphs naturally leads question self consistency underlying algorithm often crucial real world clustering problems available ground truth labels intend explore direction estimating model parameters like number clusters algorithmic parameters like size number subgraphs number hops used neighborhood subgraphs etc currently picked priori based degree considerations may also possible choose different models standard blockmodels degree corrected models dot product models etc examining model leads self consistent results leave future work conclusion algorithms best knowledge first ever type algorithms used community detection believe basic principles methods broad impact range clustering estimation algorithms computationally intensive proofs results pace proof proposition since safely replace count frobenius norm squared inf kzq perm note permutation matrices thus kzq kzq maximum eigenvalue diagonal maximum diagonal entry size largest cluster thus equals size largest cluster trivially upper bounded goes kzq therefore get nkzq squaring taking infimum permutation matrices right hand side obtain claimed inequality prove theorem proof broken two propositions first decompose cij cij nij cij nij cij note nij therefore yij cij nij cij estimate separately proposition wij yij cij application proof let wij schwartz nij nij yij cij yij yij cij wij yij cij max wij note wij nij nij yij cij max wij hand since subgraphs chosen independently using sampling scheme identically distributed therefore taking expectations get randomly chosen subgraph subgraph selection scheme proposition let nmax size largest block nmax max nij proof since cij nij cij nij taking expectations get nij max nij max nij nmax max nij proof theorem combining propositions get finally note proof corollary sampling scheme using chernoff binomial lower tail nij nij binomial finally get plugging parameter values estimates proof corollary crucial thing observe one removes root node adjacent edges neighborhood remaining ego network blockmodel structure indeed let random ego neighborhood size root arj conditional neighbors latent cluster memberships edges independently generated ajk ajk spoke edges arj independent therefore conditional one instantiation block model parameters vertices ego networks yij bernoulli nij nij total number containing notice nij nij sum independent bernoulli random variables bernoulli enij chernoff bound nij enij exp exp order apply theorem need following two ingredients work use slightly loose convenient form chernoff bounds exp exp independent binary random variables estimate estimate nij estimate note akr since akr independent chernoff inequality exp exp therefore upper bound holds particular exp exp similarly using chernoff inequality binomial upper tail show exp estimate nij recall nij nij binomial nij nij nij nij nij nij clearly nij nij given nij nij invoke chernoff inequality get nij nij nij nij nij exp exp nij nij therefore nij exp nij nij exp nij nij exp exp nij nij nij exp thus exp nij nij nij exp exp thus nij exp exp exp ready use using estimates get exp next plug estimates derived subsection get desired bound results gale clustering subgraph let arg minq perm used shorthand zsi true cluster membership matrix members define matrix requirement words kfi first analyze algorithm match recall two clusterings set agree confusion matrix diagonal matrix permutations entries diagonal corresponding cluster sizes either clusterings following lemma consider noisy version two clusterings perfect agreement lemma essentially establishes supplied two clusterings whose confusion matrix diagonal matrix permutations plus noise match recover correct aligning permutation noise large lemma let mini also let diag match returns applied confusion matrix proof lemma let diag let permutation encoded easy see muv muv whereas muv mina daa hence min muv min daa max muv hence top recall number rows elements diagonal elements thus elements learned match algorithm equivalent establish misclustering errors subgraphs equal original misclustering errors first need lemma happens align two subgraphs based intersection lemma consider two random suppose clustering algorithm outputs clusterings let size least assume match output proof lemma ease notation let write restricting get note multiplication permutation matrix change norm matrix therefore similarly finally used fact row exactly one one therefore note assumptions individual misclustering errors kfi therefore since assumption match hence apply lemma see output proposition let subsets associated estimated clusterings misclustering error consider traversal spanning tree algorithm satisfying sxi min min number nodes cluster subgraph applying gale walk let define recursively match sxi sxi visited otherwise set zxi zxi proof claim arg min zxi zxi zxi use strong induction prove claim true definition assume true representation zxi matrix visited holds induction hypothesis otherwise apply lemma two clusterings conclude permutation matrix thus indeed arg minq consequence induction complete establish upper bound error estimator gale terms errors aligned clusterings proposition let set cluster membership matrix gale proof proof similar proof theorem decompose using gale therefore gale gale analyze let noting application inequality gives max max note therefore mention auxiliary results whose proofs deferred appendix help control probability bad event defined proof theorem lemma let let probability least log recall view random nodes put edge nodes use shorthand denote edge yab graph next lemma shows fact random lemma random graph exp yab assumptions follows well connectivity threshold random graphs lemma connected probability least exp next lemma states intersection two random contains enough representatives cluster high probability lemma consider two random let log log ready prove main result gale proof theorem first construct good set sample space consider following bad events connected min min log max let let lemma exp lemma union bound gives lemma finally hypothesis individual misclustering errors union bound therefore exp choosing suitably large good set spanningtreessm traversalst hypothesis propositions satisfied note proposition also thus proposition taking noting since bound rhs depend particular spanning tree used particular traversal thereof conclude good event max max choice log log therefore event log choosing suitably large thus conclude probability least max max acknowledgments thank arash amini david blei david choi valuable comments versions work presented conferences ssm thanks aditya guntuboyina alan hammond luca trevisan bin valuable comments quals talk berkeley based paper references adamic glance political blogosphere election divided blog proceedings international workshop link discovery pages acm airoldi blei fienberg xing mixed membership stochastic blockmodels advances neural information processing systems pages amini chen bickel levina methods community detection large sparse networks annals statistics amini levina semidefinite relaxations block model arxiv preprint bhattacharyya bickel community detection networks using graph distance arxiv preprint bickel chen nonparametric view network models modularities proceedings national academy sciences binkiewicz vogelstein rohe covariate assisted spectral clustering arxiv preprint cai robust computationally feasible community detection presence arbitrary outlier nodes annals statistics charikar similarity estimation techniques rounding algorithms proceedings annual acm symposium theory computing stoc pages new york usa acm gopalan blei efficient discovery overlapping communities massive networks proceedings national academy sciences greene wellner exponential bounds hypergeometric distribution bernoulli vershynin community detection sparse networks via grothendieck inequality probability theory related fields pages guruharsha rual zhai mintseris vaidya vaidya beekman wong rhee cenaj protein complex network drosophila melanogaster cell holland laskey leinhardt stochastic blockmodels first steps social networks hollingshead elmstown youth impact social classes adolescents karrer newman stochastic blockmodels community structure networks physical review levina vershynin sparse random graphs regularization concentration laplacian arxiv preprint lei rinaldo consistency spectral clustering stochastic block models annals statistics mackey talwalkar jordan distributed matrix completion robust factorization journal machine learning research mcauley leskovec learning discover social circles ego networks nips volume pages mcsherry spectral partitioning random graphs foundations computer science proceedings ieee symposium pages ieee newman clauset structure inference annotated networks arxiv preprint newman girvan finding evaluating community structure networks physical review jordan weiss spectral clustering analysis algorithm advances neural information processing systems nowicki snijders estimation prediction stochastic blockstructures journal american statistical association rohe chatterjee spectral clustering stochastic blockmodel ann rohe qin directed graphs discover asymmetries directional communities proceedings national academy sciences shi malik normalized cuts image segmentation pattern analysis machine intelligence ieee transactions snijders nowicki estimation prediction stochastic blockmodels graphs latent block structure journal classification tang sussman priebe universally consistent vertex classification latent positions graphs annals statistics wang cheng cheng approach attributed graph clustering proceedings acm sigmod international conference management data pages acm yan sarkar convex relaxation community detection covariates arxiv preprint yan sarkar cheng exact recovery number blocks blockmodels arxiv preprint zhang levina zhu community detection networks node features arxiv preprint details dgcluster detail distance based greedy algorithm dgcluster idea behind dgcluster note community kci otherwise kci communities balanced thus expect able cluster vertices using dij namely starting root vertex threshold putting vertices satisfying cluster picking another root vertex remaining set putting vertices remaining set satisfy cluster root vertex may chosen one vertices highest degree remaining set according random sampling scheme give importance highly connected vertices also note depending threshold number blocks get vary practice start small yielding large number communities stop smallest gives blocks get blocks merge succession pairs blocks largest intersectionpin relative sizes get exactly blocks rule thumb would start small gradually increase algorithm distance based greedy clustering algorithm dgcluster input output dgcluster clustering based distances rows merging guidance necessary set cmin set know flag true naivecluster know maxi know flag true else flag false know maxi know let know compute upper triangle matrix ckl rij pick arg maxi rij merge algorithm turn makes use following two algorithms algorithm naive clustering algorithm naivecluster input threshold output naivecluster clustering based distances rows set unassigned unassigned random uniform etc index unassigned unassigned unassigned unassigned compute dij dij unassigned unassigned algorithm merge input output merge clustering blocks merged min max else auxiliary results lemma thresholding pace let min particular proof note cij ber qij qij cij cij thus cij qij cij cij cij therefore qij cij qij cij cij means cij cij min summing gives desired bound lemma rounding gale consider estimated cluster membership matrix true cluster membership matrix kround proof let markov inequality note round zik thus zik kround shall need following concentration result hypergeometric random variables lemma corollary greene wellner restated using slightly different notation consider random variable hypergeometric let exp let deduce two convenient chernoff type bounds upper lower tail hypergeometric variable lemma consider random variable hypergeometric max exp exp proof note hypergeometric using lemma get taking exp get exp exp gives desired bound lower tail upper tail handled similarly analysis pace stochastic block model recall need know algorithm performs randomly selected subgraph subgraph selection procedure discuss one obtain guarantees stochastic blockmodel require understand behavior sizes different communities randomly chosen subgraph community sizes random results subsection depend modeling assumption let random subgraph set cluster size vector clearly hypergeometric therefore lemma nmin nmin exp exp therefore union bound nmin nmin nmin exp choosing nmax log see probability least nmin nmin nmax log similarly show max taking nmax exp nmax log conclude probability least max nmax nmax log community sizes ego neighborhoods let randomly chosen ego neighborhood note size block neighborhood satisfies hard see independent conditional root also follows means nmin nmax therefore chernoff inequality nmin exp exp nmin since right hand side depend take expectations sides respect get nmin exp implies application union bound nmin choosing nmin exp nmax log conclude probability least nmin nmin nmax log one prove similarly max taking nmax exp nmax log conclude probability least max nmax nmax log analysis adjacency spectral clustering shall use community size estimates previous subsection along lemma random probability least nmax nmax log nmax nmin nmin nmax log nmin nmax log nmax nmax log note stays bounded nmaxminlog fact approaches nmaxminlog thus probability least randomness probability least randomness nmax min conclude nmax min nmin ego neighborhoods given probability least randomness max min min therefore ejs probability least nmax nmax log nmax nmax nmax nmin nmin nmin nmax log nmin nmax log nmin log nmax log nmax nmin stays bounded nmax log fact approaches note nmax log log nmax thus ejs min nmax nmin hand exp finally nmax min analysis sdp setting corollary random mink bkk let denote number pairs vertices subgraph one community community lemma cnk min probability least cnk cnk nkk min similarly show probability least nkk min let given expected variance edges say using assumptions get satisfy therefore using lemma subgraph conclude given probability least solution subgraph conclude min min proofs results gale section collect proofs auxiliary results presented section proof lemma note binomial random variables hence chernoff inequality another use chernoff gives exp setting log conclude probability least log log proof lemma first show three different nodes overlap variables yab yac independent immediate consequence following two facts yab independent follows observation yab hypergeometric given overlaps yab yac independent follows fact independent discussion yab hypergeometric using lemma get get yab exp proof lemma notice yab hypergeometric yab follows fact given yab distributed uniform yab lemma gives log yab yab yab therefore bound holds unconditional probability another application lemma yab hypergeometric gives log yab thanks satyaki mukherjee making observation makes proof considerably shorter original approach based concentration inequalities sums dependent bernoulli random variables using two bounds conclude probability least long log log large enough suffices make rhs
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apr exceptional groups maneesh thakur indian statistical institute sansanwal marg new delhi india abstract main aim paper prove simple simply connected defined field arbitrary algebraic groups tits index characteristic let group prove exists quadratic extension extension denotes group classes sense manin see consequence follows variety retract conjecture holds groups moreover projectively simple abstract group field extension monograph tits richard weiss conjectured albert division algebra field structure group str generated scalar homotheties known equivalent conjecture groups tits index settle conjucture albert division algebras first constructions affirmative results obtained corollaries main result shows albert division algebra first construction structure group algebraic group norm similarities field extension keywords exceptional groups algeraic groups albert algebras structure group conjecture introduction main aim paper prove certain exceptional algebraic groups precisely let simple simply connected algebraic group defined field see index notation groups arbitrary characteristic tits index indices exist field admits central division algebra degree reduced norm admits degree central division algebras unitary involutions quadratic extension reduced norm map see exist finite note also groups tits indices fields number fields algebraically closed fields local fields field reals however exist example function field one variable maneesh thakur classified isotopy classes albert division algebras groups index associated buildings moufang hexagons defined hexagonal system type either quadratic extension see groups classified isotopy classes albert division algebras associated index moufang sets albert division algebra see let algebraic group tits index one mentioned indices prove exists quadratic field extension field extension sense manin since quadratic extension trivial center follows index simple abstract group field extension well known see anisotropic kernels groups correspond upto central torus structure groups albert division algebras albert division algebras field obtainable two rational constructions due tits called first construction second construction prove anisotropic kernel comes first construction albert division algebra situation splits cubic extension conjecture predicts connected simple simply connected algebraic group defined isotropic field subgroup generated points unipotent radicals kparabolics see theorem gille equivalent proving thm conjecture tits richard weiss asserts albert division algebra field structure group str generated scalar homotheties defined see page conjecture see equivalent conjecture groups tits index results prove conjecture first tits construction extensive survey new results problem refer reader along classics paper reduction relative rank one case mention recent results case simply connected group settled modern version terms index veldkamp proof presented case trialitarian forms global fields settled paper also presents description whitehead group infinite field absolutely simple simply connected algebraic group type result groups general field covered arbitrary field conjecture groups tits index settled paper second series papers dedicated problem first paper proved among results conjecture true pure first constructions also proved albert division algebra group aut problem semisimple algebraic groups studied several mathematicians see example references therein connects properties algebraic groups simple simply connected algebraic group isotropic defined field variety retract see thm remark simple simply connected algebraic group tits index semisimple anisotropic kernel isomorphic isom full group isometries norm form albert division algebra anisotropic kernel split cubic extension first tits construction see proposition corollary retract rationality results also prove simple simply connected exists quadratic algebraic group defined tits index extension underlying variety retract anisotropic kernel split cubic extension retract plan paper paper always denote base field infinite arbitrary characteristic give quick introduction albert algebras tits processes structure group albert algebra description strongly inner groups type given detail followed brief introduction tits index algebraic groups end state main results consequences contain key results paper mainly concerned extension automorphisms subalgebra albert algebra automorphism also discusses subgroups aut contains results extending norm similarities subalgebras albert algebra see theorem important results proved develop results various subgroups aut str albert algebras general well special case first constructions results culminate main theorem theorem prove str first tits construction albert algebras discusses proof groups type anisotropic kernel split cubic extension proof conjecture albert division algebras arising first tits construction presented corollary anisotropic kernel split cubic discuss groups tits index extension prove groups end results reduced albert algebras retract rationality results derived prove consequence results anisotropic kernel group whose tits index split cubic extension retract preliminaries results much preliminary material albert algebras refer reader recall briefly notions need base fields considered paper assumed infinite arbitrary characteristic define notion albert algebra field arbitrary characteristics take approach via cubic norm structure defined cubic norm structures let finite dimensional vector space field cubic norm structure consists tuple base point called identity element norm structure cubic form trace form defined logn nondegenerate maneesh thakur adjoint map defined quadratic map trace hold scalar extensions denotes directional derivative polynomial function direction evaluated differential calculus rational maps related notation refer chap sect let one gets unital quadratic jordan algebra structure see sometimes denoted linear operators defined called jordan algebra element invertible case one recall called division algebra surjective equivalent list examples notation use paper example let denote separable associative algebra degree let denote norm trace unit element classical adjoint map get quadratic jordan algebra structure denote example let separable associative algebra involution second kind center unit element restriction norm get cubic norm structure substructure example let octonion algebra field norm map trace let denote canonical involution let diag xij xii algebra matrices follows form entries transpose define abc aeij aeji terms matrix units defines cubic norm structure see hence jordan algebra called reduced albert algebra call reduced albert algebra split coordinate octonion algebra split case tits process let finite dimensional associative degree norm trace let define cubic norm structure space xyz quadratic jordan algebra associated norm structure denoted regard subalgebra first summand recall see division algebra norm construction called first tits process arising parameters let quadratic extension separable associative algebra degree involution let unit denotes nontrivial define cubic norm structure space formulae quadratic jordan algebra corresponding cubic norm structure denoted note identified subalgebra first summand recall see division algebra norm construction called second tits process arising parameters note switch involution opposite algebra case second construction identified first construction albert algebras tits process described start central simple algebra get first tits construction albert algebra similarly tits process start central simple algebra center quadratic algebra involution second kind described get second tits construction albert algebra one knows albert algebras obtained via tits constructions albert algebra division algebra cubic norm anisotropic see also recall first construction albert algebra either division algebra split isomorphic split octonion algebra moreover albert division algebra arising first construction split cubic subfields center see hexagonal systems hexagonal system cubic norm structure whose norm anisotropic say hexagonal system type arises first tits construction albert division algebra type quadratic field extension maneesh thakur arises tits second construction albert division algebra corresponding center pure first second constructions let albert algebra first construction second construction respect degree central simple algebra whose center quadratic field extension call pure first construction albert algebra similarly pure second construction albert algebras obtainable first construction recall stage albert division algebra subalgebra either cubic subfield form degree central simple algebra involution second kind center quadratic extension see chap lemma norm similarities albert algebras let albert algebra denote norm map norm similarity bijective linear map since working infinite norm similarities synonimous isotopies albert algebras see chap thm thm adjoints let albert algebra norm map recall given invertible shown norm similarity fact invertible central simple algebra degree adjoint map satisfies shown also follows algebraic groups albert algebras paper field extension denote let albert algebra norm algebraic closure full group automorphisms aut aut simple algebraic group type defined simple groups type defined arise way division algebra aut anisotropic denote group points aut aut note division algebra precisely norm form anisotropic see thm full group str norm similarities called structure group connected reductive group type denote str group krational points str invertible norm similarities generate normal subgroup str called inner structure group denote instr commutator subgroup isom str full group isometries simple simply connected group type anisotropic anisotropic division algebra see note aut stabilizer str occasions need treat algebra algebraic variety particularly dealing situation confusion likely shall continue denote underlying affine space morphisms rational maps would carry similar meanings base change object defined base field extension denoted record following observations proofs proposition let albert division algebra aut str isom split degree extension first construction hence aut degree splitting field pure first construction every minimal splitting field cubic cyclic pure second construction hence aut splitting field degree proof write second construction suitable parameters decribed preliminaries first construction cubic subfield splits see thm hence splits aut statement follows let center must quadratic field extension first construction base changing argument presented cubic extension splits hence split degree extension prove note first construction cubic splitting field conversely degree splitting field invariant vanishes hence springer theorem hence first construction assertion pure first construction follows thm proof follows proposition let albert division algebra first tits construction str equivalently isom split cubic extension proof first assume albert division algebra arising first tits construction split cubic extension see thm hence thm also splits structure group str conversely str splits cubic extension thm albert algebra reduced thm stabilizer split group type equals aut hence aut invariant follows coordinatizing octonion algebra split hence springer theorem therefore must first tits construction strongly inner group type let simple simply connected algebraic group defined strongly inner type see definition isomorphic isom algebraic group norm isometries albert algebra moreover anisotropic division algebra see thm let simple simply connected algebraic group groups tits index index associated diagram tits index anisotropic kernel group simple simply connected strongly inner group type realized group isom norm isometries albert division algebra let denote tits building associated moufang hexagon associated hexagonal system type quadratic field extension written second construction first construction group group linear automorphisms automorphisms explicit realization groups type moufang hexagon due tits weiss detailed monograph maneesh thakur let simple simply connected algebraic group groups tits index tits index anisotropic kernel simple simply connected strongly inner group type isomorphic isom albert division algebra see tits index associated diagram following max koecher one realize follows fix albert divsion algebra let algebraic group birational transformations generated structure group str maps algebraic group tits index arises way type see element form str proved koecher infinite characteristic loos arbitrary chatacteristics context jordan pairs remark light proposition proposition anisotropic kernel simple split degree simply connected group defined tits index extension corollary let simple simply connected algebraic group index anisotropic kernel split cubic extension isom albert division algebra arising first tits construction proof immediate proposition discussion algebraic groups let irreducible variety field nonempty following manin define points exists sequence points rational mape defined regular connected algebraic group defined set elements normal subgroup set requivalence classes natural bijection quotient therefore carries natural group structure identify group group measures obstruction rational parameterizing points say field extensions recall variety defined said birationally isomorphic affine space well known see chap prop let connected reductive group defined assume admits image contained center let denote subgroup generated points unipotent radicals subgroups normal quotient whitehead group theorem gille thm semisimple simply connected absolutely almost simple group defined isotropic problem asks group whitehead group trivial generally compute whitehead group paper prove theorem let simple simply connected algebraic group type strongly inner type whose anisotropic kernel split cubic extension corollary let simple simply connected algebraic group defined isotropic anisotropic kernel split cubic extension field tits index conjecture holds moreover projectively simple field extension corollary let simple simply connected algebraic group defined isotropic exists quadratic field extension field tits index extension prove result steps key result proving theorem main theorem theorem let albert division algebra arising first tits construction str corollary conjecture let albert division algebra arising first tits construction str subgroup str consisting scalar homotheties automorphisms let albert algebra subalgebra denote aut closed subgroup aut consisting automorphisms fix pointwise aut denote closed subgroup automorphisms stabilizing group points groups denoted ordinary fonts example aut aut division algebra proper subalgebra either cubic subfield degree jordan subalgebra form degree central division algebra form central division algebra degree quadratic extension involution second kind thm record description subgroups aut shall use sequel refer details proposition let albert algebra suppose degree central simple write aut algebraic group norm elements let degree central simple algebra degree quadratic extension involution second kind write suitable parameters aut int maneesh thakur particular subgroups described simply connected simple type defined hence rational therefore last assertion due fact connected reductive groups absolute rank rational see algebraic group defined denote normal subgroup elements extending automorphisms let albert division algebra subalgebra certain conditions extend automorphisms automorphisms nine dimensional see thm thm need certain explicit extensions results proceed describe need following important result sequel frequently record lemma arbitrary characteristics lemma let degree central simple algebra center quadratic algebra involution second kind let satisfies exists first let subalgebra may assume degree central simple algebra center quadratic extension see suitable parameters recall automorphism form sim sim immediate fact norm similarity form see thm among automorphisms precisely norm similarities fix identity element see thm recall notation int proposition let albert algebra let aut given gxg sim let arbitrary map given gag automorphism extending proof show automorphism first note bijection hence page suffices check preserves norms first condition obvious let verify second first note formulae used computation also since compute using formulae gag gag gag gag gag gag hence automorphism extending corollary let albert algebra first construction let aut given gxg map gxg automorphism extending proof identify second construction switch involution apply proposition note sim extension constructed simplified follows let since using becomes hence expression gag pbq proposition let albert algebra quadratic extension automorphism stabilizing form pbq aut det det denotes group norm elements embedded diagonally proof let aut let aut gag sim let extend automorphism formula pbq maneesh thakur arbitrary discussed automorphism fixes pointwise hence therefore get pbq set proof first assertion complete arbitrary hence lemma exists computations done follows pxq automorphism stabilizing hence automorphism stabilizes precisely form pxq map det aut given pxq surjective homomorphism ker inducing required isomorphism two groups corollary let first tits construction albert algebra automorphism stabilizing form gxg proof identify second tits construction apply proposition norm similarities last section extended automorphisms subalgebra albert algebra automorphism albert algebra section analyse norm similarities albert algebra spirit study rationality properties subgroups str first make simple observations since norm similarity fixing identity element necessarily automorphism follows subgroup str norm similarities fix pointwise equal subgroup aut aut str recall denote normal subgroup str generated instr prove results norm similarities needed paper begin theorem let albert algebra instr str proof let instr let similarly defined let defined uat ubt uct rational map defined domain open subset consisting invertible elements clearly regular defined str hence assertion proved theorem generalization corollary field arbitrary characteristic theorem let albert division algebra aut automorphism fixes cubic subfield pointwise proof let denote subalgebra fixed points light fact proper subalgebras albert division algebras dimension dimk invariant base change may work since eigenvalues eigenvalues semisimple part may assume semisimple let maximal torus containing theorem group automorphisms fixing primitive idempotent isomorphic spin conjugate maximal torus spin conclude fixes vector hence fixes generates cubic subfield fixes pointwise following theorem generalization theorem arbitrary characteristics theorem let first tits construction albert division algebra let aut fixes cubic cyclic subfield pointwise product two automorphisms stabilizing subalgebra proof proof goes along exact lines proof given except use theorem place corollary following corollary proof exactly cor use paper arbitrary characteristics corollary let first tits construction albert division algebra aut cubic cyclic subfield product three automorphisms stabilizing subalgebra following result generalization theorem arbitrary characteristics theorem let albert division algebra arising first tits construction let cubic cyclic subfield str denotes subgroup aut generated automorphisms stabilizing subalgebras group scalar homotheties str proof proof goes along exact lines proof thm except use corollary instead extending norm similarities let albert division algebra subalgebra given element str wish extend element str though achieved using recent result prop need certain explicit extension purpose proving algebraic groups described introduction proceed describe suitable degree central division algebra quadratic extension involution second kind therefore suitable parameters purpose conjugating suitable automprphism maneesh thakur may assume str let thm str consists maps hence exists let let int lemma exists define setting clearly hence extension clearly bijective endomorphism compute substituting using rhs equation simplifying get therefore str proved theorem let albert division algebra subalgebra every element str admits extension str let suitable parameters let str given map given norm similarity extending corollary let first construction albert algebra let str given map given axb norm similarity extending let str arbitrary let extend element str according str aut hence exists write automorphism also since exists therefore therefore associated triple str arbtrary conversely triple arbitrary define formula straight forward calculation exactly follows therefore str hence proved theorem let albert division algebra written second tits construction group str consists maps arbitrary satisfies str via proof last assertion needs proved let define str calculations surjective homomorphism compute ker let ker maneesh thakur taking relation gives gives also gives therefore get equation thus substituting get qxq gives using qbq qxq hence thus therefore hence shows ker proof complete corollary let first construction albert algebra group str consists maps axb results section proceed develop results various groups also prove main theorem need version theorem yanchevskii degree division algebras arbitrary characteristics prove proposition let degree central division algebra center quadratic extension involution second kind every nonzero element admits factorization proof may assume characteristic proof characteristic sym skew dimk sym since restricts nontrivial sym hence dimk sym also hence dimk dimk sym hence sym thus find sym sym let moreover implies hence required factorization theorem let albert division algebra let subalgebra notations str proof let str since dim suitable degree central simple algebra quadratic extension involution second kind conjugating suitable automorphism may assume suitable parameters let clearly subgroup theorem proposition write satisfying since follows since scalar homotheties str constitute dimensional torus prop prop scalar multiplications str hence may assume using writing norm similarity str claim str see define let str defined rational map defined regular str settles claim hence remains show str observe since hence therefore hence also str via rational hence hence automorphism belongs str let str defined claim str see define let whenever map str xqt maneesh thakur defined open subset rational map str corresponds particular corresponds pair hence corresponds since hence str settles claim proof theorem complete corollary let albert division algebra subalgebra aut str prove theorem let first tits construction albert division algebra cubic cyclic subfield str str proof theorem str theorem follows str subgroup str consisting scalar homotheties subgroup aut generated automorphisms stabilizing subalgebras suffices prove str element product automorphisms stabilizing subalgebra let write aut subalgebra theorem aut str str hence str proof complete corollary assume first tits construction cyclic cubic subfield aut str proof aut str hence result follows theorem theorem let albert division algebra arising first tits construction aut str proof fix cubic cyclic extension possible let aut aut theorem str may assume let subalgebra generated thm hence degree central simple algebra quadratic extension involution second kind hence may write suitable parameters degree central division algebra classical theorem extends automorphism hence corollary automorphism hence fixes pointwise corollary aut str also aut hence corollary aut str follows str assume field first extend isomorphism theorem exists int since follows int therefore gxg gxg yields gxg hence centralizes hence since maximal commutative follows also hence similarly follows let define str hence gxeg therefore since first tits construction cyclic degree central division algebra see cor lemma exists hence extend element str formula theorem str hence aut therefore aut hence theorem str also str str already shown theorem subgroups str follows str lemma let cubic extension let denote first tits process let define str rnl proof proof follows calculation exactly proof lemma position prove main result theorem let albert division algebra arising first tits construction str proof let str str hence theorem aut str implies str may assume cubic separable subfield cor tits process define str lemma extend str using hence theorem str also hence aut str follows str theorem let albert division algebra arising first tits construction field isom maneesh thakur proof let isom isom str isom theorem str let str rational map connecting str let let uat uat since follows isom also rational map isom norm isometry hence proves assertion conjecture section prove conjecture albert division algebras arise first tits construction recall paper denote str group str points algebraic group str full group norm similarities denote isom full group norm isometries isom isom thm isom connected simple simply connected algebraic group type defined follows isom commutator subgroup str let denote subgroup str consisting scalar homotheties conjecture let str str equivalent kneserequivalently conjecture predicts tits conjecture groups type let simple simply connected group defined tits index subgroup generated points unipotent radicals parabolic conjecture predicts reductive anisotropic kernel structure group str uniquely upto isotopy determined albert division algebra defined see proved str hence prove conjecture need show first quotient trivial thm suffices prove fact suffices prove let maximal torus decomposition follows birational unipotent radical minimal parabolic containing well known underlying variety rational hence prop follows hence prove conjecture suffices prove second quotient trivial proceed prove note connected reductive reductive anisotropic kernel commutator subgroup simple strongly inner type hence isom first prove lemma connected center equals proof know connected reductive connected center contains almost direct product groups since simple type rank let maximal connected center torus finite hence torus dimension rank rank therefore rank rank follows light lemma well known see thm thm albert algebra str connected reductive group defined need information center lemma let albert algebra group str connected reductive center commutator str str isom sequence isom str exact str maps norm similarity factor similitude proof str connected reductive proved let str choose two degree separable say see lemma let aut since commutes must stabilize well hence stabilizes thus aut hence aut aut group type hence trivial center follows hence str str consisting scalar homotheties remaining assertions follow thm thm observe maximal tori conjugate str see hence may assume str denote center str str str str isom since split corollary decomposition hence since str str isom isom str also since follows str therefore str str maneesh thakur finally hilbert hence follows str str state prove theorem let simple simply connected algebraic group tits index whose anisotropic kernel split cubic extension proof let denote reductive anisotropic kernel discussion corollary isom albert division algebra arising first tits construction already shown str str also theorem first tits construction str let field extension split splits hence division tits index change anisotropic kernel corresponds str hence str last equality holds str since str str element str represented element str element str joined identity element image str result follows remark note proved also str adjoint form corresponding first tits construction albert algebra field corollary conjecture let first tits construction albert division algebra str proof follows immediately theorem theorem str groups index brief section wish explore issues simple groups type reason considering groups groups also anisotropic kernels structure groups albert division algebras groups type arise way see consider groups index whose anisotropic kernel structure group albert division algebra tits first construction groups studied max koecher use explicit description given max koecher fix albert division algebra let full group birational transformations generated translations every element expression particular str contained isom anisotropic kernel central torus precisely let denote center str defined let maximal torus containing connected reductive reductive anisotropic kernel since follows previous section isom str hence str shown last section str hence arguments quite analougous proof groups index theorem let simple simply connected algebraic group tits index whose semisimple anistropic kernel split cubic extension reduced albert algebras wish explore group str reduced albert algebra mainly groups recall stage result fact holds fields arbitrary characteristics theorem let reduced albert algebra field str denotes subgroup str scalar homotheties instr inner structure group proof follows result faulkner thm reduced albert algebra group isom simple modulo center given str let invertible consider str isom since instr normal subgroup str norm similarities form uar isom whenever follows instr isom hence instr isom isom center isom consists scalar homotheties therefore hence completes proof already shown instr str hence follows theorem str str inclusion independent base field hence str record theorem let reduced albert algebra field str theorem let albert algebra arising first tits construction norm similarity product norm similarities stabilizing subalgebra maneesh thakur proof first assume split let str isom str stabilizes subalgebra containing result faulkner cited isom simple group modulo center contains scalar homotheties hence isom generated norm isometries stabilize subalgebras result follows assume division algebra may assume isom proof theorem reduce case aut using arguments proof theorem may assume stabilizes cubic cyclic extension str theorem str subgroup scalar homoteties instr subgroup str generated subgroup aut generated automorphisms stabilize subalgebras hence admits required factorization theorem let reduced albert algebra field isom proof let isom isom str theorem str argument exactly proof theorem result follows retract rationality recall irreducible variety defined field retract exists open subset defined identity map factors open subset affine space exists morphisms idu see following result proved prop thm theorem let semisimple simply connected absolutely almost simple group defined isotropic infinite field following equivalent field extensions retract variety combining results corollary let simple simply connected algebraic group defined field whose anisotropic kernel splits cubic extension tits index retract proof corollary anisotropic kernel isomorphic isom first tits construction albert division algebra since first tits construction field extension either split division algebra see cor let theorem str split result conjecture str also division algebra hence theorem theorem proves assertion theorem let reduced albert algebra field isom retract rational proof note reduced albert algebra algebraic group isom simple simply connected assertion follows group settled theorem acknowledgement part work done visited linus kramer mathematics institute university muenster march stay supported deutsche forschungsgemeinschaft sfb help gratefully acknowledged thank richard weiss tom medts useful discussions suggestions immensely benefitted discussions bernhard muhlherr gopal prasad dipendra prasad thank holger petersson otmar loos encouragement interest work references borel linear algebraic groups gtm springer verlag new york lien boelaert tom medts anastasia stavrova moufang sets structurable division algebras preprint available arxiv http chetan balwe anand sawant anisotropic groups appear imrn chernousov merkurjev special unitary group algebra chernousov merkurjev spinor groups amer math vladimir chernousov vladimir platonov rationality problem semisimple group varieties reine angew math sansuc sur les tores ann scient norm serie chernousov timoshenko group classes semisimple groups arithmetic fields russian algebra analiz translation petersburg math john faulkner octonion planes characteristic bull ams john faulkner octonion planes defined quadratic jordan algebras mem joseph ferrar holger petersson exceptional simple jordan algebras galois cohomology arch vol skip garibaldi rank form veldkamp compositio math philippe gille bourbaki maneesh thakur philippe gille pour les groupes trans ams skip garibaldi holger petersson outer automorphisms algebraic groups theorem albert algebras appear documenta mathematica jacobson structure representations jordan algebras ams providence ams colloquium publications vol xxxix jacobson division algebras fields springer verlag berlin jacobson groups transformations defined jordan reine angew math jacobson groups transformations defined jordan algebras iii groups type reine angew math max koecher uber eine gruppe von rationalen abbildungen invent math knus merkurjev rost tignol book involutions ams colloquium publications vol ottmar loos algebraic groups defined jordan pairs nagoya math vol manin cubic forms amsterdam mccrimmon constructions exceptional jordan algebras trans ams merkurjev norm principle algebraic groups algebra analiz petersson structure theorems jordan algebras degree three fields arbitrary characteristic comm algebra petersson albert algebras fields institute workshop exeptional algebras groups university ottawa available url http petersson racine springer forms first tits construction exceptional jordan division algebras manuscripta math petersson racine elementary approach invariant albert algebras indag petersson racine albert algebras jordan algebras oberwolfach kaup mccrimmon eds gruyter berlin petersson racine classification algebras arising tits process algebra petersson racine jordan algebras degree tits process algebra platonov rapinchuk algebraic groups number theory pure applied mathematics vol academic press boston parimala sridharan thakur classification theorem albert algebras trans ams parimala sridharan maneesh thakur tits constructions jordan algebras bundles plane compositio math parimala tignol richard weiss conjecture groups arbitrary field transform groups gopal prasad problem triality forms comment math helv gopal prasad raghunathan problem comment math helv springer veldkamp octonions jordan algebras exceptional groups springer monographs mathematics berlin springer linear algebraic groups second edition progress mathematics birkhauser boston springer jordan algebras algebraic groups ergebnisse der mathematik und ihrer grenzgebiete band tom medts hendrik van maldeghem moufang generalized polygons topics diagram geometry quaderni matematica volume edited antonio pasini dept seconda univ napoli caserta tom medts yoav segev course moufang sets innovations incidence geometry vol maneesh thakur automorphisms albert algebras conjecture tits weiss trans ams tits groupes whitehead groupes simples sur corps bourbaki vol exp lecture notes springer york tits strongly inner anisotropic forms simple algebraic groups journal alg tits classification buildings spherical type moufang polygons survey colloquio internazionale sulle teorie combinatorie roma tomo atti dei convegni lincei accad naz lincei rome tits classification algebraic semisimple groups algebraic groups discontinuous subgroups proc sympos pure boulder borel mostow eds vol providence maneesh thakur veldkamp unitary groups projective octave planes compositio math veldkamp unitary groups planes math tits weiss moufang polygons springer monographs mathematics springer verlag voskresenskii algebraic groups birational invariants translation math monographs vol ams providence weiss structure spherical buildings princeton university press princeton weiss quadrangular algebras princeton university press mathematical notes princeton university press princeton
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aug data prism learning regime michela meister mmeister gregory valiant valiant august abstract consider simple model unreliable crowdsourced data underlying set binary variables evaluator contributes possibly unreliable adversarial estimate values subset variables learner given true value constant number variables show provided evaluators good either correct independent noise rate true values fraction underlying variables deduced long example good worker evaluates random set items noise responses accurate recovery possible provided fraction good evaluators larger result optimal large dataset contain information setting viewed instance learning model introduced explores tradeoff number items evaluated worker fraction good evaluators results require number evaluators extremely large although algorithm runs linear time given query access large dataset evaluations setting results also viewed examining general class csps planted assignment extreme parameter regime fraction reliable data small inverse exponential amount data provided source relevant number practical settings example settings one large dataset customer preferences customer specifying preferences small constant number items goal ascertain preferences specific demographic interest results show large dataset lacks demographic information leveraged together preferences demographic interest constant number randomly selected items recover accurate estimate entire set preferences even fraction original dataset contributed demographic interest inverse exponential number preferences supplied customer sense results viewed data prism allowing one extract behavior specific cohorts large mixed dataset introduction imagine access large dataset market research specifically dataset consists customer evaluations products total set products large size customer asked evaluate small perhaps randomly selected subset etc products long dataset collected suppose wish identify preferences special demographic customers students let denote lower bound fraction surveyed customers students assume demographic information set evaluations customer leverage dataset learn anything problem seems hopeless amount data contributed might swamp portion dataset contributed demographic interest nevertheless main result paper shows one could hire students evaluate constant number random products set size leverage constant amount information together large dataset return accurate evaluations preferences items claim hold provided number items evaluated customers dataset guarantees algorithm ensure high probability returned evaluations incorrect number products evaluated hired students function independent total number items particular strong success guarantee holds irrespective behavior demographics original particular could even adversarial provided single malicious entity trying disguise feedback provided setting one large dataset reflecting number demographics wishes leverage large dataset conjunction small set verified datapoints one demographic interest seems widely applicable beyond market research domain indeed many biological datasets demographic interest might trait expensive evaluate example perhaps one large database medical records wishes investigate propensity certain medical conditions subset people specific genetic mutation large dataset medical records likely contain information whether individuals mutation question nevertheless results imply accurate inferences subset people likely made long fraction people mutation large dataset minuscule one obtain small constant amount data individuals genetic mutation question example studying constant number individuals known mutation formal model formally model problem instance learning model proposed charikar steinhardt valiant suppose set boolean variables workers provide evaluation values randomly selected subset variables suppose workers reliable submit evaluations property reported values incorrect independently probability prel make assumptions evaluations submitted unreliable evaluations could biased arbitrary even adversarially chosen goal confounding learning algorithm addition large dataset also receive verified data points consist values random subset variables size goal learner return assignments variables probability least returned assignments differ true values previous work focussed regime number workers contrast allow focus interplay number variables evaluated individual fraction reliable workers throughout positive results hold number verified data points constant independent dependent summary results connections random csps main result following theorem fix failure probability accuracy parameter consider set items boolean value reviewers evaluate uniformly random subset items suppose reviewers good reviews correct independently probability least given sufficiently many reviewers accurate reviews least items inferred given true values constant independent sized random subset variables provided fraction good reviewers satisfies specifically given values random subset items size log probability least one recover accurate evaluations least items provided number reviewers additionally algorithm runs time linear number items given ability query dataset reviewers evaluated given set items constant time specifically runtime algorithm hidden constant hides exponential dependence polynomial dependence log following straightforward observation demonstrates theorem optimal relationship fraction good reviewers number items reviewed individual error rate good reviewer observation good reviewer incorrectly reviews item independently probability fraction good reviewers satisfies denotes number items evaluated reviewer remaining fraction reviewers behave every set items randomly selected reviewer distribution reviews items uniform possible review vectors hence dataset contains useful information one reason theorem surprising inverse exponential dependence number reviews per reviewers fraction good reviewers attained via usual approach matrix approximation often applied problem recommendation systems see approaches applied note matrix rows entries rank matrix exactly agrees entries intuitively factors capable representing different subset reviewers still best would result algorithm capable capturing different groups reviewers words seems extremely unlikely approaches could yield positive results setting fraction good reviewers less contrast results allow fraction setting theorem easily mapped language constraint satisfaction problem given evaluations reviewers build constraint satisfaction problem associating boolean variable items every set variables define set allowable assignments variables include review vectors constitutes fraction review vectors associated items words fraction reviewers evaluated given set items submitted vector reviews allowable assignment variables requirement guarantees every set items irrespective behaviors fraction bad reviewers randomly selected reviewer probability reviews correct strictly larger additionally requirement number reviewers ensures high probability elementary concentration bounds every set items sufficiently many reviewers assigned set items ensure number accurate ratings provided good reviewers exceeds fraction overall reviews set items hence high probability obtain constraint satisfaction problem every set variables correct assignment set allowable assignments least one possible assignments disallowed given mapping setting constraint satisfaction problems theorem follow immediately following result concerning class adversarial constraint satisfaction problems theorem consider set boolean variables planted assignment suppose subset variables subset assignments planted assignment restricted variables set given ability query planted assignment values constant number variables chosen uniformly random planted assignment recovered errors constant specifically querying values log variables probability least output assignment differs planted assignment values additionally algorithm run time simple argument together result haussler yields tighter information theoretic recovery result yielding analog theorem polynomial rather dependence specifically number verified assignments must log log approach however seems yield algorithm runtime least opposed linear time algorithms theorems practical settings algorithm seems quite important said exploring problem information theoretic perspective also worthwhile one natural question whether one achieve time algorithm polynomial dependence discuss problem section proposition theorem consider set boolean variables planted assignment suppose subset variables subset assignments planted assignment restricted variables given ability query planted assignment values log log random entries probability least one recover assignment disagrees values proof let set assignments consistent sets partial assignments specified sets dimension set since assumption every variables possible assignments variables shown haussler theorem subset boolean hypercube dimension every exists set size every point exists point agrees least coordinates let denote covering set corresponding set every distance element use log log random coordinates vector find probability least point distance simply choosing element agrees largest fraction random samples follows leveraging chernoff bound show samples fraction disagree element distance union bound chernoff bounds argue none elements distance least disagree fewer fraction indices together yields probability element agrees largest fraction random samples distance greater true assignment suitable choice constant term log log thank anonymous reviewer early version paper drawing attention one implication result boolean constraint satisfaction problem exists satisfying assignment every subset variables constraints forbid least one possible assignments must case constant number solution clusters solution cluster set assignments differ locations indeed number clusters specified theorem proposition bound number variables whose assigned value must queried achieve constant probability failure note number solution clusters independent structure satisfying assignments slightly surprising given following two simple examples first example illustrates possible csps least two extremely different satisfying assignments second illustrates possible csps sized solution size assignments cluster quite similar example consider setting underlying assignment variables every pair variables set allowable assignments based constraints two possible satisfying single verified data point sufficient distinguish two sets assignments following example illustrates general impossible guarantee learner correctly output exact assignment unless number verified datapoints example consider setting set values constraint precludes case single solution cluster consisting assignments variables variables remaining case impossible distinguish assignments significant probability using fewer verified evaluations despite examples still unclear whether information theoretic bound proposition tight particularly small constant clear extent number solution clusters grow decreases related work motivated increasing practical importance robust generally robust learning recent interest problems information theoretic computational perspective recent works tackled general problem several basic settings including robust linear regression robustly estimating mean covariance natural classes distribution including multivariate gaussians focus works largely establishing computationally efficiency algorithms tasks approach information theoretic minimax guarantees achieved naive algorithms three works focussed regime majority data assumed good distribution cohort interest case recovery guarantees require fraction good data satisfies recent works consider setting minority data good latter paper formally proposing learning model one may obtain small amount verified data drawn question former paper considers similar item evaluation setting setting consider focusses regime number evaluators order number items evaluated regime show recovery possible provided number items reviewed evaluator contrast consider regime number evaluators might significantly larger number items establish optimal tradeoff fraction good reviewers number items evaluated reviewer demonstrating surprising ability tolerate fraction good evaluators inverse exponential number items evaluated evaluator context leveraging techniques prism extract information specific demographics large mixed dataset regime seems especially significant techniques paper via local algorithms perspective also differ significantly previous approaches robust estimation rely geometric spectral structure general challenge developing algorithms estimators robust corruptions input data dates back early work led significant body work robust statistics explores number different models data corruptions largely focusses regime majority data much work orthogonal objectives paper refer reader surveys algorithm section describe simplified algorithm obtains claimed result theorem exception two key properties runtime algorithm rather algorithm require number verified samples inverse polynomial error parameter opposed nearly inverse linear dependence specified theorem algorithm theorem applies extension algorithm describe section overall structure algorithm reduce instance problem constraints sets variables instance problem constraints sets variables general true assignment might satisfy constraints derive sets variables though able leverage derived constraints discovered false begin providing intuition algorithm case section describe intuition reduction constraints sets variables constraints formally describe general algorithm section intuition restricting pessimistic constraints algorithm proceed iteratively goal iteration inspect constant number randomly sampled verified variable values return accurate guesses least constant fraction variables algorithm recursively iterate procedure remaining variables variables assigned guesses assignments last variables chosen arbitrarily begin consider setting every pair variables set allowable assignments set provides least two implications one form one form choice example assignment implications words least one value variable would imply value variable similarly hence fix variable consider implications derived sets ranges variables must assignment variable would imply values least variables refer assignment optimistic value assignment would immediately yield values least half remaining variables would done current iteration algorithm would recurse remaining variables assigned values first key idea algorithm assume variables take pessimistic values check assumption revealing true values random sample log variables values consistent pessimistic values conclude probability least least variables actually take pessimistic values hence simply output assignment however log random checks fails means found variable takes optimistic value hence one variable together constraint sets involve imply values least variables either case constant dependent number checks yielded accurate assignment least half variables simple algorithm case summarized following findassigments input set variables every pair set allowable assignments variables error parameter failure parameter output assignments variables exists variables without assignments let denote number remaining variables determine optimistic assignment would imply values least variables define variable pessimistic value opposite assignment randomly chosen variables verified assignments consider set log variables lie set variables consideration fewer log output fail verified assignments variables set agree pessimistic assignments assign variables pessimistic assignments otherwise must found variable whose verified assignment optimistic assignment assign values least variables accordingly pessimism way given algorithm case successful provided every pair variables least one forbidden assignment question reduce setting constraints sets variables setting constraints sets variables following trivial lemma key reduction lemma given set allowable assignments variables subset variables exists assignment variables would imply value rth variable proof consider tuple additional variable set allowable assignments restriction assignments variables contains possible assignments must case least one assignments unique value must assume otherwise would possible assignments restriction assignments tuple contain assignments assignment would vacuously imply value rth variable utility lemma variables considering possible additional variables exists assignment determines value least fraction variables hence designate optimistic assignment property assignment holds imply assignments least fraction remaining variables assume optimistic assignment allowed thereby reducing set allowable assignments variables size proceed inductively sense intermediate step algorithm considering sets variables allowable sets assignments considering may completely accurate verifying whether sets actually take optimistic assignments however variables actually takes values assignment either immediately imply values constant least fraction variables must subset larger tuple takes optimistic assignment two cases holds easily decided via querying values constant number random variables describe full algorithm following section basic algorithm structure algorithm described previous part takes form descending pass followed ascending pass descending pass iteratively turn constraints tuples constraints tuples tuples etc forbid optimistic assignments ensure rth level tuple allowable assignments descending phase terminates pessimistic conjectured assignments variables randomly check values discover inconsistencies conjectured values safely conclude conjectured values correct discovered inconsistencies begin ascending phase investigates checks discovered optimistic assignments one minor wrinkle trust fraction values appear implied optimistic assignment set variables implications might result forbidding optimistic assignment larger tuple nevertheless randomly check implications either verify accuracy implications found optimistic assignment tuple sense ascending phase either terminate upon satisfactorily verifying significant constant sized subset set output assignments found optimistic assignment tuple implications tuples based directly given set constraints valid assumption hence phase algorithm return assignments constant least fraction variables findassigments input set variables integer every tuple distinct variables set allowable assignments variables error parameter failure parameter output assignments least variables exists variables without assignments run descend set unassigned variables corresponding sets allowable assignments descend input set assignments variables ascendandverify set assigned values variable else every tuple create set assignments find optimistic assignment would determine least fraction variables existence assignment guaranteed lemma set run descend set corresponding sets assignments size ascend verify input proposed assignments variable set variables integer indicating size tuples whose constraints generated proposed assignments assignment provided implication access sets allowable assignments corresponding tuples size constant log log randomly sample verified variable assignments verified variable assignments agree proposed assignments permanently assign proposed assignments otherwise let denote variable whose assignment disagrees proposed assignment hence assignment together constraints tuples must imply least fraction variable assignments denote assignments new run ascendandverify new efficient algorithm variant basic algorithm described previous section hinges two observations first given rather consulting constraints determine optimistic assignment one determine assignment implies least fraction variable values high probability via sampling constant independent dependent number constraints note sampling look verified variable samples constraints consider formalize ability efficiently determine optimistic assignment via following subroutine following lemma characterizing performance find optimistic assignment input set variables ability query constraints ability find optimistic assignments tuples probability failure output optimistic assignment would probability least imply assignments least fraction variables via constraints define return constraint else select log variables uniformly random variables compute via recursive call indoptimisticassignment robf ailure define assignment lexicographically first assignment via constraints imply least fraction variables note assignment exists since least one possible assignment would imply value call optimistic assignment tuple store following two lemmas quantify performance algorithm first lemma characterizes probability failure proof follows immediately standard chernoff tail bounds lemma probability least optimistic assignment returned algorithm findoptimisticassignment input property least fraction variables assignment together constraint set would computed algorithm input tuple implies value variable proof letting denote true fraction variables whose assignments implied recall chosen based independent samples yielding empirical estimate standard tail bounds yield yielding lemma since log lemma given query access constraint sets tuples satisfying tuple algorithm findoptimisticassignment input probability failure returns runs log independent size variable set proof note computing calls log computations called error parameter smaller obtained via single query expanding recursion yields lemma second observation underpins efficient need determine optimistic assignments form constraints phase algorithm returns assignments constant fraction unassigned least suffices find single tuple takes optimistic assignment indeed tuple definition takes values imply assignments constant fraction remaining variables variables whose assignment implied assignment tuple value variable determined constant time consulting constraint observation clarified following algorithm adaptation algorithm described previous section finally highlight fact algorithm proceeds iteratively given initial set variables intermediate step algorithm let denote set variables yet output assignment algorithm terminate goal current step algorithm output assignments least fraction variables fraction assignments incorrect bounded log given bound fraction incorrect returned phase algorithm total fraction errors bounded assignments first bound error due arbitrary assignments log last variables benefit target accuracy increase decreases given verified samples drawn uniformly random check proposed assignment set target accuracy need least verified samples set ignoring logarithmic dependence probability failure guarantee number verified samples obtained verified samples using trick desired set need draw accuracy degrade decreases phase algorithm set log verified samples opposed samples would required fixed target error rate rounds algorithm efficient find assignments input set variables integer every set allowable assignments error parameter probability failure output set assignments set log least unassigned variables let denote set unassigned variables let log denote target accuracy round set log take verified samples revealing planted assignment values variables let denote subset variables set let denote verified assignment variable output fail determine via findoptimisticassignments failure parameter every variable compute output assignment otherwise let denote variable efficientascend run efficient ascend input set variables integer tuple verified assignments tuple parameter output output subset variables set output fail take verified samples let denote intersection set variables verified assignments denoting verified assignment variable determine via call findoptimisticassignment ailurep rob let denote subset variables constraint output fail together implies value holds every variable compute output assignment implied otherwise let denote variable implies assignment run efficientascend proposition algorithm efficientfindassignments run error parameter probability failure following properties algorithm require log verified samples drawn uniformly random set variables probability least algorithm output assignments variable fraction assignments disagree planted assignment algorithm runs time hidden constant function proof high level outline execution algorithm efficientfindassignments step outer loop assignment least fraction remaining unassigned variables output continues point remaining variables assigned arbitrary labels algorithm terminates hence log iterations loop iteration conducted unassigned variable set goal return assignn ments fraction returned assignments incorrect log total number initial variables provided accuracy goals met step algorithm overall fraction errors bounded log first term errors due arbitrary assignment remaining variables additionally number verified samples required iteration log hence total number verified samples across log iterations bounded log claimed analyze run loop efficientfindassignments recursive calls efficientascend high level recursive call efficientascend either assignment least fraction remaining unassigned variables returned via implications verified optimistic assignment tuple found tuple verified assignments variables assignment optimistic assignment case subsequent call efficientascend considers strictly larger tuple bound runtime algorithm note run algorithm requires constant time dependent independent number variables point algorithm assignment output step efficientascend point algorithm computational expense assignment constant fraction least remaining variables output algorithm repeated remaining unassigned variables hence overall runtime algorithm linear number variables bound probability given run loop fails successfully output assigny ment least variables meets target accuracy log leverage union bound number standard chernoff tail bounds first note probability efficientfindassignments outputs fail step given round algorithm bounded probability sum random variables hence probability bounded specified efficientfindassignments bound number exp calls efficientascend bounds number runs loop given probability assignment output step efficientfindassignments meet target accuracy log bounded remaining probability failure stems execution efficientascend algorithm failure stem three different issues constant number constraints computed via findoptimisticassignment prior step efficientascend erroneous fail imply desired fraction assignments probability bounded sufficient guarantee every optimistic set computed execution algorithm accurate implies desired fraction assignments aside constraints computed assignment output efficientascend step efficientfindassignments efficientascend never output fail step constraints corresponding constraints satisfied assumption final potential failure mode algorithm step efficientascend random set verified assignments insufficiently large verify target accuracy given potential set assignments implied optimistic assignment via given assignment tuple optimistic guaranteed validity findoptimisticassignments described probability failure also trivial application standard chernoff bounds guaranteeing step efficientascend deviates lower bound expectation random variable factor union bound probabilities failure runs efficientascend algorithm yields desired proposition future work work shows possible tolerate fraction good data inverse exponential sparsity datapoint number evaluations submitted per reviewer provided number sufficiently large ensure set items evaluated significant number good reviewers algorithm runs time linear number items review provided ability query summary statistics set reviewers evaluated given sets items uses constant number verified reviews independent total number items review depends inverse linearly desired error logarithmic factors one natural question prompted results provide efficient algorithms regime poly number reviewers linear number items reviewed uses constant dependent verified reviews possible achieve polylog number reviewers linear grows significantly slowly require leveraging constant number verified reviews end algorithm ever considers implications proposed assignments assignment set variables considered optimistic directly implies values significant fraction variables easy imagine extending definition also consider longer chains implication perhaps specific assignment variables would imply values additional variables turn would imply values variables etc indeed basic setting approach realized yield algorithm requires constraints random subset size opposed constraints assumed work computational perspective seems unlikely approach could pushed yield efficient algorithm regime fewer sets variables nontrivial constraints indeed even random instances sat planted solution efficient algorithms threshold elusive see example recent related work random csps planted assignments purely information theoretic picture entirely clear either contrast random csps setting complicated adversarial nature constraints placed even setting tuples chosen random adversary chooses constraints place random tuples immediately clear analyze extent implications propagate second difficulty goal setting find satisfying assignment find something close specific planted assignment results imply setting consider constant number solution clusters seems interesting investigate extent holds csps fewer constraints perhaps constraints threshold constant number clusters references kush bhatia prateek jain purushottam kar robust regression via hard thresholding advances neural information processing systems pages emmanuel candes yaniv plan matrix completion noise proceedings ieee charikar steinhardt valiant learning untrusted data symposium theory computing appear ilias diakonikolas gautam kamath daniel kane jerry ankur moitra alistair stewart robust estimators high dimensions without computational intractability foundations computer science focs ieee annual symposium pages ieee vitaly feldman perkins santosh vempala complexity random satisfiability problems planted solutions proceedings annual acm symposium theory computing pages acm frank hampel elvezio ronchetti peter rousseeuw werner stahel robust statistics approach based influence functions volume john wiley sons david haussler sphere packing numbers subsets boolean bounded vapnikchervonenkis dimension journal combinatorial theory series peter huber robust statistics springer raghunandan keshavan andrea montanari sewoong matrix completion entries ieee transactions information theory kevin lai anup rao santosh vempala agnostic estimation mean covariance foundations computer science focs ieee annual symposium pages ieee prasad raghavendra satish rao tselil schramm strongly refuting random csps spectral threshold arxiv preprint jacob steinhardt gregory valiant moses charikar avoiding imposters delinquents adversarial crowdsourcing peer prediction advances neural information processing systems pages john tukey survey sampling contaminated distributions contributions probability statistics
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diverse benchmark dataset facial beauty prediction beauty prediction fbp significant visual recognition problem make assessment facial attractiveness consistent human perception tackle problem various models especially deep learning techniques introduced benchmark dataset become one essential elements achieve fbp previous works formulated recognition facial beauty specific supervised learning problem classification regression ranking indicates fbp intrinsically computation problem multiple paradigms however fbp benchmark datasets built specific computation constrains limits performance flexibility computational model trained dataset paper argue fbp computation problem propose new diverse benchmark dataset called achieve facial beauty prediction dataset totally frontal faces diverse properties ages diverse labels face landmarks beauty scores within beauty score distribution allows different computational models different fbp paradigms facial beauty model evaluated dataset fbp using different combinations feature predictor various deep learning methods results indicates improvement fbp potential applications based ntroduction assessing facial beauty seems natural human absolute definition facial beauty remains elusive recently facial beauty prediction fbp attracted evergrowing interest pattern recognition machining learning communities aims achieve automatic facial attractiveness assessment computational model application potential facial makeup image retrieval aesthetic surgery face beautification computational perspective fbp still challenging problem involved formulation visual work supported part national natural science foundation china nsfc grant grant part gdstp grant grant grant part national key research development plan china grant part fundamental research funds central universities corresponding author lianwen jin beau jan lingyu liang luojun lin lianwen jin duorui xie mengru south china university technology guangzhou china lianglysky dataset download url https asia female asia male caucasian female caucasian male fig images different facial properties beauty scores proposed benchmark dataset dataset download url shown title representation predictor abstract concept facial beauty tackle problem various models introduced fbp one line works follows classic pattern recognition process constructs fbp system using combination features shallow predictors related feature derived visual recognition includes geometric features like geometric ratios landmark distances texture features like features shallow fbp predictoris trained extracted feature statistical manner another line works advanced reviving neural networks especially deep learning hierarchial structure deep learning model allows build fbp system automatically learns representation predictor facial beauty simultaneously data many works indicate fbp table epresentative databases facial eauty rediction database image num beauty class face property face landmarks publicly available eisenthal caucasian female chen unknown asian gunes female fan generated female redi multiple sampled ava asian female based deep learning superior shallow predictors facial feature current fbp models makes benchmark dataset become one essential elements fbp many works benchmark datasets involved fbp datasets focus specific problem specific computation constrains shown table yan regarded fbp ranking problem proposed dataset images gathered social networks fan focused geometry analysis fbp proposed dataset containing computergenerated faces different facial proportions northeast china database shanghai database database ava database face subset ava database databases involved fbp northeast china shanghai database limited geometric facial beauty analysis without attractiveness ratings database focuses fbp ava database originally designed aesthetic analysis entire images facial attractiveness previous work xie published benchmark dataset led many fbp models especially hierarchial fbp models deep learning despite prevalent usage contains asian female faces may limit performance datademanded model fbp find fbp formulated recognition facial beauty specific supervised learning problem classification regression ranking indicates fbp intrinsically computation problem multiple paradigms previous databases built specific computation constrains would limit performance flexibility computational model trained dataset difficult compare different models derived dataset specific computation paradigm therefore paper argues fbp computation problem proposes new diverse benchmark dataset called achieve facial beauty prediction dataset totally frontal faces diverse properties ages diverse labels face landmark beauty score beauty score distribution allows different computational model different fbp paradigms facial beauty model furthermore diverse faces beauty scores gathered different labelers lead many interesting research facial beauty analysis personalized fbp automatic face beautification shallow prediction model feature deep learning models evaluated dataset results indicates improvement fbp potential applications main contributions paper summarized following dataset propose new benchmark dataset totally frontal faces diverse properties diverse labels allows construction fbp models different paradigms benchmark analysis analyze samples score labels labelers facial landmarks statistically visualization data illustrates properties facial beauty prediction evaluation shallow prediction model feature deep learning models trained evaluation results indicates improvement fbp based proposed dataset better diversity onstruction dataset face images collection dataset contains frontal unoccluded faces aged neutral expression divided four subsets different races gender including asian females asian males caucasian females caucasian males images collected internet portions asian faces datatang caucasian faces adult database table utlier umber ortion eauty cores aucasian female aucasian male sian female sian male subset total score num outlier num outlier portion score distribution score distribution score distribution score distribution fig gaussian fitting yellow curve piecewise fitting red blue curve visualization beauty score distribution caucasian female caucasian male asian female asian male fig distribution standard deviations caucasian female asian female caucasian male asian male respectively box figure box figure box figure box figure facial beauty scores facial landmarks images labeled beauty scores ranging totally volunteers aged average beauty score means attractive developed gui system obtain facial beauty scores labeling system deployed ali cloud labeling tasks distributed volunteer crowdsourcing manners four subset asian caucasian labeled separately face subset randomly shown volunteer volunteer asked select beauty scores within face reduce variance labeling process faces recurred randomly correlation coefficient two beauty score faces less volunteer would asked rate face decide final score allow geometric analysis facial beauty facial landmarks located significant facial components images eyes eyebrows nose mouth gui landmarks location system developed original location landmarks initialized active shape model asm trained dataset detected landmarks asm modified manually volunteers ensure accuracy fig box figures standard deviations caucasian female asian female caucasian male asian male respectively iii enchmark nalysis made benchmark analysis beauty scores labelers face landmarks different gender races including asian female asian male caucasian female caucasian male distribution beauty scores visualize distribution beauty scores respectively obtain better visualization preprocess data filter outliers beauty scores regard average score labelers score specific labeler face differs score treated outlier removed distribution visualization table iii orrelation oefficients ale emale abelers eauty core aucasian female aucasian male sian female sian male female labelers male labelers labelers faces distributions original data preprocessed data mostly similar therefore visualized score distribution using preprocessed data four subset respectively two distribution fitting schemes used one gaussian fitting yellow curve piecewise fitting red blue curve shown fig results indicates beauty scores four subset approximately fitted mixed distribution model two gaussian components standard deviation beauty scores female labeler male labeler labeler faces calculate standard deviation scores gathered different labelers illustrate results histogram fig box figure fig observe distribution standard deviations similar gaussian distribution standard deviations within reasonable range correlation labelers fig correlation coefficient male female labelers beauty score faces subsection investigate correlation male female asian labelers beauty scores shown table iii fig observe correlation asian faces persistently larger caucasian consistent psychological research human better facial beauty perception faces race pca analysis facial geometry sigma mean sigma pca analysis visualize face landmarks dataset using principle component analysis pca fig illustrate mean five first principle component facial geometry asian female asian male landmarks data caucasian share similar distribution asian faces observe face shape one main component influence face geometry beauty consistent related psychological research previous works fbp valuation via rafted eature hallow redictor section evaluate using feature shallow predictor next section introduce deep learning model achieve fbp geometric feature shallow predictor extract ratio feature vector faces formulate fbp based different regression mod pca analysis els linear regression gaussian regression fig pca analysis face landmarks asian female asian support vector regression svr comparison performed caucasian asian male subsets performance different model measured using pearson correlation coefficient maximum abthe number portion outliers four subset solute error mae root mean square error rmse listed table small portion outlier indicate folds cross validation results listed table reliability labeling process beauty score table regarded baseline geometric since outlier portion beauty score tiny analysis fbp sigma mean sigma table facial beauty prediction using geometric feature shallow models subsets different races gender mae rmse mae rmse asian female svr caucasian female svr asian male svr caucasian male svr table facial beauty prediction using eometric eature table vii omparison lex measurement mae rmse fold cross validation alexnet mae alexnet rmse alexnet average average average shallow models whole dataset linear regression gaussian regression svr mae rmse table facial beauty prediction using abor feature two sampling scheme whole dataset mae rmse svr svr fig two sampling schemes extract appearance feature fbp left one method right one unisamplepoint method appearance feature shallow predictor extract gabor feature maps every original image five directions eight angles obtain appearance feature fbp using two different sampling schemes extracts component gabor feature maps following sample gabor feature maps obtain feature vector shown right fig use obtain feature vector shown left fig finally use pca reduce extracted feature dimension train predictor results appearancebased shallow predictors data shown table table viii omparison lex measurement mae rmse training testing mae rmse alexnet fbp valuation via eep redictor evaluate three recently proposed cnn models different structures fbp including alexnet cnn models trained initializing weights using networks imagenet dataset evaluation performed two different experiment settings following models trained tested using cross validation means fold containing samples images accuracy fold average fold shown table vii models trained using samples images tested rest images results shown table viii results illustrates deepest model obtains best performance comparing alexnet experiment setting observed deep cnn model superior shallow predictor geometric feature table appearance feature table indicates effectiveness powerfulness feature learning deep model fbp comparing results table vii table viii also find accuracy cross validation slightly higher results split training testing one reasons may due amounts diversity training samples since cross validation use samples train models observation indicates data augmentation techniques may improve performance deep fbp model merits exploring future onclusion paper introduce new diverse benchmark dataset called achieve facial beauty prediction dataset totally frontal faces diverse properties ages diverse labels face landmarks beauty scores within beauty score distribution benchmark analysis made beauty scores landmarks visualization data shows statistical properties dataset since designed adapted different fbp models different tasks like appearancebased model facial beauty evaluated using different combinations feature predictor deep learning models results indicates reliability dataset eferences xie liang jin benchmark dataset facial beauty perception proc ieee smc liang xie jin lin scattering convolution networks facial beauty prediction proc ieee icip jin liang feng xie mao facial attractiveness prediction using psychologically inspired convolutional neural network proc ieee icassp liang jin liu label propagation mobile facial enhancement cloud ieee trans circuits systems video technology vol liang jin facial skin beautification using adaptive mask ieee trans cybernetics vol chao liu shu yan deep face beautification proc acm international conference multimedia leyvand dror lischinski enhancement facial attractiveness acm trans gunes survey perception computation human beauty proc laurentini bottino computer analysis face beauty survey comput vision image vol zhang chen computer models facial beauty analysis springer international publishing switzerland liu xing liu zhou yan wow beautiful today acm transactions multimedia computing communications applications vol scherbaum ritschel hullin blanz seidel facial makeup comput graph forum vol computational facial attractiveness prediction aestheticsaware features neurocomputing vol lecun bengio hinton deep learning nature vol eisenthal dror ruppin facial attractiveness beauty machine neural computation vol murray marchesotti perronnin ava database aesthetic visual analysis proc cvpr laurentini bottino computer analysis face beauty survey computer vision image understanding vol stephen law smith stirrat perrett facial skin coloration affects perceived health human faces international journal primatology vol kagian dror leyvand ruppin humanlike predictor facial attractiveness proc nips mao jin automatic classification chinese female facial beauty using support vector machine proc ieee smc zhang zhao chen quantitative analysis human facial beauty using geometric features pattern recognition vol chen zhang benchmark geometric facial beauty study medical biometrics fan chau wan zhai lau prediction facial attractiveness facial proportions pattern recognition vol yan ordinal regression fully automatic facial beauty assessment neurocomputing altwaijry belongie relative ranking facial attractiveness ieee workshop wacv chen mao jin novel method evaluating facial attractiveness ieee proc icalip chiang lin huang wan cluster assessment facial attractiveness using fuzzy neural network classifier based moir features pattern recognition vol kalayci ekenel gunes automatic analysis facial attractiveness video proc icip gan zhai liu deep learning facial beauty prediction neurocomputing wang shao attractive beauty prediction encoders robust late fusion acm multimedia ren geng sense beauty label distribution learning proc ijcai white eden maire automatic prediction human attractiveness berkeley vol gray gong predicting facial beauty without landmarks proc eccv redi rasiwasia aggarwal jaimes beauty capturing faces rating quality digital portraits ieee workshops afgr gunes piccardi assessing facial beauty proportion analysis image processing supervised learning international journal studies vol whitehill movellan personalized facial attractiveness prediction proc afgr fan liu fan guo samal wan stan label distribution based facial attractiveness computation deep residual learning ieee transactions multimedia online davis lazebnik analysis human attractiveness using manifold kernel regression proc icip datatang url http bainbridge isola oliva intrinsic memorability face journal experimental psychology general vol krizhevsky sutskever hinton imagenet classification deep convolutional neural networks proc nips zhang ren sun deep residual learning image recognition proc cvpr xie girshick aggregated residual transformations deep neural networks arxiv preprint
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degrees freedom broadcast channel hybrid csi transmitter receivers sep mohamed fadel student member ieee aria nosratinia fellow ieee abstract general different links broadcast channel may experience different fading dynamics potentially unequal hybrid channel state information csi conditions faster fading shorter fading block length often link needs trained estimated receiver likely csi stale unavailable transmitter disparity link fading dynamics presence csi limitations modeled broadcast channel link fading block lengths well dissimilar link conditions paper investigates miso broadcast channel receivers experience longer coherence intervals static receivers csir receivers experience shorter coherence intervals dynamic receivers enjoy free csir consider variety csit conditions mentioned model including csit delayed csit hybrid csit investigate degrees freedom region employ interference alignment beamforming along product superposition allows simultaneous transmission pilots data different receivers outer bounds employ extremal entropy inequality well bounding performance discrete memoryless multiuser multilevel broadcast channel several cases inner outer bounds established either partially meet gap diminishes increasing coherence times index terms broadcast channel channel state information coherence time coherence diversity degrees freedom fading channel multilevel broadcast channel product superposition authors department electrical engineering university texas dallas richardson usa aria work presented part ieee international symposium information theory isit germany june march draft ntroduction performance broadcast channel depends channel dynamics well availability quality channel state information csi two ends link two issues csi channel dynamics practically related faster fading often channel needs training thus consuming channel resources slow fading link requires infrequent training therefore slow fading models often assume csir available due cost training small amortized time practice broadcast channel links may fade faster slower others recently shown degrees freedom broadcast channel affected disparity link fading speeds existing studies focused simple uniform csi conditions neither csit csir available user paper studies broadcast channel links experience disparate fading conditions well hybrid csi conditions review relevant literature follows perfect instantaneous csi degrees freedom broadcast channel increase minimum transmit antennas total number receivers antennas however due nature channel feedback impairments perfect instantaneous csi csit may available also csi csir assumed channels broadcast channel perfect csir investigated variety csit conditions including imperfect delayed csit absence csit huang vaze varanasi showed degrees freedom collapse since receivers stochastically equivalent respect transmitter miso broadcast channel lapidoth conjectured long precision csit finite degrees freedom collapse unity conjecture recently settled positive davoodi jafar moreover miso broadcast channel perfect delayed csit tse showed using retrospective interference alignment degrees freedom number transmit antennas also number receivers scenario mixed csit investigated transmitter partial knowledge current channel addition delayed csi potential variation quality feedback links led model hybrid march draft csit csit respect different links may identical miso broadcast channel perfect csit receivers delayed others studied tandon amuru davoodi jafar showed miso broadcast channel perfect csit one user csit degrees freedom collapse unity tandon considered miso broadcast channel alternating hybrid csit perfect delayed csit respect different receivers mentioned earlier investigation broadcast channels unequal link fading dynamics fairly recent achievable degrees freedom region one static one dynamic receiver given via product superposition producing gain known coherence diversity coherence diversity gain investigated broadcast channel neither csit csir also broadcast channel investigated receivers mimo fading links experience nonidentical spacial correlation paper consider multiuser model hybrid csir scenario group receivers denoted static receivers assumed csir another group shorter link coherence time denoted dynamic receivers free csir consider model variety csit conditions including csit delayed csit two hybrid csit scenarios conditions analyze degrees freedom region new tools introduced inner outer bounds derived partially meet cases results paper cataloged follows absence csit outer bound degrees freedom region produced via bounding rates discrete memoryless multilevel broadcast channel applying extremal entropy inequality achievable degrees freedom region meets outer bound limiting case coherence times static dynamic receivers delayed csit use outdated csi model used tse fading assuming global csir nodes noting model uniform csir produced technique alignment utilize outdated csit merge together product superposition reuse pilots dynamic receivers purpose transmission static receivers moreover develop outer bound suitable channels different coherence times appropriately enhancing march draft static dynamic fig broadcast channel multiple static multiple dynamic users channel broadcast channel applying extremal entropy inequality one static one dynamic receiver achievable degrees freedom partially meet outer bound furthermore gap decreases dynamic receiver coherence time hybrid csit analyze two conditions first consider perfect csit static receivers csit respect dynamic receivers achievable degrees freedom case obtained using product superposition dynamic receiver pilots reused beamforming static receivers avoid interference second consider perfect csit respect static receivers delayed csit respect dynamic receivers achievable transmission scheme proposed via combination beamforming interference alignment product superposition methodologies outer bounds two cases based constructing enhanced physically degraded channel applying extremal entropy inequality one static receiver perfect csit one dynamic receiver delayed csit gap achievable outer sum degrees freedom ystem odel consider broadcast channel multiple receivers transmitter equipped antennas expressions receiver user employed without distinction throughout paper indicating receiving terminals broadcast channel channels users modeled rayleigh block fading channel coefficients remain constant block change independently across blocks shown fig users partitioned two sets based channel availability length coherence interval one set dynamic users another set static users former contains march draft table otation static users dynamic users miso channel gains received signals continuous dmc receive variables transmission rates number users messages degrees freedom coherence time general variables transmit signal ratio auxiliary random variables set channel gains vertex degrees freedom region canonical coordinate vector dynamic users coherence time free latter contains static users coherence time perfect instantaneous csir consider transmitter equipped antennas number dynamic static users received signals static user dynamic user respectively time instant cnt transmitted signal denote corresponding additive gaussian noise users cnt cnt denote channels static user dynamic user whose coefficients stay time instances means cost knowing csi receiver channel estimation ignored march draft respectively distributions globally known transmitter csir value available instantaneously perfectly static user furthermore static user obtains outdated version dynamic users channels also dynamic user obtains outdated version static users channel completely stale csit user take one following perfect csit channel vectors available transmitter instantaneously perfectly delayed csit channel vectors available transmitter change independently following block completely stale csit channel vectors known transmitter consider broadcast channel private messages users common messages specifically assume independent messages associated rates communicated transmitter static user dynamic user respectively ratio degrees freedom static dynamic users achieving rates defined log lim log lim degrees freedom region defined lim log log lim capacity region ratio sum degrees freedom defined csum log dsum lim csum max also coherence times channels globally known transmitter users march draft fig discrete memoryless multiuser multilevel broadcast channel sequel study degrees freedom miso broadcast channel different csit scenarios could perfect delayed csit iii csit sers section study broadcast channel defined section csit users particular give outer achievable degrees freedom regions section section respectively outer degrees freedom region based construction outer bound rates multiuser multilevel discrete memoryless channel given section multiuser multilevel broadcast channel multilevel broadcast channel introduced borade broadcast discrete memoryless broadcast channel two users degraded respect capacity channel degraded message sets established nair gamal study multiuser multilevel broadcast channel two sets degraded users see fig one set contains users received signal user set contains users received signal user therefore march draft form two markov chains consider broadcast channel private messages common message outer bound multilevel broadcast channel given following theorem theorem rate region multilevel broadcast channel two sets degraded users outer bounded intersection pmf forms markov chains proof see appendix remark theorem extension outer bound theorem two users recovers bound march draft remark multiuser multilevel broadcast channel characterized establish capacity degraded message sets appendix one common message communicated receivers one private message communicated one receiver outer degrees freedom region sequel give outer bound degrees freedom broadcast channel defined section csit users outer bound development depends results theorem section theorem outer bound degrees freedom region fading broadcast channel characterized without csit otherwise offset two coherence intervals proof equations outer bounds broadcast channel whose users either homogeneously static homogeneously dynamic remainder proof dedicated establishing enhance channel giving users global csir follows directly csit channel belongs class multiuser multilevel broadcast channels section use two outer bounds developed multilevel broadcast channels generate two degrees freedom bounds merge get desired result begin outer bound described combine equations obtain partial bounds static dynamic receivers march draft log log set channel vectors follows chain rule log follows since received signals static users statistics also using theorem log log log follows chain rule follows since statistics follows since log log define received signal user time instance obtain bound rates log log log log log log log log march max log log log draft log follow chain rule conditioning increase differential entropy follows extremal entropy inequality order bound use specialization lemma follows lemma consider two random matrices covariance matrix log log log min min proof lemma omitted directly follows lemma lemma yields max following outer bound degrees freedom repeat exercise bounding sum rates deriving degrees freedom time starting following bounding steps parallel adding yields outer bound completing proof theorem achievable degrees freedom region theorem fading broadcast channel described achieve following degrees freedom without csit proof achievable scheme uses product superposition transmitter uses one antenna send super symbol two users one dynamic one static symbol intended static user march draft pilot super symbol intended dynamic user since degrees freedom analysis insensitive additive noise omit noise component following hxs hxs dynamic user estimates equivalent channel first time instance decodes coherently based channel estimate static receiver utilizes received signal first time instance gxs knowing channel gain static receiver decode achievable degrees freedom two users proceed prove degrees freedom region characterized achieved via combination product superposition strategies outlined strategies clarity exposition refer describes degrees freedom constraints dynamic receivers bound coherent bound degrees freedom restricts intersection coherent bound orthant orthant vertices bound hyperplane partitions simplex vertices one side bound therefore intersection simplex bound produces polytope illustration see fig showing degrees freedom two static users fig one static user verify vertices achieved either strategy via product superposition strategy vertices corresponding transmission static user achieving one degree freedom verified simple counting exercise involving number edges simplex cross bound march draft bound bound coherent bound fig achievable degrees freedom region one dynamic two static users coherent bound fig achievable degrees freedom region one static two dynamic users vertices corresponding transmission dynamic user achieving degrees freedom vertices corresponding product superposition applied possible pairs static dynamic users achieving degrees freedom one static user degrees freedom one dynamic user one trivial vertex origin corresponding transmission achieving zero degrees freedom users hence number vertices completes achievability proof theorem remark static dynamic users coherence time inner outer bounds degrees freedom coincide case degrees freedom optimal serve time one dynamic one static elayed csit users delayed csit transmitter knows channel gain longer valid condition also known outdated csit begin proving inner outer bounds march draft transmitting static users dynamic users one static one dynamic user synthesize collection bounds overall degrees freedom region transmission static users theorem degrees freedom region fading broadcast channel characterized delayed csit static users dynamic users proof special case fast fading discussed tse achievability established retrospective interference alignment aligns interference using outdated csit converse proved generating improved channel without csit tight degrees freedom region tdma according results achievability established employing retrospective interference alignment presented super symbols length converse proved following procedures generate improved channel without csit identical coherence intervals length according results tdma tight degrees freedom region improved channel transmission dynamic users theorem fading broadcast channel characterized delayed csit dynamic users static users achieve degrees freedom outer bound degrees freedom region proof achievability part proved follows beginning super symbol pilots sent channel estimation retrospective interference alignment super symbols employed remaining instances achieve march draft converse part proved giving users global csir applying theorem moreover bound dynamic user proved follows single user delayed csit feedback increase capacity consequently assumption delayed csit removed hence bound dynamic user delayed csit bound without csit transmission one static one dynamic user theorem fading broadcast channel characterized delayed csit one static one dynamic user achieve following degrees freedom furthermore achievable degrees freedom region convex hull degrees freedom pairs proof section product superposition achieves pair require csit two users remainder proof dedicated achievability pair provide transmission scheme based retrospective interference alignment along product superposition transmitter first emits intended static user occupies time instances following structure diagonal matrix contain symbols intended static user components march draft static user definition knows decode yields degrees freedom remaining involve observations unknowns require independent observations reliable decoding components equivalent channel estimated dynamic user dynamic user saves interference cancellation upcoming steps transmitter sends second super symbol intended dynamic user diagonal includes independent symbols intended static user contains independent symbols intended dynamic user components equivalent channel estimated dynamic user dynamic user saves includes unknowns hence additional components independent observations observations needed decode equivalent channel estimated static user static user saves upcoming steps knowing static user achieves degrees freedom decoding march draft transmitter emits third super symbol consisting linear combination signals generated first second super symbols diagonal contains independent symbols intended static user hence static user achieves degrees freedom static user cancels saved second super symbol obtains includes additional independent observations needed decoding therefore static user achieves degrees freedom dynamic user estimates equivalent channel cancels saved contains additional observations needed first super symbol obtains decoding hence dynamic user achieves degrees freedom aggregate time instants static dynamic user achieve degrees freedom completes proof theorem theorem outer bound degrees freedom region fading broadcast channel characterized one static one dynamic user delayed csit proof inequality represents outer bound prove bound follows enhance original channel giving users global csir addition channel output dynamic user given static user therefore channel outputs time instant static user dynamic user enhanced channel physically degraded hence removing delayed csit reduce capacity also march draft auxiliary random variable forms markov chain therefore log log log log max log log log log log follows since log log follows extremal entropy inequality follows lemma hence bound proved similar argument role two users reversed leads bound remark inner outer bounds obtained case partially meet gap diminishing coherence time dynamic user shown fig fig respectively transmission arbitrary number static dynamic users theorem fading broadcast channel characterized delayed csit achieve multiuser degrees freedom characterized vectors canonical coordinate vector convex hull characterized achievable degrees freedom region proof achievability proved section via multiuser transmission static users achievability proved section via transmission pair march draft achievable region outer region fig one static one dynamic delayed csit achievable region outer region fig one static one dynamic delayed csit march draft show achievability via retrospective interference alignment along product superposition super symbol length consider following transmission diagonal includes independent symbols intended static user super symbol containing independent symbols intended dynamic users according retrospective interference alignment therefore static user decodes thus time instants static user achieves degrees freedom dynamic users achieve hence achieved theorem outer bound degrees freedom fading broadcast channel characterized delayed csit proof inequalities represent bounds static dynamic users respectively remainder proof dedicated establishing bounds enhance channel providing global csir well allowing full cooperation among static users full cooperation among dynamic users enhanced channel equivalent broadcast channel two users one static equipped antennas one dynamic equipped antennas define received signals static dynamic respectively enhanced channel enhance channel giving static user generating physically degraded channel since forms markov chain feedback including delayed csit effect capacity therefore remove consideration subsequently utilize outer bound march draft therefore applying extremal entropy inequality lemma log log log therefore bound proved similarly prove bound using steps switching roles two users enhanced channel ybrid csit erfect csit tatic sers csit dynamic sers theorem fading broadcast channel characterized perfect csit static users csit dynamic users achieve following multiuser degrees freedom therefore convex hull also achievable proof achieved inverting channels static users transmitter every static user achieves one degree freedom achieved using product superposition along channel inversion follows transmitted signal instants symbol intended static user contain independent symbols intended dynamic user static users receive signal first time instant achieving one degrees freedom dynamic user estimates equivalent channel first time instant decodes remaining time instants march draft theorem outer bound degrees freedom fading broadcast channel characterized perfect csit static users csit dynamic users proof inequalities represent bounds static users outer bound dynamic users established remains prove follows enhance channel giving global csir users allowing full cooperation static users gives rise equivalent static user antennas receiving equivalent channel noise point system csit available respect one user others bound performance system another similar system csit use local statistical equivalence property developed used first draw according distribution independent enhance channel providing static receiver receivers provide transmitter csit respect according log therefore remove enhanced channel without reducing degrees freedom new equivalent channel one user antennas receiving users receiving csit enhanced channel form multilevel broadcast channel studied section hence using theorem enhanced channel removal transmitter receivers still share information random variable independent remaining transmit receive variables march draft dynamic receiver received signals distribution following bounding steps parallel log log therefore log log log log log log last inequality follows applying extremal entropy inequality lemma concludes proof bound ybrid csit erfect csit tatic sers elayed csit dynamic sers begin inner outer bounds one static one dynamic user extend result multiple users transmitter knows channel static users perfectly instantaneously outdated version channel dynamic users transmitting one static one dynamic user theorem fading broadcast channel characterized one static one dynamic user perfect csit static user delayed csit dynamic user achievable degrees freedom region convex hull vectors march draft proof degrees freedom achieved product superposition discussed section iii without csit proceed prove achievability consider complex matrix containing symbols intended static user intended dynamic user beamforming vector addition define using components transmitter constructs transmits length whose value time note carry information either user serves pilot received super symbol static user received super symbol dynamic user bvt dynamic user estimates equivalent channel received value first time instant remaining terms include symbols intended dynamic user plus interference whose cancellation subject next step transmitter next sends second super symbol length symbol intended static user hence dynamic user estimates equivalent channel first time instant acquires interference therefore using dynamic user solves achieving degrees freedom furthermore static user solves achieving one degree freedom also uses solve achieving degrees freedom march draft summary instants static user achieves degrees freedom dynamic user achieves degrees freedom shows achievability theorem fading broadcast channel characterized one static one dynamic user perfect csit static user delayed csit dynamic user outer bound degrees freedom region proof inequalities represent outer bounds remains prove outer bound follows enhance channel giving global csir users also give static user enhanced channel physically degraded static user dynamic user physically degraded channel causal feedback including delayed csit affect capacity remove delayed csit respect dynamic user use another enhancement motivation remove remaining csit noncausal respect static user accomplished similar theorem via local statistical equivalence property following manner create channel noise distribution independently true channel noise signal genie give static receiver receivers shown log therefore remove enhanced channel without reducing degrees freedom enhanced channel still physically degraded therefore log auxiliary random variable forms markov chain therefore log march draft achievable region outer region fig one static one dynamic user hybrid csit log log last inequality follows extremal entropy inequality lemma concludes proof bound remark broadcast channel hybrid csit achievable sum degrees freedom dsum outer bound sum degrees freedom dsum gap decreases dynamic user coherence time see fig multiple static dynamic users theorem fading broadcast channel characterized perfect csit static users delayed csit dynamic users achieve following degrees freedom march draft achievable region outer region fig one static one dynamic user hybrid csit achievable region consists convex hull vectors proof achieved inverting channel static users transmitter providing one degree freedom per static user achievability established section proved section without csit dynamic user remains achievable delayed csit achieved retrospective interference alignment along product superposition follows transmitted signal instants contains independent symbols intended static users sent inverting channels static users therefore first time instants static user receives signal achieves degree freedom furthermore dynamic users estimate equivalent channels remaining time instants dynamic receiver march draft obtains coherent observations transmit symbols combined according retrospective interference alignment techniques accordingly dynamic receiver achieves degrees freedom theorem outer bound degrees freedom region fading broadcast channel characterized perfect csit static users delayed csit dynamic users proof inequalities represent outer bounds static dynamic users respectively according theorem represents outer bound dynamic users remains prove follows original channel enhanced giving users global csir furthermore assume full cooperation static users dynamic users resulting enhanced channel broadcast channel two users one static user equipped antennas received signal channel noise noise one dynamic user equipped antennas received signal channel noise enhance channel giving static user constructing physically degraded channel enhanced channel static receiver equipped antennas received signal channel noise since causal feedback including delayed csit affect capacity physically degraded channel delayed csit dynamic receiver removed use another enhancement motivation remove remaining csit respect static user create artificial channel noise distribution independent signal march draft genie give static receiver receivers shown log therefore remove enhanced channel without reducing degrees freedom enhanced channel physically degraded without csit therefore hence log log log last inequality follows extremal entropy inequality lemma since log log concludes proof bound vii onclusion multiuser broadcast channel studied receivers experience longer coherence intervals csir receivers experience shorter coherence interval enjoy free csir degrees freedom studied delayed csit hybrid csit csit among techniques employed interference alignment beamforming along product superposition inner bounds outer bounds involved bounding rate region multiuser discrete memoryless multilevel broadcast channel highlights results one static one dynamic user delayed csit achievable degrees freedom region partially meets outer bound one static user perfect csit one dynamic user delayed csit gap achievable outer sum degrees freedom inversely proportional dynamic user coherence time considered csi conditions inner outer bounds also found arbitrary number users results conclude broadcast channel coherence diversity delivers gains distinct augment gains beamforming interference alignment march draft authors anticipate tools results paper helpful future studies hybrid networks ppendix roof heorem recall messages users respectively enhance channel assuming user knows messages user knows messages using fano inequality chain rule data processing inequality bound rates static user denotes received signal user time instant forms markov chain rate static user bounded march draft similarly nri define leads markov chain using chain rule sum identity obtain bound introducing auxiliary random variable defining march draft establish similarly follow steps prove switching role two sets variables completes proof theorem ppendix ultilevel roadcast hannel egraded essage ets study capacity multiuser multilevel broadcast channel characterized degraded message sets particular communicated receivers furthermore communicated receiver threereceiver special case studied nair gamal idea indirect decoding introduced capacity set rate pairs min pmf sequel give generalization nair gamal multiuser multilevel broadcast channel theorem capacity multiuser multilevel broadcast channel characterized degraded message sets set rate pairs min pmf proof converse parallels proof converse case studied nair gamal replacing respectively particular compactness expression refer receiver variable denoting received signal march draft defined follows let random variable uniformly distributed set independent set completes converse part proof achievability part uses superposition coding indirect decoding follows rate splitting divide private message two independent messages rate rate codebook generation fix pmf randomly independently generate sequences according randomly conditionally independently generate sequences according pair randomly conditionally independently generate sequences according encoding send message pair encoder transmits decoding users decoder declares sent unique message yin hence law large numbers packing lemma probability error tends zero min last equality follows applying data processing inequality markov chain decoding decoder declares sent unique message triple hence law large numbers packing lemma probability error tends zero march draft decoding users decoder decodes indirectly declaring sent unique message zjn hence law large numbers packing lemma probability error tends zero min min last two equalities follow applying chain rule data processing inequality markov chain combining bounds substituting eliminating procedure proof achievability completed eferences fadel nosratinia broadcast channel hybrid csit csir ieee international symposium information theory isit june huang jafar shamai vishwanath degrees freedom region mimo networks without channel state information transmitters ieee trans inf theory vol vaze varanasi regions mimo broadcast interference cognitive radio channels csit ieee trans inf theory vol lapidoth shamai wigger capacity fading mimo broadcast channels imperfect transmitter arxiv preprint jafar blind interference alignment ieee sel topics signal vol june fadel nosratinia broadcast channel unequal coherence intervals ieee international symposium information theory isit july coherence disparity broadcast multiple access channels ieee trans inf theory vol march draft caire shamai achievable throughput multiantenna gaussian broadcast channel ieee trans inf theory vol july weingarten steinberg shamai capacity region gaussian broadcast channel ieee trans inf theory vol davoodi jafar aligned image sets channel uncertainty settling conjecture lapidoth shamai wigger collapse degrees freedom finite precision csit arxiv preprint tse completely stale transmitter channel state information still useful ieee trans inf theory vol july gou jafar optimal use current outdated channel state information degrees freedom miso mixed csit ieee commun vol july tandon tulino poor shamai fading broadcast channels partial channel state information transmitter international symposium wireless communication systems iswcs amuru tandon shamai miso broadcast channel hybrid csit ieee international symposium information theory isit tandon jafar shamai poor synergistic benefits alternating csit miso broadcast channel ieee trans inf theory vol july nosratinia product superposition mimo broadcast channels ieee trans inf theory vol coherent product superposition downlink multiuser mimo ieee trans wireless vol fadel nosratinia coherent broadcast channels multiuser degrees freedom ieee international symposium information theory isit june coherence disparity time frequency proc ieee global telecommunication conference globecom zhang fadel nosratinia spatially correlated mimo broadcast channel analysis overlapping correlation eigenspaces ieee international symposium information theory isit june borade zheng trott multilevel broadcast networks ieee international symposium information theory isit june nair gamal capacity region class broadcast channels degraded message sets ieee trans inf theory vol liu viswanath extremal inequality motivated multiterminal problems ieee trans inf theory vol may liu liu poor shamai vector generalization costa inequality applications ieee trans inf theory vol apr marzetta hochwald capacity mobile communication link rayleigh flat fading ieee trans inf theory vol zheng tse communication grassmann manifold geometric approach noncoherent multipleantenna channel ieee trans inf theory vol march draft marton coding theorem discrete memoryless broadcast channel ieee trans inf theory vol may yang kobayashi gesbert degrees freedom time correlated miso broadcast channel delayed csit ieee trans inf theory vol yang gesbert kobayashi degrees freedom region temporally correlated mimo networks delayed csit ieee trans inf theory vol shannon zero error capacity noisy channel ieee trans inf theory vol bergmans random coding theorem broadcast channels degraded components ieee trans inf theory vol mar simple converse broadcast channels additive white gaussian noise ieee trans inf theory vol mar gamal feedback capacity degraded broadcast channels corresp ieee trans inf theory vol may telatar capacity gaussian channels european transactions telecommunications vol weingarten liu shamai steinberg viswanath capacity region degraded compound broadcast channel ieee trans inf theory vol mukherjee tandon ulukus secure degrees freedom region miso broadcast channel alternating csit ieee trans inf theory vol apr information theory coding theorems discrete memoryless channels budapest gamal kim network information theory march cambridge university press draft
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aug coordination level modeling analysis parallel programs using petri nets francisco heron departamento universidade federal ufc fortaleza brazil rafael dueire lins centro universidade federal pernambuco recife brazil march abstract last fifteen years high performance computing hpc community claimed parallel programming environments reconciles generality higher level abstraction portability efficiency parallel computing platforms hash component model appears alternative addressing hpc community claims fitting requirements paper presents foundations enable parallel programming environment based hash model address problems debugging performance evaluation verification formal properties parallel program means powerful simple widely adopted formalism petri nets introduction haskell parallel extension haskell widely used pure functional programming language makes possible coordination set functional processes written haskell configuration language called hcl haskell configuration language thus haskell separate programming task two levels computation level functional processes written haskell coordination level functional processes coordinated coordination media haskell also called hash component model recent works generalized hash component model order support programming languages haskell computation level functional processes units possibly written programming language supported programming environment complies hash component model paper still suppose units functional processes haskell coordination level haskell hash component model designed order make possible translate coordination media haskell programs onto petri nets paper addresses issue presenting translation schema demonstrating use formal analysis involving verification formal properties use petri nets allows reusing existing automatic tools based formalisms reasoning hash programs pep ina extensions could make possible performance evaluation using timed stochastic petri nets variants rest paper use hash component model refer coordination level haskell assumed functional processes written haskell functional modules communicate either singleton communication channels link input output ports concepts present general definition hash component model additional introduction paper comprises following sections section presents additional details hash component model section presents translation schema hash programs onto petri nets section demonstrates petri net models hash programs may used verification formal concurrency properties hash programs hash component model distributed parallel programs may viewed collections processes interact exchanging messages execution current programming models provide ability describe computation processes augmenting common languages notations explicit message passing however provide ability modularize concerns appear design parallel applications including concern parallelism scattered across implementation processes advocate key feature integrating advanced software engineering techniques development environment hpc parallel applications sequential programming focus modularization concerns since unique conceptual process efficiency requirements less restrictive essential difference makes sequential programming actually suitable current software engineering techniques large scale applications current parallel programming hash component model may viewed new paradigm developing message passing programs may viewed two orthogonal perspective dimensions dimension processes dimension components process correspond related notion derived conventional message passing programming thus agents perform computational tasks communicating communication channels conceptually hash channels like occam synchronous typed unidirectional bounded buffers also supported disciplined use channels feature makes possible formal analysis parallel programs using petri nets main topic paper component abstract entity address functional concern application execution environment parallel program component describe role set processes respect given concern sets components respectively implement set concerns may overlap allowing modular separation concerns interlaced across implementation processes concerns separation concerns active research area programming large scale applications hash program defined main component address overall application concern common examples concerns appears hpc applications placement processes onto processors secure policies accessing computing resources grids schemes applications parallel debugging execution timing hash programming performed perspective components instead processes resulting specification topology network parallel processes components may composed simple composed components programmed using hash configuration language hcl built hierarchical composition components called inner components hcl may viewed language gluing orchestrating components connector language distinguished compositional languages supports composition parallel components overlapping concerns address conventional compositional languages allows nested composition sequential components simple components addresses functional concerns implemented using host language supposed sequential simple components atoms functionality hash programs constituting leaves component hierarchy hash component model supports parallel programming skeletons without additional language support partial topological skeletons may expose topological patterns interaction processes hash program may used produce efficient code specific architectures execution environments implemented composed components parameterized addressed concerns hash component model origins haskell host language used program simple components haskell haskell enables separation coordination computation code attaching lazy streams communication channels coordination level avoiding use communication primitives communication code paper focused haskell next section described composed components skeletons programmed using hash configurations programming simple components haskell described section programming message passing programming efficient portable expressive structured high level abstraction program recv recv send compiler code generation compiler unfolding send computable computable computable send send send send recv recv recv send send unit send process process process view component view process view concerns scattered among processes possible factorize components concerns encapsulated components send recv send process slice port channel recv recv requires programmer intervention component recv recv recv concerns scattered among processes possible factorize components figure component perspective versus process perspective programming composed components composed components define coordination media haskell programs parallelism concerns addresses without mention entities computation levels computations specified composed components written hcl hash configuration language also define core hash component model supported haskell configuration composed component specifies collection units agents perform specific tasks interact means typed unidirectional communication channels addressing given parallel programming concern unit instantiated interface associated component figure latter specifies task performed unit since functional modules describe addressed concerns former specifies unit interacts coordination medium unit associated simple component called process unit associated composed component called cluster interface unit defined set typed input output ports protocol protocol interface specifies order ports may activated execution units instantiated interface means embedded language whose constructors semantic equivalence regular expressions controlled semaphores formalism equivalent petri nets allowing formal property analysis simulation performance evaluation programs using available petri net tools pep ina port activated time becomes ready perform assignmennt instantiation interface declaration unit virtual unit interface interface class unit declaration assign declaration figure instantiating configuring unit unification cde factorization figure communication operation time completes operation according communication mode channel port connected units interact communication channels connect output port unit transmitter input port another one receiver types connected ports must supported communication modes inspired mpi synchronous buffered ready unit specification interface ports replicated form groups groups may two kinds according semantics activation activation group ports kind implies activation port members activation group ports kind implies one port members put ready communication one complete communication chosen port one activated ports whose communication pairs also activated instant internal perspective unit groups treated indivisible entities perspective coordination medium port members referred directly order forming channels input output ports groups individually unit interface must mapped arguments return points component assigned respectively wire functions useful necessary transform values boundary ports points particularly useful use wire functions aggregate data received input ports belonging group ports kind unique value passed associated argument similarly wire functions allow value produced exit point mapped onto collection values order sent port members associated group output ports kind wire functions replication figure illustrative example replication torus sqmatmult pipeline overlapping farm topological reorganization mmshift figure topology matrix multiplication using torus increases changes reusing component resolving possible conflicts two operations defined units unification factorization unification allows unify collection units forming single unit factorization inverse unification allowing units divided many virtual units replication third operation applied units allows network induced collection units replicated operations assume units fully connected behavioral connectivity preserving restrictions applied formalized connectivity restrictions imply possibility replicate ports whenever necessary adjust topological connectivity operation figures present illustrative examples operations virtual units skeletons allow overlapping components support skeletons notion virtual unit introduced unit virtual whenever component associated terms task performed virtual unit defined components partially parameterized addressed concern means placing virtual units constitution component comprises least one virtual unit called abstract component partial topological skeleton terms used synonyms programmer abstract component specification hash program must assign components virtual units comprising abstract components may instantiate applications necessary describe computation performed constituent virtual units assignment operation used allows associate component virtual unit making unit also superseding operation allows take unit replacing virtual unit topology behavioural compatibility restrictions unit replaced virtual unit guarantees sequence communication actions figure composing form systolic mesh valid unit remains valid virtual unit superseding operation syntactic sugar hcl since may implemented using unification assignment figure shows skeleton describing systolic mesh processes implemented overlapping collection skeleton instances abstract components pipeline used describe interaction processes placed mesh lines columns unit systolicmesh abstract component formed two slices one described unit comes vertical pipeline component described unit comes horizontal one programming simple components simple components also called functional modules atoms functionalities hash programming collection simple components hash program describes computation media simple components might programmed virtually general purpose language called host language needed define host language constructions correspond arguments return points underlying functional module preferred extensions host language necessary purpose keeping transparency coordination computation media simple components may overlapped configuring composed components goal implement really approach parallel programming proposed use cca common component architecture recent standard proposed integrating components written different languages parallel environment another possibility use heterogeneous implementations mpi recently proposed facilitating task since cluster environments hash programs compiled mpi since translation schema onto petri nets defined top coordination component cpipeline component sqmatmult iterator range interface icpipe ports protocol repeat seq unit pipe ports icpipe iterator range use torus farm use mmshift interface isqmatmult ports itorus protocol seq repeat seq counter connect pipe pipe buffered component torus unit torus assign torus torus unit farm assign farm farm use iterator range interface itorus ports icpipe icpipe protocol repeat seq par par unit vpipe assign cpipeline vpipe unit hpipe assign cpipeline hpipe unify sqmm ports isqmatmult unify sqmm sqmm root ports isqmatmult protocol seq assign mmshift sqmm unify vpipe hpile node ports itorus module mmshift main component farm main num main matmult unit distributor ports job unit worker ports job result protocol seq job result unit collector ports result connect synchronous connect synchronous matmult num matmult matmult matmult replicate worker figure configuration code matrix multiplication torus module tracking main import track import tallies import mcp types main user spec info particle seed event int main user info particle list let events map particle list events tally bal event lists particle create source track user info particle figure functional module component component unit unit channel channel unit unit port port typeu typeu repetitive behavior protocol idsem idsem action action skip seq action action par action action alt action action repeat action repeat counter action repeat forever action signal wait activate port port direction multiplicity typep nesting actor multiplicity single group typeg port port direction input output typep stream typeg channel connect idport idport chanmode chanmode synchronous buffered ready figure abstract hash configuration language syntax media abstracting computation media concerns specific details programming simple components provided paper illustration figure presents example functional module program written haskell abstract representation hash components figure defined simplified syntax abstract representation hash configurations named abstract hash abstract hash configuration language ahcl captures information strictly relevant translation schema onto petri nets presented example interface declarations operations units unifications factorizations replications assign operation represented ahcl supposed operations resolved translation process ahcl simplification indeed compilation process hash configurations ahcl correspond intermediate code generated compiler module serves input modules developed generating pnml petri net markup language code hash configuration following paragraph describes structure ahcl component composed set units set channels units described identifier collection ports port unit repetitive typeu interface unit defined collection ports protocol described means embedded language specifies valid orders activation ports language constructors equivalent combinators regular component unidades repetitivas unit unit unit unidades unit figure translating component expressions controlled balanced semaphores port described identifier direction direction type stream nesting factor additionally port multiplicity specifies port single port group ports notice group port two types port identifiers assumed distinct abstract hash programs channel connect two ports associated mode synchronous buffered ready notice abstract hash syntax force two ports opposite directions restriction implicitly assumed translating hash programs petri nets section schema translating hash programs petri nets introduced order make translation schema easier understand informally described using diagrams possibility making intuitive visual descriptions interesting feature petri nets translation schema specified inductively hierarchy components coordination medium hash program thus simple components ignored overall steps translation procedure hash program interlaced petri net translating units unit comprising component unit declarations interface used yielding interlaced petri net describing activation order interface ports hash configuration language defined embedded language interface declarations whose combinators correspondence operators regular expression controlled semaphores formalism proved expressiveness interface interface class assignmennt instantiation interface declaration unit virtual unit unit declaration assign declaration figure unit interface instantiation component assignment equivalence petri nets according formal language theory unit cluster composed component assigned unit necessary generate petri net corresponds assigned component using information mapping points ports unit possible synchronize behavior unit behavior component way compatible synchronize units petri net exists describing communication behavior traces unit communication channels connect declarations may used coordinating synchronized behavior petri nets units synchronize streams port carrying stream petri net describing protocol stream synchronization overlapped petri net produced last step semantics stream communication described separately increases complexity generated interlaced petri net making computationally hard analysis thus hash programming environment programmer may decide include stream synchronization protocol obviously information may lost may necessary useful analysis next sections provide details translation steps also discussed information encompassed skeletons may used simplify generated network modelling components figure petri net resulted application translation function configuration component illustrated translation function applied unit comprising component generating petri net describes communication behavior resulting petri nets connected order model parallel execution units places process started process finished number units correspond start places stop places petri nets modelling units respectively token placed process started unit ready initiate execution token placed process finished unit group ports port group port individual port port group port single figure translating unit ports finished transitions process restart number repetitive units allows repetitive units return back initial state finalization introduced petri net generated unit place program end ready receives mark units terminates case tokens places process restart enabled removed preventing repetitive processes execute state program terminates repetitive processes also terminate causes transition processes join fired token deposited program end modelling units section intends describe individual units translated petri nets firstly figure presents interlaced petri nets model activation ports interface unit mark place port prepared indicates port prepared communication firing transition port send recv models communication causing deposit mark place port complete indicating communication completed port possible two ports active time groups ports places group prepare group complete connected places port prepared port complete port belonging group model local preparation completion communication ports belonging group according semantics groups ports kind individual ports activated consequence activation group groups ports kind one port chosen among ports ready communication completion obeying semantic restriction firing transition port send recv port group kind causes removal marks places port prepared places belonging action skip figure translating null action way ports complete communication firing transition port send recv order marks placed places port prepared controlled interlaced petri net models protocol unit following sections discusses primitive actions action combinators behavior expressions translated petri nets primitive actions skip null action wait increment semaphore primitive signal decrement semaphore primitive activation input port activation output ports combinators actions seq sequential actions par concurrent actions alt choice among actions conditional choice two actions null action skip skip combinator communication effect known null action one place needed start stop place figure interlaced petri net generated sequencing seq seq combinator describes total ordering execution set actions sequential execution represented may modelled sequential composition petri nets induced action figure concurrency par par combinator describes concurrent interleaving execution set actions represented may modelled parallel composi action seq action action figure sequence actions action action par action action action figure interleaving concurrency among actions tion petri nets induced action figure choice alt alt combinator describes conceptually choice among set actions represented may modelled composing petri nets induced action using conflict place alt begin alt label start place figure firing transitions alt select branch alt label models choice streams conditions checking stream termination two combinators modelled next sections repeat requires testing condition order choose next action performed condition defined logical predicate disjunctive normal form dnf action alt action action action figure choice among actions logical variables references ports carry streams section attempts formalize notion streams hash component model valuation logical variables conditions streams stream defined sequence semantically related data items terminated special mark transmitted channel making analogy conventional message passing programming using mpi trivial example stream sequence data items transmitted calls specific occurrence mpi send primitive context iteration iteration item stream transmitted termination iteration modelled hash program end mark carried stream communication channels carry streams may implemented using persistent communication objects underlying messaging passing library may reduce communication overhead hash streams may nested streams streams nesting depth may defined stream nesting factor stream positive integer indicates order nested stream stream may defined following value eos value data item eos termination value nesting level notice termination values carry integer indicating nesting level stream terminated feature nested streams appeared consequence design haskell streams coordination level must associated lazy lists computation media experience haskell programming shown laziness haskell nested lists may useful applications feature analogous communication operations occurs context nested iterations mpi parallel programming hash configuration language streams declared placing symbols identifier port declaration interfaces number indicates nesting factor stream carried port ports carrying streams nesting levels may connected channel defined port transmit single value nesting level zero notion streams defined possible define syntax semantics predicates testing synchronized termination streams necessary feature combinators repeat kind predicate referred stream predicate syntactically stream predicate logical predicate disjunctive normal form logical operators supported logical logical disjunctions may enclosed delimiters logical variables references interface ports unit formal syntax stream predicates shown haskell simple components functional modules written haskell stream predicate sync conjunction sync conjunction sync conjunction simple conjunction simple conjunction simple conjunction port port port idn let depth nesting occurrence repeat combinator relation outermost occurrence exists exists port carrying streams nesting factor equal less appear termination condition possible define semantics stream predicates defining values logical variables may inferred execution units instance let port carrying stream value false whenever data value value ending value nesting level eos transmitted sent received last activation otherwise true value stream predicate may evaluate true false fail value true obtained evaluating stream predicate ignoring semantics angle brackets delimiters fail obtained negation conjunction enclosed angle brackets evaluates true assuming following identity defines angle brackets semantics stream predicate evaluates neither true error value stream predicate false angle brackets delimiters used ensuring synchronization nature values transmitted streams whenever necessary order make possible model test stream predicates petri nets firstly necessary model stream communication using formalism particularly necessary introduce petri net hash program places remember kind value transmitted last activation ports carry streams hash program port carries stream nesting factor two sets places referred stream flags stream flags stream flags stream port flag stream flags stream port flag dual stream port places set stream flags form split binary semaphore flags also mutually exclusive corresponding places stream flags stream port flag stream port flag dual port carrying stream nesting factor places stream flags stream flags used remember kind value transmitted last activation possibilities action repeat action figure stream controlled repetition ending marker nesting level eos data value places stream port flag respectively associated ending marks eos port carrying stream nesting factor place stream port flag associated data value assuming restrictions mark stream port flag place value corresponding kind transmitted last activation port restrictions guaranteed petri net protocol synchronization streams introduced section next two sections present respectively model repeat combinators assuming existence sets places stream flags stream flags repetition controlled stream predicates repeat combinator used model repeated execution action termination condition may provided counter clause later introduced next section former uses stream predicate testing termination repetition iteration petri net modelling repeat combinator clause assumed existence modelling stream communication introduced section figure illustrated petri net resulted translation repeat combinator clause check termination conflict place checking conditions models decision terminate action repeat counter action figure repeated action fixed number times action repeat forever action figure infinite repetition repeat action else action action else figure conditional iteration values logical variables corresponding ports stream predicate tested using respective stream flags stream flags places arrangement places transitions figure allows testing value stream predicates iteration mutual exclusive firing transitions terminate fail loop correspond respectively values true execute one iteration false termination error abort program stream predicate bounded repetition combinator repeat termination condition defined counter clause models repeated execution action fixed number times translation petri nets illustrated figure certain fixed bounded repetition weight arcs enter remaining performed exit define number repetitions act ion infinite repetition whenever termination condition defined given occurrence repeat combinator given action repeated infinitely petri net models kind repeat combinator illustrated figure conditional choice combinator describes conditional choice two actions translation petri nets illustrated figure reader may notice action signal action wait figure signal left wait right semaphore primitives analogy construction petri net diagram petri net generated repeat combinator mainly concerning test stream predicate semaphore primitives wait signal balanced counter semaphore primitives par combinator make behavior expressions comparable labelled petri nets descriptive power absence semaphore primitives regular patterns unit behavior could described semantic semaphore primitives defined using notation introduced concurrent synchronization wait await signal angle brackets model mutual exclusion atomic actions concurrent processes await statement models condition synchronization wait primitive causes process delay value semaphore greater zero order decrement value thus value semaphore greater zero instant execution unit concurrent systems synchronization controlled balanced counter semaphores like behavior expressions defined value semaphore must initial final states system signal wait primitives modelled using petri nets described figure given semaphore number marks place sem counter models semaphore value given state port activation hash primitive actions models respectively send receive primitives message passing programming hash programs cause activation groups ports individual ports member group section petri net slice model individual ports groups ports illustrated figure individual port prepared communication whenever mark deposited place port prepared group ports kind prepared whenever ports prepared group kind prepared whenever ports prepared communication completed whenever place deposited place port complete group complete activation port defined time preparation port completion communication format petri net slices induced translation occurrences primitives illustrated figure firing transition activate start prepares port communication firing transition activate stop occurs whenever port completes communication notice whenever mark deposited place activate port active communication units translation component petri net slices resulted translation units form interlaced petri net models asynchronous execution information regarding synchronization units means communication channels yet included figure petri net slices model respectively three kinds channels may occur hash program synchronous buffered ready presented translation function applied communication channel component generating petri net slices according translation schema illustrated figure petri net slices overlapped petri net slices models behavior units order model synchronous execution units synchronous channels communication pairs must active time communication complete implementing necessary unify respective transitions modelling completion communication respective pairs buffered channels bounded buffers sender need wait completion communication operation resuming execution whenever sender port channel activated transition port send must activated mark action activate port port translation ports figure activation ports chan nel chan nel mode ready mode buffered mode synchronous connect mode chan nel fusion translation ports figure modelling communication channels place chan buffer free models number empty slots buffer place chan buffer used receives mark activation port send models number used slots buffer notice sender blocks whenever empty slots buffer channels supporting ready mode require complex protocol place chan ready open ensures communication proceeds activation sender port precedes activation receiver notice whenever receiver activated sender sender proceed causing deadlock may detected petri net verification tool using approach possible verify example certain parallel program using ready channels may fail program state execution ready communication mode may improve communication performance mpi programs unfortunately hard ensure communication semantics safe arbitrary parallel programs modelling communication semantics petri nets may overcome difficulties debugging synchronization streams protocol section introduced two sets places must exist stream port stream flags stream flags additionally restrictions introduced markings allow check kind transmitted value last activation port making possible check stream termination conditions occurs repeat combinators section presents protocol updating marking places stream flags stream flags way restrictions introduced section obeyed figure illustrated network presented figure activation ports may enriched order introduce protocol updating places stream flags stream flags arbitrary stream port nesting factor petri net slice introduced transitions clear flag set flag main components arranged way transitions inside sets mutually output ports input ports receiver may decide value received completion communication activation sender may decide kind value going transmitted figure activation stream ports clusive firing transition clear flag clears set places stream flags moving mark corresponding cleared place corresponding place set stream flags places stream flags exactly one mark places stream flags zero mark sequence one transition set set flag fired causing moving mark one places stream flags chosen corresponding place stream flags sequence actions models test kind value transmitted current activation port notice moment choice different input output ports input port updates stream flags communication completed accordance implementation semantics receiving value receiver may check kind transmitted value ensuring consistency communication channel communication semantics imposes kind value transmitted sender given activation stream port kind value receiver receives corresponding activation figure shown restriction ensured channels synchronous ready mode communication following description consider individual ports groups containing one port thus consider channel connecting sender stream port receiver stream port nesting factor groups contained respectively necessary create arc links place stream port flag transition set flag way consistency stream flags stream flags forced sender side receiver side channel kind received value must kind sent value figure ensuring consistency streams connected synchronous channels munication completion two different senders connected ports belonging group kind decide transmit values different nesting levels give activation deadlock occurs possible introduce petri net slice detecting event bounded buffered communication imposes complicated approach illustrated figure consider channel buffer size buffer slot channel connecting ports nesting factor set places buf slot flag remember kind value stored buffer slot communication channel essentially firing transition set flag marking stream port flag moment activation group saved buf slot flag number next available buffer slot slots filled sender blocks slot freed receiver group kind notice groups one port kind arc place buf slot flag transition set flag ensures marking stream flags reflects kind oldest value placed buffer sender semantics buffered communication imposes group kind necessary introduce petri net slice shown figure ensures copy marking places buf slot flag chosen receiver port mutually exclusive places group port activated belonging remember port chosen enable appropriate set transitions group copy flag connected places buf slot flag channel chosen port connected marking set places copied places group copied flag connected transitions set flag sender side buffered channel buffer management receiver side kind received value must kind oldest sent value figure ensuring consistency streams connected buffered channels activation groups ports kind figure copying protocol groups input ports kind arrangement places buf slots locked buf slots unlocked transition buf slots unlock avoids accesses buffer updated transmission firing transition buf slot select means selection next empty slot whenever buffer full marks deposited place chan buffer used sender must block waiting receiver consume contents first slot entry copying kind value first slot buffer discarded shift operation occurs mark deposited place buf slots shifting allows save marking places buf slot flag buf slot flag ensuring consistency order kind transmitted values following example illustrates needs imposing one restriction petri net slice stream control protocol instance consider nested haskell list int nesting factor correspondent stream hash component model transmit following values activation corresponding port eos eos eos eos eos eos eos eos eos eos eos eos eos eos eos eos notice transmitting value eos possible transmit value eos since enclosing stream nesting level finalized yet case values eos eos eos data item may transmitted consider general case transmission given end marker nesting level next value transmitted may end markers nesting level greater data item stream finalized attempt read value finalized stream considered error figure shown petri net slice presented figure output port may enriched order support restriction stated last paragraph mark placed place order fail whenever attempt activate finalized stream port kind last transmitted value eos occurs nesting level four places two transitions controls consistency order transmitted value place flag open receives value whenever value eos may sent mutually exclusive dual place named flag open dual allows resetting marking flag open activation resetting procedure implemented using next elements described place cleaning flag mark whenever resetting procedure enabled port nesting level corresponding place cleaned flag mark resetting procedure finishes transition clean resets place flag open stream output port figure consistency order kind transmitted stream values original state zero marks transition keep cleaned fires whenever place flag open already cleaned notice transitions keep cleaned clean mutual exclusive since flag open flag open dual complexity generated petri net overlapping petri net slice models stream communication semantics allowing make precise analysis behavior hash programs coordination level petri net simple hash programs may become large large petri nets may turn impossible programmers analysis without help automatic higher level means also makes hard memory consuming computations performed underlying petri net tools computation reachability coverability graphs place transition invariants etc difficulties comes transitory technological limitations since processing power memory amount machines increased rapidly recent years expected continue increase next decades also possible use parallel techniques perform high computing demanding analysis approach exploited designers petri net tools reasonable assumption parallel programmers access parallel computer despite facts desirable provide ways helping programmers work large petri nets simplifying generated petri net following techniques proposed process analysis programmer may decide augment petri net hash program protocol modelling stream communication approach makes sense whenever component problem class num procs max key num buckets total keys max iterations max procs test array size define parameters params problem class num procs max key num buckets total keys max iterations max procs test array size iterator range num procs use skeletons collective allreduce alltoallv use functional module unit comm assign allreduce num procs mpi sum mpi integer comm unit comm assign alltoallv num procs comm unit shift assign rshift num procs shift interface iis iallreduce uarray int int ialltoallv int ptr int rshift int behaviour seq repeat seq unify peer iis assign parameters peer figure configuration code program mation provided protocol always necessary analysis conducted reason separation stream communication protocol rest petri net hash program another approach proposed works build higher level environments analysis hash programs top petri net tools instead programmers manipulate petri net components manipulate hash program elements abstraction analysis transparently automatically translated proving sequences using ina tool specific translation schemas skeletons might specified order simplify generated petri net approach illustrated next section instance defined information provided use collective communication skeletons might used translation process hash programs petri nets modelling collective communication skeletons message passing libraries mpi support special primitives collective communication hash programming environment defined library skeletons implement pattern communication involved collective communication operations supported mpi bcast scatter scatterv reduce scatter scan gather gatherv reduce allgather allgatherv alltoall alltoallv allreduce next paragraphs use collective communication skeletons illustrated means example figure code hash version program npb nas parallel benchmarks presented parallel implementation bucket sort algorithm originally written section used collective port join transitions protocol modelling stream communication semantics shared among collective ports figure modelling collective communication skeleton semantics illustrate use collective communication skeletons hash program motivating definition specific strategy translating collective communication patterns interaction among units hash program line figure declared allreduce alltoallv used configuration lines three units declared named comm comm shift first two collective communication skeletons assigned respectively allreduce alltoallv since skeletons composed components units clusters units interact using collective communication patterns described skeleton line unification correspondent units comprise clusters comm comm shift forms units comprise topology unification virtual units distinct cluster allows overlap skeletons behavior virtual units result unification partial topological skeletons composed components least one unit virtual case collective communication skeletons units virtual named peer num procs specified interface iis simple component assigned defining computation process program notice spmd program task performed processes defined simple component interface iis declared composition iallreduce ialltoallv irshift interfaces interface slices iis interface slices respectively identified use combinator abbreviation avoids rewrite behavior interface slices thus relates sequence actions encapsulated specification interface iallreduce using conventions overlapping collective communication skeletons order form complex topologies simple define specific translation rule patterns collective communication interactions identifiers interface slices collective communication skeletons might viewed special kinds ports collective ports operator activation operator notice interface slices may used termination conditions streams like line collective ports direction since processes participate communication communication operations synchronous figure illustrates petri net slice models collective communication operation involved ports collective ports correspond interface slices units participates certain collective communication operation defined cluster units defines figure illustrated augment petri net slice shown figure protocol modelling stream communication semantics important notice set places stream flags stream flags shared collective ports involved collective operation petri net analysis formal properties section solutions two well known synchronization problems implemented hash approach illustrates use petri nets analyzing hash programs verifying formal properties dining philosophers dining philosophers problem one relevant synchronization problems concurrency theory since originally proposed dijkstra widely used exercising ability concurrent languages models providing elegant solutions avoiding deadlocks concurrent programs dining philosophers problem stated following way five philosophers sited around table dinner philosophers spend times eating thinking philosopher wants eat takes two forks table philosopher phil phil phil phil phil phil phil phil phil phil figure hash topologies dining philosophers problem wants think keeps two forks available table however five forks available requiring philosopher share two forks neighbors thus whenever philosopher eating neighbors thinking solution dining philosopher problem establishes protocol ordering activity philosophers figure hash topologies solutions dining philosophers problem presented first one whose code presented figure anarchical solution philosophers free decide think eat solution reader may observe use buffered channels groups ports kind composing network topology buffered channel allows model fact fork may table waiting philosopher acquire occurs whenever one message pending buffer beginning interaction philosophers fork release philosopher releases fork semantics group ports kind ensures philosopher waiting fork obtain fork immediately otherwise possible philosopher released fork chance acquire fork solution satisfy enunciated requisites good solution instance critical section problem example philosophers acquire right left forks deadlock also possible philosopher never get chance obtain forks eventual entry second solution whose code presented figure ensure requisites additionally ensures maximal parallelism state execution two philosophers eating figure presents petri nets model respective behaviors individual philosophers first second solutions figure presents petri nets modelling interaction among five philosophers modelling communication channels figures illustrative since networks number components intractable simple visual inspection component diningphilosophers index range interface iphil ports get get put put protocol seq put repeat seq par get get par put put get get put put unit phil ports iphil grouping get neighbor self get neighbor self put neighbor self put neighbor self connect connect connect connect phil put neighbor phil put self phil put neighbor phil put self phil mod get neighbor phil get self phil mod get neighbor phil get self buffered buffered buffered buffered component diningphilosophers index range interface iphil ports get get put put protocol repeat seq put get get put interface iphil ports get get put put protocol repeat seq get put put get unit phil ports iphil mod connect phil put phil mod get buffered connect phil put phil mod get buffered figure hash code first solution dining philosophers problem figure petri net modelling behavior one philosopher proving properties dining philosophers solution section ina integrated network analyzer used underlying engine verifying formal properties hash solutions dining philosophers problem ina allows perform several structural dynamic analysis petri net induced two solutions among possible analysis approaches ina provides model checking facilities allows check validity ctl constructive tree logic formulae describing properties hash program reachability graph corresponding petri net ctl suitable formalism temporal logics expressing verifying safety invariance liveness properties dynamic systems allows temporal operators quantify paths possible given state exist superset ctl named augments expressive power ctl allowing express fairness constrains allowed expressed ctl however ina restricts ctl model checking algorithms ctl efficient linear formula size exponential formula size let introduce relevant properties may proved solutions dining philosophers problems model properties using ctl dining philosophers problem may thought instance critical section problem fact forks critical sections since taken one philosopher good solution instance critical section problem must ensure three properties mutual exclusion two adjacent philosophers obtain figure petri nets dining philosophers hash solutions macros channels sender prepared receiver prepared rendezvous buffer full buffer empty sender blocked sender blocked receiver blocked receiver blocked macros ports port pair prepared port pair prepared port prepared port prepared sender ready receiver ready buffer free buffer used sender ready ready sender ready buffer full receiver ready ready receiver ready buffer empty port prepared port prepared macros groups ports group prepared group prepared port prepared port prepared table useful formula macros hash programming fork enter critical section absence deadlock deadlock occurs whenever least one active philosopher active terminated philosophers blocked classical deadlock situation dining philosophers problem occurs philosophers acquire right left forks state philosophers may proceed finished thus deadlock absence unnecessary delay philosopher demands right left fork right left neighbor thinking philosopher prevented obtaining fork eventual entry philosopher demands fork eventually obtain following paragraphs properties characterized using ctl formulae intending facilitate concise modular specification complex ctl formulas define notion macro macro new kind form hmacro namei hmacro namei macro name qualifiers ctlformulae macros may expanded flat applying recursively definitions instance macro defined using following syntax hmacro namei hmacro namei name macro qualifier variables possibly making reference macros philosophers phil demands phil demands phil posseses phil posseses phil eating phil posseses phil posseses phil thinking eating phil waiting phil demands phil demands phil finished process finished phil phil finished phil finished macros forks fork free phil thinking phil thinking fork use right phil posseses fork use left phil posseses fork use fork use right fork use left table useful formula macros dining philosophers macros flat appears ctlformula macro qualifier variables used qualifiers place transition identifiers references enclosed macros table useful macros defined simplifying specification ctl formulae restrict domain hash programs table macros defined restrict domain dining philosophers problem application oriented notice macros table defined top defined table unlike later macros implementation former ones sensitive modifications underlying translation schema illustrates transparency provided use macros environment proof analysis formal properties used ina proving three properties enunciated dining philosophers first three ones safety properties may proved negating predicate describing state reached execution bad state last one liveness property validity predicate must checked possible states proof mutual exclusion safety property one valid formulation corresponding bad state bad fork use right fork use left proof absence deadlock corresponding bad state bad safety property one valid formulation phil waiting phil finished phil finished proof absence unnecessary delay mulation corresponding bad state safety property one valid transmitter bit await asl bit err bit err bit bit err bit sender side ack receiver side receiver figure hash topology abp alternating bit protocol bad fork free phil demands phil demands proof eventual entry liveness property one valid formulation corresponding good state good phil waiting phil eating alternating bit protocol alternating bit protocol abp simple effective technique managing retransmission lost messages low level implementations message passing libraries given transmitter process receiver process connected stream channel abp ensures whenever message sent lost retransmitted hash implementation described based functional implementation described figure illustrates topology component abp might used implementing abp protocol virtual units transmitter receiver model processes involved communication units implement protocol units transmitter await corrupt ack implement sender side abp protocol units receiver ack corrupt send implement receiver side await process may retransmit message repetitively message received process ack retransmissions modelled using streams nesting factor streams streams elements nested stream correspond retransmission attempts given value correct arrive message performed inspecting value received port processes corrupt ack corrupt send verify occurrence errors messages arrive sender receiver respectively modelling unreliable nature communication channel hash configuration code presented figure implements abp component petri net induced translating abp component presented figure component abp use await corrupt ack interface abp transmitter ports protocol repeat interface abp receiver ports protocol repeat interface ports bit protocol repeat seq interface await ports bit bit bit protocol repeat seq repeat seq interface corrupt ports bit bit protocol repeat seq interface ack ports bit protocol repeat seq interface ports bit protocol repeat seq repeat unit transmitter ports abp transmitter unit receiver ports abp receiver unit unit unit unit unit unit await corrupt ack ack corrupt send connect connect connect connect connect connect connect connect ports ports ports ports ports ports corrupt corrupt corrupt iout iawait icorrupt iin iack icorrupt grouping corrupt corrupt assign assign assign assign assign assign await corrupt ack corrupt buffered buffered figure abp component await corrupt ack ack corrupt send transmitter await receiver ack figure petri net induced abp component references andrews concurrent programming principles practice addison wesley armstrong gannon geist keahey kohn mcinnes parker smolinski towards common component architecture scientific computing ieee international symposium high performance distributed computing ieee bailey harris shapir van der wijngaart woo yarrow nas parallel benchmarks technical report nasa ames research center december http best esparza grahlmann melzer rmer wallner pep verification system workshop formal design safety critical embedded systems femsys carvalho junior lima lins coordinating functional processes haskell acm press editor acm symposium applied computing track coordination languages models applications pages march carvalho junior lins haskell parallel programming made simple efficient journal universal computer science august carvalho junior lins implementation spmd applications using haskell brazilian symposium computer architecture high performance computing ieee press november carvalho junior lins topological skeletons haskell international parallel distributed processing symposium ipdps ieee press april pages carvalho junior lins lima parallelising evaluating haskell parallel programming environment unb editor brazilian symposium computer architecture computing september carvalho junior lins lima translating haskell programs petri nets lecture notes computer science vecpar cole algorithm skeletons structured management paralell computation pitman djikstra structure multiprogramming system communications acm november dybjer sander functional programming approach specification verification concurrent systems formal aspects computing german spnl processes building blocks stochastic petri nets conference computer performance evaluation modelling techniques tools pages springer verlag gischer shuffle languages petri nets grammars communications acm september grahlmann best pep petri net tool lecture notes computer science tools algorithms construction analysis systems second int workshop tacas passau germany volume pages springer verlag march inmos occam programming manual hoare series editor ito nishitani universality concurrent expressions synchronization primitives theoretical computer science kiczales lamping menhdhekar maeda lopes loingtier irwin programming lecture notes computer science programming european conference ecoop volume pages november peyton jones editors hughes report programming language haskell purely functional language february roch starke manual integrated net analyzer version shaw software descriptions flow expressions ieee transactions software enginnering may squyres lumsdaine component architecture proceedings european users group meeting number lecture notes computer science venice italy september squyres lumsdaine george hagedorn devaney interoperable message passing interface impi extensions proceedings mpidc march thompson haskell craft functional programming addisonwesley publishers zimmermann freiheit german hommel petri net modelling performability evaluation timenet int conf modelling techniques tools computer performance evaluation tools pages lecture notes computer sciente formal syntax hcl follows described grammar hcl haskell configuration language whose syntax programming abstractions informally presented section examples hcl configurations meanings presented sections notation employed similar used describing syntax haskell indeed grammar reused haskell code appears hcl configurations faced italic bold minor difference notation resides use instead describing optional terms simplicity notation indexed notation ignored description formal syntax hcl may resolver parsing definitions configuration header static parameter list component interface declaration header declaration declaration component static parameter list component interface idn ports naming iterator decl interface decl import decl use decl unit decl assign decl replace decl channel decl unify decl factorize decl replicate decl bind decl haskell code use declaration use decl use use spec use spec spec use spec use spec import declaration import decl impdecl iterator declaration iterator decl iterator idn range numeric exp numeric exp interface declaration interface decl interface context tyvar tyvar interface spec interface spec interface ports spec interface inheritance behavior behavior expression interface ports description interface ports spec port spec list port spec list port spec list port spec port spec port spec port spec atype interface composition interface slice interface slice interface inheritance interface slice ports naming composition ports naming composition ports naming ports naming ports naming ports naming port naming list port naming list port naming list idn interface behavior behavior expression action condition disjunction sync conjunction simple conjunction unit declaration unit decl unit spec unit interface setup group spec group type wire function sem idn action par action action seq action action alt action action repeat action condition condition action else action signal wait disjunction counter numeric exp sync conjunction sync conjunction simple conjunction simple conjunction idn unit unit spec unit interface wire setup setup ports naming composition interface spec group type group spec wire function idn numeric exp exp assignment declaration assign decl assign assigned component assigned unit assigned component actual parameter list ports naming composition actual parameter list numeric exp numeric exp assigned unit qid ports naming composition replace declaration replace decl replace qid ports naming composition operand unit channel declaration channel decl connect qid qid qid qid comm mode comm mode synchronous buffered numeric exp ready unification declaration unify operand unit operand unit unit spec adjust wire setup setup operand unit qid interface pattern interface pattern interface pattern port pattern list port pattern list port pattern list pattern pattern pattern pattern qid unify decl factorization declaration factorize decl factorize operand unit unit spec unit spec adjust wire setup setup replication declaration replicate decl replicate operand unit operand unit numeric exp adjust wire setup setup bind declaration bind declaration bind qid qid bind qid qid miscelaneous haskell code topdecls qid idn qid idn foundations notations section discussed formalisms comprise formal framework development work concerning modelling communication behavior processes according hash component model design principles formal languages theory formal languages employed framework study patterns communication interaction hash process parallel program main interest investigate relations descriptive power concurrent expressions petri nets order define language expressing communication behavior processes embedded hash language definition alphabet alphabet finite set indivisible symbols denoted definition word word finite sequence symbols alphabet symbol denotes empty word whose length zero definition kleene closure alphabet kleene closure alphabet denoted defined word thus sequence symbols including belongs common define definition given alphabet formal language defined follows labelled petri nets formal languages notations definitions concerning petri nets presented definition petri net petri net directed bipartite graph formalized quadruple finite set places store unlimited number marks finite set transitions defines set arcs way aturais thus arc transition place place transition number associated arc indicating weight simplicity weight omitted one relation defines initial marking number marks stored place initial state petri net initial marking petri net defines set reachable markings marking reachable obtained initial marking firing sequence transitions according firing rules formalized follows definition enabled transition petri net transition enabled thus transition enabled number marks one input places greater equal weight arc links transition definition firing rule reachable markings petri net marking transition marking new marking side effect firing enabled transition represented relation indicates new marking input places transition indicates new marking output places transition fire enabled effect firing remove marks input places add marks output places according weights arcs link places necessary generalize definition cover concept transitively reachable marking thus marking reachable mark sequence firing transitions notation abbreviated symbol denotes sequence firings transitions definition labelled petri nets alphabet labelled petri net petri net function associate transitions symbol transitions labelled called silent transitions labelled petri nets extension petri nets generating formal languages two classes petri net languages definition petri net language given labelled petri net define formal language generated definition petri net terminal language given petri net final marking define terminal language generated respect given labelled petri net language generated defines possible sequence firing traces initial marking terminal language generated respect final marking differs language traces considered definition classes petri nets languages denote class petri net languages class petri net terminal languages simple demonstrable interlaced petri nets interlaced petri nets alternative extension labelled petri nets introduced article simplifying specification translation schema hash programs labelled petri nets like hierarchic petri nets interlaced petri nets allows complex large scale petri nets implemented simpler modular way modify descriptive power labelled petri nets however hierarchic petri nets allow nesting composition petri nets interlaced petri nets also allow overlapping given set petri net slices one addressing different concerns may overlapped form interlaced petri net interlaced petri nets may also viewed petri net slices composing higher level interlaced petri nets definition interlaced petri nets interlaced petri net defined inductively base tuple represents labelled petri net function maps petri net nodes places transitions onto list qualifiers represents simple interlaced petri net induction let interlaced petri nets called petri net slices context hypothesis composed interlaced petri net induction step anything may formed application inductive rules interlaced petri net definition unfolding composite interlaced petri nets unfolded interlaced petri net obtained interlaced petri net applying transformation function defined composite correspond petri net union operator defined binary operator qualifiers used identify components interlaced petri nets places transitions must treated component components said equivalent components may belong distinct slices next qualifiers identification rules formalized definition qualifier let finite set symbols set informally defined qualifiers induced set qualifier defined following qualifier primitive qualifier tuple qualifier composed qualifier assuming primitive qualifier following rule teaches identify equivalent vertices interlaced petri net using qualifiers definition identification vertices places transitions let interlaced petri net consider unfolded variant let collection components following equivalence relation defined two vertices regular expressions controlled balanced semaphores scre following definitions presented order introduce class regular expressions controlled balanced semaphores scre generalization regular expressions descriptive power comparable labelled petri nets used model language specification communication behavior hash processes definition regular expressions regular expression alphabet inductively defined follows regular expression regular expression regular expression regular expression thus also regular expressions definition regular language regular expression formal language generated lre defined following way lre lre lre para lre lre lre lre lre lre lre lre lre lre class regular languages denoted previous version hash language used regular expressions model communication traces processes however petri nets far expressive simple regular expressions describing communication traces fact motivated generalize adopted approach using class synchronized concurrent expressions shown equivalent petri nets concurrent expressions extension regular expressions defined model concurrency authors refer languages shuffle languages definition concurrent expressions alphabet concurrent expression defined following rules regular expression concurrent expression concurrent expressions concurrent expressions operators distinguish least syntactically concurrent expressions language generated concurrent expressions defined definition concurrent languages concurrent expression concurrent language lce defined according following rules lce lre regular expression lce lce lce lce lce class concurrent languages denoted clear every regular expression concurrent expression strongest important result relates concurrent regular languages theorem relating concurrent regular expressions concurrent expression makes use operator lce regular another important result gives bounds expressiveness concurrent expressions theorem bounds concurrent expressions expressiveness concurrent expression lce language order increase expressivity concurrent expressions allowing express recursively enumerable languages synchronized concurrent expressions proposed extend concurrent expressions synchronization mechanisms used extensively analyse expressiveness synchronization mechanisms concurrency mainly based semaphores definition synchronized concurrent expressions alphabet set symbols denote synchronization primitives disjoint concurrent expression called synchronized concurrent expression language said synchronization mechanism synchronized concurrent expression adopting synchronization mechanism denoted class synchronized concurrent expression denoted sce next definition defines language synchronized concurrent expression gives meaning synchronization mechanism definition synchronized concurrent languages synchronized concurrent expression language lsce defined lsce lce homomorphisms defined follwing class synchronized concurrent languages uses synchronization mechanism denoted scek definition parameterizes adopted synchronization mechanism common based semaphores whose significative examples presented following paragraphs set synchronization primitives com following synchronization mechanisms defined counter semaphore lce wherec semaphore lce binary semaphore lce whereb semaphore lce denote sets synchronization primitives number primitives notation generalized synchronization mechanism semaphores let besthe family synchronization mechanisms primitives thus special cases concurrent expression controlled semaphore system called semaphore controlled concurrent expression many important results expressive power synchronized concurrent expressions presented work one deserve special attention establishes equivalence synchronized regular expressions uses semaphores synchronization protocol petri nets synchronized regular expressions defined definition regular expressions controlled balanced semaphores regular expressions controlled balanced semaphores recbs defined synchronized concurrent expressions make use operator class recbs denoted recbs remember theorem guarantees concurrent expressions without operator equivalent regular expressions however presence operator semaphores guarantees kind expression may generate richer class formal languages simple regular expressions fact enunciated following theorem theorem equivalence recbs petri nets class languages generated recbs result theorem convenient purpose make descriptive power hash language equivalent descriptive power petri nets use recbs avoids use operator shaw introducing flow expressions make software descriptions gave two interpretations parallel loop sequential loop creates process fork iteration neither interpretation practical hash language assumes static parallelism
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trinder chechina papaspyrou sagonas thompson scaling reliably may scaling reliably improving scalability erlang distributed actor platform phil trinder natalia chechina nikolaos papaspyrou konstantinos sagonas simon thompson stephen adams stavros aronis robert baker eva bihari olivier boudeville francesco cesarini maurizio stefano sverker eriksson viktoria fordos amir ghaffari aggelos giantsios rockard green csaba hoch david klaftenegger huiqing kenneth lundin kenneth mackenzie katerina roukounaki yiannis tsiouris kjell winblad release strep project april abstract distributed actor languages effective means constructing scalable reliable systems erlang programming language influential model erlang model conceptually provides reliable scalability inherent scalability limits force developers depart model scale article establishes scalability limits erlang systems reports work release project improve scalability understandability erlang reliable distributed actor model systematically study scalability limits erlang address issues virtual machine language tool levels specifically evolved erlang virtual machine work effectively large scale multicore numa architectures made important changes architectural improvements widely used release designed implemented scalable distributed erlang libraries address scalability issues provided validated set semantics new language constructs make large erlang systems easier deploy monitor debug developed made open source releases five complementary tools specific erlang throughout article use two case studies investigate capabilities new technologies tools distributed hash table based orbit calculation ant colony optimisation aco chaos monkey experiments show two versions aco survive random process failure hence erlang preserves erlang reliability model report measurements range numa cluster architectures key scalability experiments conducted athos cluster hosts cores even programs global recovery data maintain erlang partitions network reduce network traffic hence improves performance orbit aco benchmarks hosts aco measurements show maintaining global recovery data dramatically limits scalability however scalability recovered partitioning recovery data exceed established scalability limits distributed erlang reach limits erlang benchmarks scale hosts cores introduction stant messaging server erlang actors termed processes managed sophisticated virtual machine single multicore numa host distributed erlang provides relatively transparent distribution networks vms multiple hosts erlang supported open telecom platform otp libraries capture common patterns reliable distributed computation pattern process supervision largescale system needs scalable persistent storage following cap theorem erlang uses indeed implements dynamostyle nosql dbms like riak cassandra erlang distributed actor model conceptually provides reliable scalability inherent scalability limits indeed distributed erlang systems must depart distributed erlang paradigm order scale maintaining fully connected graph hosts release project set establish address scalability limits erlang reliable distributed actor model outlining related work section benchmarks used throughout article section iii investigate scalability limits seeking identify specific issues virtual machine language persistent storage levels section report release project work address issues working following three levels distributed programming languages frameworks central engineering large scale systems key properties include scalability reliability scalability mean performance increases hosts cores added large scale mean architectures hundreds hosts tens thousands cores experience high performance data centre computing shows reliability critical scales host failures alone account around one failure per hour commodity servers approximately cores usable programming languages employed must supported suite deployment monitoring refactoring testing tools work scale controlling shared state way build reliable scalable systems state shared multiple units computation limits scalability due high synchronisation communication costs moreover shared state threat reliability failures corrupting permanently locking shared state may poison entire system actor languages avoid shared state actors processes entirely local state interact sending messages recovery facilitated model since actors like operating system processes fail independently without affecting state actors moreover actor supervise actors detecting failures taking remedial action restarting failed actor erlang beacon language reliable scalable computing widely emulated distributed actor model influenced design numerous programming languages like clojure many languages actor frameworks kilim java cloud haskell akka scala erlang widely used building reliable scalable servers ericsson telephone exchange switch facebook chat server whatsapp designed implemented set scalable distributed erlang libraries address reliability scalability issues operational semantics provided key new construct implementation validated semantics section evolved erlang virtual machine work effectively multicore numa architectures improved shared trinder chechina papaspyrou sagonas thompson scaling reliably ets tables time management load balancing schedulers improvements included release currently downloaded approximately times month section facilitate development scalable erlang systems make maintainable developed three new tools devo sdmon wombatoam enhanced two others visualisation tool percept refactorer wrangler tools support refactoring programs make scalable easier deploy large scale hundreds hosts easier monitor visualise behaviour tools freely available open source licences wombatoam deployment monitoring tool commercial product section vii throughout article use two benchmarks investigate capabilities new technologies tools computation symbolic algebra specifically algebraic orbit calculation exploits distributed hash table ant colony optimisation aco parallel search program section iii report reliability scalability implications new technologies using orbit aco benchmarks use chaos monkey instance randomly kills processes running system demonstrate reliability benchmarks show erlang preserves erlang reliability model report measurements range numa cluster architectures specified appendix key scalability experiments conducted athos cluster hosts cores established scientifically folklore limitations around connected distributed erlang systems section key result show erlang benchmarks exceed limit reach limits athos ter section viii contributions article first systematic presentation coherent set technologies engineering scalable reliable erlang systems developed release project section presents first scalability study covering erlang language storage scalability indeed believe first comprehensive study distributed actor language scale hosts around cores individual scalability studies erlang scaling language storage scaling appeared language level design implementation validation new libraries section reported piecemeal included completeness improvements made erlang virtual machine section thoroughly reported conference publications others reported first time sections iii section vii wombatoam sdmon tools described first time revised devo system visualisation tools profiling debugging refactoring developed project previously published piecemeal first unified presentation performance results section viii entirely new although comprehensive study erlang performance available recent article context scalable reliable programming models plethora shared memory concurrent programming models like pthreads java threads models like openmp simple high level however synchronisation costs mean models generally scale well often struggling exploit even cores moreover trinder chechina papaspyrou sagonas thompson scaling reliably liability mechanisms greatly hampered shared state example lock becomes permanently unavailable thread holding fails high performance computing hpc community build core distributed memory systems using facto standard mpi communication libraries increasingly hybrid applications combine mpi openmp unfortunately mpi suitable producing general purpose concurrent software low level explicit message passing moreover widely used mpi implementations offer fault part computation fails entire computation fails currently issue addressed using hoped highly reliable computational networking hardware intense research interest introducing reliability hpc applications server farms use commodity computational networking hardware often scale around cores host failures routine typically perform rather constrained computations big data analytics using reliable frameworks like google mapreduce hadoop idempotent nature analytical queries makes relatively easy frameworks provide implicit reliability queries monitored failed queries simply contrast actor languages like erlang used engineer reliable general purpose computation often recovering failed stateful computations actor languages actor model concurrency consists independent processes communicating means messages sent asynchronously processes process send message process address erlang process identifier pid remote process may reside different host notion actors originated fault tolerance provided less widely used mpi implementations like used widely general metaphor concurrency well incorporated number niche programming languages recently come back prominence rise multicore chips also distributed programming data centres cloud concurrency data isolation actors natural paradigm engineering reliable scalable systems model two main concepts actors unit computation messages unit communication actor contains addresses actors aware addresses either locations memory direct physical attachments network addresses pure actor language messages way actors communicate receiving message actor following send messages another actor create new actors iii designate behaviour handle next message receives model impose restrictions order actions must taken similarly two messages sent concurrently received order features enable actor based systems support indeterminacy quasicommutativity providing locality modularity reliability scalability actors one paradigm languages libraries related message passing paradigms recent example languages include rust provide explicit channels similar actor mailboxes probably famous message passing library mpi apis many languages widely used clusters high performance computers however arguable important contribution actor model asynchronous communication messages coupled sender neither transferred synchronously temporary container transmission takes place buffer queue mailbox message sent trinder chechina papaspyrou sagonas thompson scaling reliably receiver entity responsible message erlang programming language based actor model history use production systems initially developer ericsson widely open source adoption actor frameworks many languages include akka scala cloud haskell parley python termite scheme currently active use development moreover recently defined rust language version actor model built albeit imperative context iii erlang support concurrency erlang actors termed processes virtual machines termed nodes key elements actor model fast process creation destruction lightweight processes enabling concurrent processes single host ram fast asynchronous message passing copying semantics process monitoring strong dynamic typing selective message reception default erlang processes addressed process identifier pid spawn fun spawns process execute anonymous function given argument spawn primitive binds new process identifier new process execute function defined subsequent call finish sends messaged finish process identified alternatively processes given names using call form register http http registers process name node process name table already present subsequently names used refer communicate corresponding processes send message hello distributed erlang system executes multiple nodes nodes freely deployed across hosts located different hosts help make distribution transparent programmer two nodes connect transitively sharing sets connections without considerable care quickly leads fully connected graph nodes process may spawned explicitly identified node spawn fun remote process addressed local significant burden programmer identify remote nodes large systems return sections scalability erlang systems reliability erlang designed solve particular set problems namely building telecommunications infrastructure systems need scalable accommodate hundreds thousands calls concurrently soft systems need highlyavailable reliable robust case failure come software hardware faults given inevitability latter erlang adopts let fail philosophy error handling encourage programmers embrace fact process may fail point rely supervision mechanism discussed shortly handle failures figure illustrates erlang support concurrency multicores distribution erlang node represented yellow shape trinder chechina papaspyrou sagonas thompson scaling reliably figure conceptual view erlang concurrency multicore support distribution rectangle represents host address red arc represents connection erlang nodes node run multiple cores exploit inherent concurrency provided done automatically user intervention needed typically core associated scheduler schedules processes new process spawned core process spawns work moved different scheduler allocation algorithm scheduler allows process ready compute fixed number computation steps switching another erlang functions bifs implemented start project run completion scheduled causing performance responsiveness problems bif long execution time scaling erlang provided two different ways possible scale within single node means multicore virtual machine exploiting concurrency provided multiple cores numa nodes also possible scale across multiple hosts using multiple distributed erlang nodes reliability erlang actor languages process private state preventing failed failing process corrupting state processes messages enable stateful interaction contain deep copy value shared references pointers senders internal state moreover erlang avoids type errors enforcing strong typing albeit dynamically connected nodes check liveness heartbeats monitored outside erlang operating system process however important way achieve reliability supervision allows process monitor status child process react failure example spawning substitute process replace failed process supervised processes turn supervise processes leading supervision tree supervising supervised processes may different nodes different hosts hence supervision tree may span multiple hosts nodes provide reliable distributed service registration global namespace maintained every node maps process names pids mean talk reliable system one named process distributed system restarted without requiring client processes also restarted name still used communication see global registration action consider pong server process global clients server send messages registered name global finish server fails supervisor spawn replacement server process new pid register name thereafter client messages delivered new server process return discuss scalability limitations maintaining global namespace section ets erlang term storage erlang pragmatic language actor model supports pure erlang processes besides communicating via asynchronous message passing also share data public memory areas called ets tables trinder chechina papaspyrou sagonas thompson scaling reliably erlang term storage ets mechanism central component erlang implementation used internally many libraries underlies databases ets tables stores store erlang tuples one positions tuple serves lookup key ets table type may either set bag implemented hash table currently implemented avl tree main operations ets supports table creation insertion individual entries atomic bulk insertion multiple entries table deletion lookup entry based key destructive update operations implemented functions erlang code snippet creates set ets table keyed first element entry atomically inserts two elements keys updates value associated table entry key looks entry table ets new set public keypos ets insert table tuple value ets table another tuple value key value ets lookup table ets tables heavily used many erlang applications partly due convenience sharing data programming tasks also partly due fast implementation shared resource however ets tables induce contention become scalability bottleneck shall see section iii benchmarks scalability reliability two benchmarks use throughout article orbit measures scalability without looking reliability ant colony optimisation aco allows measure impact scalability adding global namespaces ensure reliability source code benchmarks together detailed documentation available https release project team also worked improve reliability scalability erlang programs including substantial approximately lines erlang code simdiasca simulator instant messenger typical erlang applications cover systematically orbit orbit computation symbolic algebra generalises transitive closure computation compute orbit given space list generators applied initial vertex creates new values generator functions applied new values new value generated orbit suitable benchmark number aspects characterise class real applications core data structure maintains set distributed environments implemented distributed hash table dht similar dhts used replicated form nosql database management systems also distributed mode uses standard techniques like termination algorithm choosing orbit size benchmark parameterised specify smaller larger computations suitable run single machine section many nodes sectioni moreover hundred lines code shown figure computation initiated master creates number workers single node scenario benchmark workers correspond processes workers also spawn processes apply generator functions subset input values thus creating intraworker parallelism distributed version trinder chechina papaspyrou sagonas thompson scaling reliably figure distributed erlang orbit architecture workers mapped nodes benchmark processes spawned master node worker nodes maintaining dht fragment newly spawned process gets share parent credit returns completion computation finished master node collects credit workers completed ant colony optimisation aco ant colony optimisation metaheuristic applied large number combinatorial optimisation problems purpose article applied scheduling problem known single machine total weighted tardiness problem smtwtp number jobs given lengths arranged single linear schedule goal minimise cost schedule determined certain constraints aco method attractive point view distributed computing benefit multiple cooperating colonies running separate compute node consisting multiple ants ants simple computational agents figure distributed erlang ant colony optimisation architecture rently compute possible solutions input problem guided shared information good paths search space also certain amount stochastic variation allows ants explore new directions multiple colonies increases number ants thus increasing probability finding good solution implement four distributed coordination patterns aco computation follows implementation individual colonies perform number local iterations generations ants report best solutions globallybest solution selected reported colonies use update pheromone matrices process repeated number global iterations aco single master node collects colonies best solutions distributes overall best solution back colonies figure depicts process node placements cluster nodes master process spawns colony processes available nodes next step colony process spawns ant processes local node ant iterates times returning result colony master colony iterates times reporting best solution receiving solution master process validated implementation applying number standard smtwtp instances obtaining good results cases confirmed number perfect solutions increases increase number colonies aco master node receives messages colonies thus could become bottleneck addresses tree submasters figure node bottom level collecting results small number colonies fed tree nodes higher levels selecting best solutions children globally reliable aco single colony fails report back trinder chechina papaspyrou sagonas thompson scaling reliably figure distributed erlang ant colony optimisation architecture tem wait indefinitely adds fault tolerance supervising colonies faulty colony detected restarted allowing system continue execution scalable reliable aco also adds using supervision within new section architecture discussed detail scaling erlang single host investigate erlang scalability built bencherl extensible open source benchmark suite web bencherl shows application performance changes resources like cores schedulers added options control resources change number nodes number erlang vms used typically multiple hosts erlang scalability limits section investigates scalability erlang language persistent storage levels aspect choose explore security large scale systems example one might imagine providing enhanced security systems multiple clusters cloud instances connected wide area network assume existing security mechanisms used virtual private network number cores per node number schedulers threads execute erlang processes parallel binding topology cores underlying computer node release flavor arguments used start erlang nodes information bencherl available http trinder chechina papaspyrou sagonas thompson scaling reliably parallelism without parallelism time schedulers parallelism without parallelism speedup schedulers figure runtime speedup two configurations orbit benchmark using using bencherl investigated scalability initial set twelve benchmarks two substantial erlang applications using single erlang node machines cores including bulldozer machine specified appendix set experiments reported confirmed programs scaled well recent release time also revealed language level scalability bottlenecks figure shows runtime speedup curves orbit benchmark master workers run single erlang node configurations without parallelism configurations program scales runtime continuously decreases add schedulers exploit cores speedup benchmark without intraworker parallelism without spawning additional processes computation green curve almost linear cores increases less rapidly point see similar clearly visible pattern configuration red curve performance improvement beyond schedulers due asymmetric characteristics machine consists modules couple two conventional cores share early pipeline stages floating point unit cache rest module benchmarks however scale well experienced significant slowdowns run many schedulers threads example benchmark multiple processes accessing shared ets table figure shows runtime speedup curves eight cores hyperthreading intel machine shows runtime increases beyond two schedulers program exhibits slowdown instead speedup many benchmarks obvious reasons poor scaling like limited parallelism application contention shared resources reasons poor scaling less obvious benchmarks exactly chosen study detail subsequent work simple example parallel bencherl benchmark spawns processes creates list timestamps checks timestamp list strictly greater previous one sends result parent figure shows eight cores additional core leads slowdown thereafter small speedup obtained cores slowdown small aspect benchmark easily overlooked explains poor scalability benchmark creates timestamps using erlang function whose implementation acquires global lock order return unique timestamp two calls erlang even different processes guaranteed produce monotonically increasing values lock precisely trinder chechina papaspyrou sagonas thompson scaling reliably time time schedulers schedulers speedup speedup schedulers schedulers figure runtime speedup bencherl benchmark using figure runtime speedup bencherl benchmark called parallel using tleneck limits scalability benchmark describe work address timing issues figure iii tion need discover nodes design works well small numbers nodes however emergent server architecture scales hundreds nodes design becomes expensive system architects must switch default erlang model need start using hidden nodes share connection sets investigated scalability limits imposed network connectivity costs using several orbit calculations two large clusters kalkyl athos specified appendix kalkyl results discussed figure section shows representative results distributed erlang computing orbits elements athos cases performance degrades beyond scale nodes orbit nodes orbit figure illustrates additional network traffic induced fully connected network allows comparison discussion investigations identified contention shared ets tables shared resources like timers key scalability issues section outlines addressed issues recent releases distributed erlang scalability network connectivity costs normal distributed erlang nodes communicate share connection sets typically leads fully connected graph nodes system nodes maintain connections relatively expensive tcp connections continual maintenance traffic design aids transparent trinder chechina papaspyrou sagonas thompson scaling reliably tween number packets sent fully connected network sent network partitioned using new paper investigates impact data size computation time calls independently combination scaling properties common generic server processes global information costs maintaining global information known limit scalability distributed systems crucially process namespaces used reliability global investigate scalability limits imposed distributed erlang global information designed implemented open source parameterisable scalable benchmarking framework measures throughput latency distributed erlang commands cluster erlang nodes design influenced basho bench benchmarking tool riak instance acts peer providing scalability reliability eliminating central coordination single point failure evaluate scalability distributed erlang measure adding hosts increases throughput total number successfully executed distributed erlang commands per experiment figure shows parameterisable internal workflow debench three classes commands commands function tunable argument size computation time run remote node include spawn rpc synchronous calls server processes global commands entail synchronisation across connected nodes global global iii local commands executed independently single node look local name table benchmarking conducted host configurations kalkyl cluster steps measures throughput successful commands per second minutes one erlang host one instance full focus impact different proportions global commands mixing global local commands figure shows even low proportion global commands limits scalability distributed erlang global commands limits scalability around nodes figure reports latency commands shows latencies local commands stable scale latency global commands increases dramatically scale results illustrate impact global operations throughput latency distributed erlang system severe explicit placement network connectivity global information impact performance scale investigations also identified explicit process placement programming issue scale recall section iii distributed erlang requires programmer identify explicit erlang node spawning process identifying appropriate node becomes significant burden large dynamic systems problem exacerbated large distributed systems hosts may identical different hardware capabilities different software installed communication times may may fast send message vms host slow vms different hosts large distributed system factors make difficult deploy applications especially scalable portable manner moreover programmer may able use knowledge decide spawn processes enable application run efficiently application deployed different platform platform changes hosts fail added becomes outdated trinder chechina papaspyrou sagonas thompson scaling reliably figure internal workflow figure scalability percentage global commands distributed erlang discussion investigations confirm three scalability limitations erlang developer folklore maintaining fully connected network erlang nodes limits scalability example orbit typically limited nodes global operations crucially global operations required reliability maintain global namespace seriously limit scalability distributed erlang systems explicit process placement makes hard built performance portable applications large architectures issues cause designers reliable large scale systems erlang depart standard erlang model using techniques like hidden nodes ing pids data structures section develop language technologies address issues iii persistent storage large scale system needs reliable scalable persistent storage studied scalability limits erlang persistent storage alternatives envisage typical large server around cores around hosts reviewed requirements scalable available persistent storage evaluated four popular erlang dbms requirements target scale trinder chechina papaspyrou sagonas thompson scaling reliably figure latency commands number erlang nodes increases around hosts mnesia couchdb unsurprisingly suitable however nosql dbms like cassandra riak potential investigated current scalability limits riak nosql dbms using basho bench benchmarking framework cluster nodes independent disks found scalability limit riak version nodes kalkyl cluster study placed public scientific domain previously anecdotal developer experience also shown resources like memory disk network limit scalability riak instrumenting global otp libraries identified specific riak remote procedure call fails scale outline later releases riak refactored eliminate scalability bottlenecks discussion conclude nosql dbmss potential deliver reliable persistent storage erlang target scale approximately hosts specifically erlang cassandra interface available riak already provides scalable available persistent storage nodes moreover scalability riak much improved subsequent versions improving language scalability section outlines scalable distributed erlang libraries designed implemented address distributed erlang scalability issues identified section erlang introduces two concepts improve scalability partition set nodes erlang system reduce network connectivity partition global data section placement alleviates issues explicit process placement large heterogeneous networks section two features independent used separately combination overview erlang section outline semantics validation sections iii respectively erlang design reduce number connections node maintains size name spaces minimise global information specifically names registered synchronised nodes within following parameters name trinder chechina papaspyrou sagonas thompson scaling reliably list nodes list registered names node belong many none node belongs behaves usual distributed erlang node library defines functions shown table functions manipulate provide information creating providing list nodes given remaining functions manipulate names registered provide information names example register process pid name name sgroupname use following function name registered process executed node belongs given neither name pid already registered group sgroupname name pid yes illustrate impact scalability repeat global operations experiment section figure erlang experiment partition set nodes containing ten nodes hence names replicated synchronised ten nodes nodes distributed erlang results figure show global operations throughput distributed erlang stops growing nodes throughput erlang continues grow linearly connection topology extremely flexible may organised hierarchy arbitrary depth branching could multiple levels tree see figure moreover necessary create hierarchy example constructed orbit implementation using ring given flexible way organising distributed systems key questions design erlang system following structured depending reason nodes grouped reducing number connections reducing namespace freely structured tree ring topology large smaller mean communication synchronisation state nodes constrains maximum size found constraint serious restriction example many either relatively small internal terminal elements topology leaves nodes tree nodes different communicate two nodes communicate erlang system minimise number connections communication nodes different typically routed via gateway nodes belong avoid single points failure reliability minimise communication load multiple gateway nodes processes may required information make design choices provided tools section vii benchmarking challenge systematically refactor distributed erlang application erlang outlined section detailed discussion distributed system design refactoring erlang provided recent article illustrate typical erlang system designs showing refactorings orbit aco benchmarks section iii distributed erlang computation starts master node actual computation done worker nodes distributed erlang version nodes interconnected messages transferred directly sending node receiving node figure contrast erlang version nodes grouped messages transferred different via submaster nodes figure fragment code creates node follows master groupname case groupname groupname nodes trinder chechina papaspyrou sagonas thompson scaling reliably table summary functions function description sgroupname nodes creates new consisting nodes sgroupname deletes sgroupname nodes adds list nodes sgroupname nodes removes list nodes returns list known node returns list tuples node belongs returns list nodes node belongs sgroupname returns list nodes given info returns state information sgroupname name pid registers name given sgroupname name pid name changes registration given sgroupname name unregisters name given node node returns list registered names given node sgroupname returns list registered names given sgroupname name return pid name registered given node sgroupname name send sgroupname name msg send message name registered given send node sgroupname name msg master groupname nodes format exception message end distributed erlang erlang orbit aco presented section viii similarly introduce benchmark section create scalable reliable aco see figure apart reducing number connections also reduce global namespace information instead registering name pid globally nodes names registered nodes comparative performance evaluation placement recall section iii distributed erlang spawns process onto explicitly named erlang node spawn fun also recall portability programming effort issues associated explicit placement large scale systems discussed section trinder chechina papaspyrou sagonas thompson scaling reliably throughput number nodes distributed erlang erlang figure global operations distributed erlang erlang represents process represents colony process master node level master process represents ant process represents level represents node colony nodes level level figure erlang aco architecture address issues developed placement library enables programmer select nodes spawn processes based information properties nodes example process performs lot computation one would like spawn node considerable computation power two processes likely communicate frequently would desirable spawn node nodes fast interconnect ated hosts total currently available ram installed software hardware configuration etc second deals notion communication distances models communication times nodes distributed system therefore instead specifying node use attr function define target node implemented two erlang libraries support placement first deals node attributes describes properties individual erlang vms report investigation communication latencies range numa cluster architectures demonstrate effectiveness placement libraries using spawn attr params fun trinder chechina papaspyrou sagonas thompson scaling reliably figure erlang architecture benchmark athos cluster semantics precise specification basis validation provide operational semantics operations figure defines state erlang system associated abstract syntax variables abstract syntax variables left defined members sets denoted turn may contain tuples denoted sets particular process name pid nis set state system modelled four tuple comprising set set set set nodes type group associated nodes namespace additionally name whereas consists one node hidden node simultaneously acts node group group namespace share node free normal hidden groups names uniquely defined nodes associated therefore group names either nogroup set namespace set name process pid pairs replicated nodes associated group node following four parameters identifier either hidden normal connections names groups node belongs node belong either list one free groups type free group defined node type connections set transitions semantics form state command value meaning executing command node state returns value transitions semantics presented detail illustrate function section function returns list names registered node belongs empty list otherwise figure denotes disjoint set union issgroupnode returns true node member false otherwise outputnms returns set process names registered namespace iii semantics validation semantics concrete readily made executable erlang lists replacing sets throughout executable trinder chechina papaspyrou sagonas thompson scaling reliably grs fgs fhs nds state node grs namespace fgs namespace fhs namespace nds node connections nogroup namespace name pid connections normal hidden nogroup figure erlang state grs fgs fhs nds grs fgs fhs nds nms grs fgs fhs nds nis grs nms outputnms issgroupnode grs otherwise issgroupnode grs nis grs outputnms figure erlang semantics function tics allows users engage understand semantics behaves vis vis library giving opportunity assess correctness library semantics better still automatically assess system behaves comparison executable semantics executing lockstep guided constraints operations possible point building abstract state machine model library generate random sequences traces model appropriate library data ated random generation supported quickcheck testing system architecture testing framework shown figure first abstract state machine embedded module derived executable semantic specification state machine defines abstract state representation transition one state another operation applied test case data generators defined control test case generation includes automatic generation eligible operations input data trinder chechina papaspyrou sagonas thompson scaling reliably operations test oracles encoded postcondition operations testing test command applied abstract model library application test command abstract model takes abstract model current state new state described transition functions whereas application test command library leads system new actual state actual state information collected node distributed system merged normalised format abstract state representation test successful execution test command test oracles specified command satisfied various test oracles defined operations instance one generic constraints applies operations operation normalised system state equivalent abstract state thousands tests run three kinds errors subsequently corrected found errors library implementation found including one error due synchronisation nodes related operation erroneously raised exception also found couple trivial errors semantic specification missed manual examination finally found situations figure testing using quickcheck inconsistencies semantics library implementation despite states equivalent example particular values returned functions certain errors overall automation testing boosted confidence correctness library implementation semantic specification work reported detail improving scalability section reports primary library improvements designed implemented address scalability reliability issues identified section improvements erlang term storage ets tables heavily used erlang systems focus scalability improvements start describing redesign including improvements release project work prior historical improvements relevant scalability study form basis subsequent changes improvements point got support multiple cores release single lock ets table optional fine grained locking ets tables set bag tables introduced adding locks hash buckets reader groups minimise read synchronisation overheads introduced key observation single count multiple readers must synchronised across many cache lines potentially far away numa system maintaining reader counts multiple local caches makes reads fast although writes must check every reader count number bucket locks default number reader groups upgraded illustrate scaling properties trinder chechina papaspyrou sagonas thompson scaling reliably figure runtime ets operations releases updates left updates right shows time seconds lower better number threads schedulers figure runtime ets operations varying numbers reader groups updates left updates right shows runtime seconds lower better number schedulers ets concurrency options using bencherl benchmark intel numa machine hyperthreaded cores specified appendix benchmark inserts items table records time perform operations operation either lookup insert delete experiments conducted set ets table different percentages update operations insertions deletions figure shows runtimes seconds operations different versions varying number schedulers xaxis reflecting scalability ets tables improved recent releases figure shows runtimes seconds operations ets table different numbers reader groups varying number schedulers see one reader group sufficient updates two updates beyond different numbers reader groups little impact benchmark performance except using groups updates slightly degrades performance explored four extensions redesigns ets implementation better scalability allowing programmer control number bucket locks hashbased tables programmer reflect number schedulers expected access pattern using trees get better scalability ets tables described using queue delegation locking improve scalability adopting schemes completely eliminating locks meta table complete discussion work ets found papers outline work trees tree monitors contention different parts trinder chechina papaspyrou sagonas thompson scaling reliably operations microsecond set ordset tree tree number threads number threads operations microsecond set ordset tree tree figure throughput tree variants updates left updates right data structure introducing routing nodes locks response high contention removing response low contention experimental purposes two variants tree implemented represent virtual machine one extends existing avl trees erlang uses treap data structure figure compares throughput tree variants set number schedulers increases unsurprising trees scale much better protected single lock surprising also scale better set due hash tables using locking fixed granularity trees adapt number locks current contention level also parts key range contention occurring improvements schedulers erlang scheduler responsible executing multiple processes concurrently timely fair fashion making optimal use hardware resources implements preemptive multitasking soft realtime guarantees erlang processes normally scheduled reduction count basis one reduction roughly equivalent function call process allowed execute either blocks waiting input typically message process executed quota reductions erlang usually started one scheduler per logical core available host machine schedulers implemented threads erlang process spawned placed run queue scheduler parent waits queue scheduler allocates slice core time work stealing used balance load cores idle scheduler may migrate process another run queue scheduler run queues visualised figure default load management mechanism load compaction aims keep many scheduler threads possible fully loaded work attempts ensure scheduler threads run work developed new optional scheduler utilisation balancing mechanism available new mechanism aims balance scheduler utilisation schedulers strive equal scheduler utilisation schedulers scheduler utilisation balancing mechanism performance impact system enabled hand enabled results changed timing system normally small overhead due measuring utilisation calculating balancing information depends underlying primitives provided operating system new balancing mechanism results better distribution processes schedulers trinder chechina papaspyrou sagonas thompson scaling reliably reducing probability core contention together improvements interruptable bifs garbage collection results lower latency improved responsiveness hence reliability soft applications iii improvements time management soon start release project time management erlang became scalability bottleneck many applications illustrated parallel benchmark figure issue came prominence severe bottlenecks eliminated subsection motivates outlines improvements time management made incorporated new api time time warping old api still supported time writing use deprecated original time api provides erlang returns erlang system time time since epoch micro second resolution time basis time internally erlang many scalability problems erlang stem specification written time erlang documentation promises values returned strictly increasing many applications ended relying example applications often employ erlang generate unique integers erlang system time align operating system view time since epoch system time however system time freely changed forwards backwards erlang system time without invalidating strictly increasing value guarantee erlang therefore contains mechanism slowly adjusts erlang system time towards system time align one problem time adjustment deliberately presents time accurate frequency required align erlang system time system time smoothly two deviated case clock shifts leap seconds inserted deleted another problem erlang system time system time differ long periods time new api resolve using common technique monotonic time zero point unspecified point time monotonic time allowed make leaps forwards backwards system time allowed erlang system time thus dynamically varying offset erlang monotonic time time retrieval retrieval erlang system time previously protected global mutex made operation thread safe scaled poorly erlang system time erlang monotonic time need run frequency otherwise time offset would constant common case monotonic time delivered operating system solely based machine local clock changed system time adjusted using network time protocol ntp run different frequencies linux exception monotonic clock ntp adjusted runs frequency system time align frequencies erlang monotonic time erlang system time adjust frequency erlang monotonic clock done comparing monotonic time system time delivered calculating adjustment achieve scalably one thread calculates time adjustment use least minute adjustment needs changed new adjustment information published used calculate erlang monotonic time future thread needs retrieve time reads monotonic time delivered time adjustment information previously published calculates erlang monotonic time preserve monotonicity important threads read trinder chechina papaspyrou sagonas thompson scaling reliably time schedulers speedup monotonic time map exactly erlang monotonic time requires synchronisation updates adjustment information using lock lock adjustment information changed means vast majority cases lock allows multiple readers run concurrently prevent bouncing lock use bespoke reader optimised lock implementation reader threads notify presence counters separate concept similar reader indicator algorithm described fig alternatives include counter used snzi algorithm timer wheel bif timer timer wheel contains timers set erlang processes original implementation protected global mutex scaled poorly increase concurrency scheduler thread assigned timer wheel used processes executing scheduler implementation timers uses built function bif operations bif also protected global mutex besides inserting timers timer wheel bif timer implementation also maps timer references timer timer wheel improve concurrency provide bif timer servers erlang processes keep information timers private ets tables insert one timer time timer wheel benchmarks measured several benchmarks bulldozer machine eight dual cpu amd opteron present three first micro benchmark compares execution time erlang receive receive specifies timeout see release project deliverable http schedulers figure bencherl parallel benchmark using erlang erlang different releases runtimes left speedup obtained using erlang right provides default value receive sets timer process blocks receive cancels message arrives total execution time standard timers longer without timers using improved implementation total execution time optimised timers longer without timers second micro benchmark repeatedly checks system time calling erlang calling erlang erlang adding results machine uses schedulers default release times faster release third benchmark parallel bencherl trinder chechina papaspyrou sagonas thompson scaling reliably benchmark section figure shows results executing original version benchmark uses erlang create monotonically increasing unique values using three releases also measure version benchmark call erlang substituted call erlang graph left shows performance time management remained roughly unchanged releases prior improved time management make time management less likely scalability bottleneck even using erlang new time api using erlang friends provides scalable solution graph right side figure shows speedup modified version parallel benchmark achieves vii scalable tools section outlines five tools developed release project support scalable erlang systems tools developed scratch like devo sdmon wombatoam others extend existing tools like percept wrangler include tooling transform programs make scalable deploy scalability monitor visualise tools freely available open source licences devo wrangler wombatoam commercial product tools used profiling refactoring aco orbit benchmarks section iii erlang tool ecosystem consists small tools tracing profiling debugging erlang systems used separately together appropriate solving problem hand rather single monolithic tools presented designed used part ecosystem complement already available functionality rather duplicate erlang runtime system support tracing many types events infrastructure forms basis number tools tracing profiling typically tools build specialise services offered erlang virtual machine number functions recently since release project planned application gives comprehensive overview many data basis actor frameworks languages see section recently become widely adopted commercially tool support remains relatively immature generic nature tools support language rather distinctively concurrent aspects given widespread use erlang tools developed point way tools actor languages frameworks example many erlang tools use tracing support provided erlang actor frameworks akka use jvm monitoring system similarly tools actor languages frameworks could use data derived tracing frameworks probes show section erlang provided host language tracing hooks appropriate infrastructure refactoring scalability refactoring process changing program works without changing done readability testability prepare modification extension case order improve scalability refactoring involves transformation source code typically performed using machine support refactoring tool number tools support refactoring erlang release project chosen extend tools include tidier http http http https http trinder chechina papaspyrou sagonas thompson scaling reliably refactorerl supporting api migration erlang libraries modify erlang library becoming new library result erlang programs using refactored use kind api migration problem uncommon software evolves often changes api library rather simply extend wrangler refactoring perform particular operation instead added framework automatic generation api migration refactorings adaptor module approach automatic api migration works way api function interface changed author api function implements adaptor function defining calls old api terms new definition automatically generate refactoring transforms client code use new api refactoring also supplied api writer clients library upgrade allowing users upgrade code automatically refactoring works generating set rules fold adaptation client code resulting code works directly new api details design choices underlying work technicalities implementation found paper support introducing parallelism introduced support parallelising explicit list operations map foreach process introduction complete computationally intensive task parallel introducing worker process deal call handling erlang generic server parallelise tail recursive function discuss turn details practical examples refactorings appear conference paper describing work uses map foreach list processing among obvious places parallelism introduced added small library wrangler called provides parallel implementations map foreach transformation explicit use sequential use parallel counterparts straightforward even manual refactoring would problem however operation could also implemented differently using recursive functions list comprehensions etc identifying kind implicit usage done using wrangler code inspection facility refactoring turns implicit explicit also specified using wrangler transformation api computations two tasks depend executed parallel introduce new process refactoring implemented wrangler used spawn new process execute task parallel parent process result new process sent back parent process consume needed order block computations depend result returned new process receive expression placed immediately point result needed list processing functions refactored explicit map operation many due data dependencies instance example might perform recursion list accumulating results accumulator variable situation possible float computations parallel computations done certain dependency constraints satisfied done program slicing discussed support program slicing program slicing general technique program analysis extracting part program also called slice influences influenced given point interest slicing criterion static program slicing generally based program dependency including control dependency data dependency backward trinder chechina papaspyrou sagonas thompson scaling reliably slicing used refactorings described also useful general made available wrangler inspector menu work compared tool developed another project also wrangler front end work concentrates skeleton introduction work using static analysis slicing transforming programs make suitable introduction parallel structures scalable deployment developed wombatoam provide deployment operations maintenance framework erlang distributed systems systems typically consist number erlang nodes executing different hosts hosts may different hardware operating systems physical virtual run different versions prior development wombatoam deployment systems would use scripting erlang shell state art actor frameworks would possible adapt wombatoam approach frameworks straightforward way architecture architecture wombatoam summarised figure originally system problems addressing full scalability role played central master node current version additional layer middle managers introduced allow system scale easily thousands deployed nodes diagram shows northbound interfaces web dashboard provided restful connections master operations master delegated middle managers http wombatoam https commercial tool available erlang solutions engage directly managed nodes managed node runs collection services collect metrics raise alarms forth describe services wombatoam designed collect store display various kinds information event running erlang systems data accessed managed web dashboard include following metrics wombatoam supports collection hundred metrics including instance numbers processes node message traffic node regular basis erlang vms running hosts also collect metrics defined users within metrics collection frameworks interface tools log display information metrics displayed histograms covering different windows last fifteen minutes hour day week month notifications well metrics support monitoring collection notifications running nodes notifications one time events generated using erlang system architecture support libraries sasl part standard distribution lager logging displayed logged occur alarms alarms complex entities alarms state raised dealt cleared also identities alarm may raised node multiple times instance need folsom collects publishes metrics erlang api https https https trinder chechina papaspyrou sagonas thompson scaling reliably wombat web dashboard wombat cli rest rest northbound interface metrics api notif api alarms api orchestration api node manager metrics notif alarms orchestration node manager metrics notif alarms orchestration topology api erlang master core core including topology orchestration data topology erlang middle manager core erlang libcloud provision vms rest provision software ssh infrastructure provider agents metrics agent virtual machine agent lager agent alarms agent managed erlang node figure architecture wombatoam dealt separately alarms generated sasl lager notifications topology topology service handles adding deleting discovering nodes also monitors whether accessible notifies services periodically tries reconnect nodes available also notifies services middle manager part talk nodes directly instead asks node manager service node manager service maintains connection managed nodes via erlang distribution protocol loses connection towards node periodically tries reconnect also maintains states nodes database connection towards node lost node manager changes node state raises alarm node manager rest api since node states provided via topology service rest api orchestration service deploy new erlang nodes already running machines also provision new virtual machine instances using several cloud providers deploy erlang nodes instances communicating trinder chechina papaspyrou sagonas thompson scaling reliably cloud providers orchestration service uses external library called erlang solutions written open source erlang wrapper called elibcloud make libcloud easier use wombatoam note wombatoam orchestration provide platform writing erlang applications provides infrastructure deploying deployment mechanism consists following five steps registering provider wombatoam provides interface different cloud providers support openstack standard amazon api wombatoam also provides interface using fixed set machines wombatoam backend implemented two driver modules elibcloud driver module uses elibcloud libcloud libraries communicate cloud providers ssh driver module keeps track fixed set machines uploading release release either proper erlang release archive set erlang modules important aspect wombatoam point view start stop commands explicitly specified enable wombatoam start stop nodes needed defining node family next step creating node family entity refers certain release contains deployment domains refer certain providers contains information necessary deploy node defining deployment domain step deployment domain created specifies providers used unified cloud api https provisioning machines username used wombatoam connects hosts using ssh node deployment deploy nodes wombatoam user needs specify number nodes node family nodes belong nodes dynamically added removed system depending needs application nodes started wombatoam ready initiate run application iii monitoring visualisation key aim designing new monitoring visualisation tools adapting existing ones provide support systems running parallel distributed hardware specifically order run modern erlang systems particular erlang systems necessary understand single host multicore distributed nature need able understand systems run erlang multicore virtual machine scheduler associated core manages run queue processes migrate queues work stealing algorithm time understand dynamics distributed erlang program user explicitly spawns processes builds existing percept tool provide post hoc offline analysis visualisation erlang systems designed allow users visualise tune parallel performance erlang systems single node single manycore host visualises erlang application level concurrency identifies concurrency bottlenecks uses contrast multicore programmers control processes spawned however still need gain insight behaviour programs tune performance https trinder chechina papaspyrou sagonas thompson scaling reliably erlang built tracing profiling monitor process states waiting running runnable free exiting waiting suspended process considered inactive running runnable process considered active program runs percept events collected stored file file analysed results stored ram database data viewed interface process offline profiling distributed erlang application using shown figure percept generates zoomable concurrency graph showing number active processes point profiling dips graph represent low concurrency lifetime bar process also included showing points lifetime process active well information extends percept number ways detailed including importantly distinguishing running runnable time process apparent process runnability comparison shown figure orange represents runnable green represents running shows clearly potential concurrency exploited showing scheduler activity number active schedulers time profiling recording information execution including migration history process run queues statistics message passing processes number messages average message size process accumulated runtime accumulated time process running state presenting process tree hierarchy structure indicating relationships processes recording dynamic function call hierarchy structure showing calling relationships functions program run amount time spent function tracing activities distributed system unlike global group allows dynamic changes structure distributed erlang system order support erlang also extended allow profiling related activities dynamic changes structure distributed erlang system captured also improved percept follows enabling control profiled profiling port activities schedulers activities message passing process migration garbage collection activities profiling process runnability indicated proc flag always enabled selective function profiling processes built version fprof measure function execution time measures everything else fprof measures eliminating measurement function execution time gives users freedom profiling function calls invoked program execution example choose profile functions defined applications code libraries improved dynamic function callgraph dynamic function callgraph user able understand causes certain events heavy calls particular function examining region around node function including path root graph edge callgraph annotated trinder chechina papaspyrou sagonas thompson scaling reliably figure offline profiling distributed erlang application using figure showing processes running runnable queues number times target function called source function well information finally also improved scalability percept three ways first parallelised processing trace files multiple data files processed time also compressed representation call graphs cached history generated web pages together make system responsive scalable tracing dtrace provides dynamic tracing support various flavours unix including bsd mac systemtap linux allow monitoring live running systems minimal overhead used administrators language developers application developers alike examine behaviour applications language implementations operating system development even live production systems comparison similar tools instrumentation frameworks relatively lightweight require special recompiled versions software examined special tools create meaningful information data gathered using probs possible identify bottlenecks trinder chechina papaspyrou sagonas thompson scaling reliably run queues figure dtrace runqueue visualisation bang benchmark schedulers time xaxis applications bottlenecks identified using large number probes inserted example explore scheduler runqueue lengths probes used measure number processes per scheduler number processes moved work stealing number attempts gain lock many succeed immediately etc figure visualises results monitoring shows size run queues vary execution bang bencherl benchmark schedulers application bottlenecks identified alternative based instead erlang builtin tracing mechanism implementation reuses existing infrastructure far possible uses different mechanism collecting information erlang programs different format trace files storage infrastructure presentation facilities devo designed provide online visualisation single node multiple cores https ple nodes grouped aspects erlang erlang systems visualisation within browser web sockets providing connections javascript visualisations running erlang systems instrumented trace tool builder ttb figure shows visualisations devo modes side single compute node shown consists two physical chips upper lower halves diagram six cores hyperthreading gives twelve virtual cores hence run queues total size run queues shown colour height column process migrations illustrated fading arc queues within circle green arc shows migration physical core grey one chip blue shows migrations two chips side figure visualisation action node graph represents erlang node colours red green blue orange used represent node belongs evident three nodes central triangle belong multiple groups act routing nodes nodes colour arc joining two nodes represents current intensity communication nodes green quiescent red busiest tool specifically designed monitoring systems purpose accomplished means shadow network agents collect data running system example deployment shown figure blue dots represent nodes target system nodes make infrastructure network deployed basis configuration file describing network architecture terms hosts erlang nodes global group partitions tracing performed trinder chechina papaspyrou sagonas thompson scaling reliably figure devo low visualisations tored nodes also specified within configuration file agent started master node free node configured tracing applied every monitored node traces stored binary format agent file system shadow network follows system changes agents started stopped runtime required shown figure changes persistently stored last configuration reproduced restart course shadow network always updated via user interface agent takes care free node tries get contact nodes apply tracing stated master binary files stored host file system tracing internally used order track operations happening runtime asynchronous message sent master whenever one changes occurs since process traced single process time node included belonging one controlled one agent node removed group group deleted another agent takes shown figure agent stopped traces controlled nodes switched monitoring network also supervised order take account network fragility agent node goes another node deployed play role also periodic consistency checks system whole inconsistency detected part system restarted monitor activities one node time particular messages displayed runtime soon agent stopped related tracing files fetched across network master made available readable format master file system provides facilities online visualisation data well post hoc offline analysis figure shows real time messages sent data trinder chechina papaspyrou sagonas thompson scaling reliably figure architecture also used input animated devo visualisation illustrated side figure viii figure evolution top eliminating bottom systemic evaluation preceding sections investigated improvements individual aspects erlang system ets tables section section analyses impact new tools technologies sections vii concert investigating deployment reliability scalability orbit aco benchmarks section iii experiments reported representative similar experiments show consistent results range several benchmarks substantial approximately lines erlang case study several state art numa architectures four clusters specified appendix coherent presentation many results available article release project bulk experiments reported conducted athos cluster using see deliverable available online http trinder chechina papaspyrou sagonas thompson scaling reliably figure online monitoring associated erlang libraries experiments cover two measures scalability orbit fixed size computation scaling measure relative speedup strong scaling speedup relative execution time single core work aco increases compute resources weak scaling appropriate measure benchmarks also evaluate different aspects orbit evaluates scalability impacts network connections aco evaluates impact network connections global namespace required reliability performs better small number nodes communication direct rather via gateway node number nodes grows however delivers better speedups beyond nodes case orbit elements beyond nodes case orbit elements increase size orbit beyond version fails due fact vms exceed available ram contrast experiments run successfully even orbit elements orbit figure shows speedup benchmarks section measurements repeated seven times plot standard deviation results show ant colony optimisation aco deployment deployment monitoring aco large lines erlang simulation using wombatoam section athos cluster detailed trinder chechina papaspyrou sagonas thompson scaling reliably relative speedup number nodes cores figure speedup distributed erlang erlang orbit elements strong scaling example experiment deploys erlang nodes without enabling monitoring hence allocates three nodes per core nodes athos hosts figure shows wombatoam deploys nodes approximately nodes per second common least one core per erlang node related experiment nodes deployed one per core nodes athos hosts nodes per second crucially cases deployment time linear number nodes deployment time could reduced logarithmic number nodes using standard techniques however demand erlang systems long running servers measurement data shows two important facts shows wombatoam scales well deployment base erlang nodes wombatoam overhead monitored node typically less effort node conclude wombatoam capable deploying monitoring substantial distributed erlang erlang programs experiments remainder section use standard distributed erlang configuration file deployment reliability erlang changes organisation processes recovery data language level seek show changes disrupted erlang reliability mechanisms level changed exercise erlang reliability mechanisms managing node failures network congestion etc detailed study erlang reliability including use replicated databases recovering instant messenger chat sessions finds similar results evaluate reliability two aco versions using chaos monkey service kills processes running system random recall provides reliability registering names critical processes globally registers within section chaos monkey runs every erlang node master submasters colony nodes killing random erlang process every second aco versions run completion recovery failure frequency measurable trinder chechina papaspyrou sagonas thompson scaling reliably time sec number nodes experimental data best fit figure wombat deployment time execution time number nodes cores figure aco execution times release weak scaling trinder chechina papaspyrou sagonas thompson scaling reliably impact runtime processes recovered within virtual machine using globally synchronised local recovery information example common platform typical erlang process recovery times around around less unix process recovery time platform conducted detailed experiments instant messenger benchmark obtained similar results conclude sraco reliable erlang preserves distributed erlang reliability model remainder section outlines impact maintaining recovery information required reliability scalability scalability figure compares runtimes versions aco section release outlined section maintains fully connected graph nodes registers process names reliability hence scales significantly worse unreliable conclude providing reliability standard distributed erlang process registration dramatically limits scalability provide reliability hence register process names maintains fully connected graph nodes limits scalability maintains connections registers process names within scales best figure illustrates maintaining process namespace fully connected network impacts performance reinforces evidence orbit benchmarks others partitioning network erlang nodes significantly improves performance large scale investigate impact erlang network traffic measure number sent received packets gpg cluster three versions aco figure shows total number sent packets highest traffic red line belongs lowest traffic belongs dark blue line shows erlang significantly reduces network traffic erlang nodes even name registration less network traffic global name registration iii evaluation summary shown wombatoam capable deploying monitoring substantial distributed erlang erlang programs like aco chaos monkey experiments show reliable hence erlang preserves distributed erlang reliability model orbit scales better sraco scales better sraco significantly less network traffic conclude even global recovery data maintained partitioning fullyconnected network reduces network traffic improves performance distributed orbit instances reach scalability limits around nodes orbit scales nodes erlang limited input size still scaling well nodes cores hence exceeded node scaling limits distributed erlang identified section reached scaling limits erlang architecture comparing scalability curves shows maintaining global recovery data process name space dramatically limits scalability comparing graco scalability curves shows scalability recovered partitioning nodes hence maintaining recovery data within relatively small group nodes results consistent experiments discussion distributed actor platforms like erlang scala akka common choice system architects model trinder chechina papaspyrou sagonas thompson scaling reliably figure number sent packets automatic reliability mechanisms makes extremely easy engineer scalable reliable systems targeting emergent server architectures hundreds hosts tens thousands cores report systematic effort improve scalability leading distributed actor language preserving reliability work vade mecum addressing scalability reliable actor languages frameworks also high impact downloads improved running month undertaken first systematic study scalability distributed actor language covering language persistent storage levels developed bencherl tools purpose key scalability issues identify include contention shared ets tables shared resources like timers key language scalability issues costs maintaining network maintaining global recovery information explicit process placement unsurprisingly scaling issues distributed actor language common distributed parallel languages frameworks paradigms like cilk legion establish scientifically folklore limitations around connected nodes distributed erlang section actor model panacea still scalability problems algorithms write either within single actor way structure communicating actors range pragmatic issues also impact performance scalability actor systems including memory occupied processes even quiescent mailboxes filling etc identifying resolving problems tools like wombatoam needed however many scalability issues arise erlang departs private state principle actor model maintaining shared state ets tables shared global process namespace recovery designed implemented set scalable distributed erlang libraries address scalability issues key constructs partitioning network global process namespace process placement deploying distributed erlang applications large heterogeneous architectures portable way provided state transition operational semantics new validated trinder chechina papaspyrou sagonas thompson scaling reliably library implementation semantics using quickcheck section improve scalability erlang libraries improved implementation shared ets tables time management load balancing schedulers following systematic analysis ets tables number bucket locks reader groups increased developed evaluated four new techniques improving ets scalability programmer control number bucket locks tree data structure iii queue delegation locking eliminating locks meta table introduced new scheduler utilisation balancing mechanism spread work multiple schedulers hence cores new synchronisation mechanisms reduce contention time management mechanisms june majority changes included primary releases scalable actor language implementation thoughtful design engineering required schedule large numbers actors hosts many cores minimise contention shared resources section facilitate development large erlang systems make understandable developed range tools proprietary wombatoam tool deploys monitors large distributed erlang systems multiple possibly heterogeneous clusters clouds made open source releases four concurrency tools detects concurrency bad smells wrangler provides enhanced concurrency refactoring devo tool enhanced provide interactive visualisation erlang systems new tool monitors erlang systems anticipate tools guide design tools large scale distributed actor languages frameworks section vii report reliability scalability implications new technologies using range benchmarks consistently use orbit aco benchmarks throughout article report measurements range numa cluster architectures key scalability experiments conducted athos cluster hosts cores even global recovery data maintained partitioning network reduces network traffic improves performance orbit aco benchmarks hosts crucially exceed node limit distributed erlang reach scalability limits erlang cores chaos monkey experiments show two versions aco reliable hence erlang preserves erlang reliability model however aco results show maintaining global recovery data global process name space dramatically limits scalability distributed erlang scalability however recovered maintaining recovery data within appropriately sized results consistent experiments benchmarks architectures section viii future work plan incorporate release technologies along technologies generic framework building performant large scale servers addition preliminary investigations suggest erlang ideas could improve scalability actor example akka framework scala could benefit semiexplicit placement cloud haskell partitioning network appendix architecture specifications specifications clusters used measurement summarised table also use following numa machines amd bulldozer cache ram four amd opteron ghz bulldozer cores giving total cores intel numa ram four intel xeon see deliverable http available online trinder chechina papaspyrou sagonas thompson scaling reliably table cluster specifications ram per host cores per name max hosts host total avail kalkyl tintin athos gpg processor ram intel xeon intel xeon interconnection ethernet infiniband amd opteron scribed dozer qdr infiniband intel xeon infiniband eight hyperthreaded cores giving total cores bulldozer microarchitecture https microarchitecture acknowledgements apache liblcoud https would like thank entire release project team technical insights administrative support roberto aloi enrique fernandez casado contributed development measurement wombatoam work supported european union grant science symbolic computing infrastructure europe release paradigm reliable server software engineering physical sciences research council grant hpcgap high performance computational algebra discrete mathematics aragon seidel randomized search trees proceedings annual symposium foundations computer science pages october joe armstrong programming erlang software concurrent world pragmatic bookshelf joe armstrong erlang commun acm gul agha actors model concurrent computation distributed systems phd thesis mit stavros aronis nikolaos papaspyrou katerina roukounaki konstantinos sagonas yiannis tsiouris ioannis venetis scalability benchmark suite torben hoffman john hughes editors proceedings eleventh acm sigplan workshop erlang pages new york usa september acm gul agha overview actor languages sigplan thomas arts john hughes joakim johansson ulf wiger testing telecoms references trinder chechina papaspyrou sagonas thompson scaling reliably software quviq quickcheck proceedings acm sigplan workshop erlang pages new york usa acm robert baker peter rodgers simon thompson huiqing visualization concurrent distributed computation erlang visual languages computing vlc international conference distributed multimedia systems dms luiz barroso jimmy clidaras urs datacenter computer morgan claypool edition symposium tfp revised selected papers volume lncs pages springer irina calciu dave dice yossi lev victor luchangco virendra marathe nir shavit locks proceedings acm sigplan symposium principles practice parallel programming pages new york usa acm francesco cesarini simon thompson erlang programming concurrent approach software development reilly media edition basho technologies riakdocs basho bench http rohit chandra leonardo dagum dave kohr dror maydan jeff mcdonald ramesh menon parallel programming openmp morgan kaufmann san francisco usa beasley distributing test problems electronic mail journal operational research society datasets available http natalia chechina huiqing amir ghaffari simon thompson phil trinder improving network scalability erlang parallel distrib cory bennett ariel tseitlin chaos monkey released wild netflix blog robert blumofe christopher joerg bradley kuszmaul charles leiserson keith randall yuli zhou cilk efficient multithreaded runtime system proceedings fifth acm sigplan symposium principles practice parallel programming pages acm olivier boudeville technical manual simulation engine edf kozsik judit melinda refactorings enable parallelization jurriaan hage jay mccarthy editors trends functional programming international natalia chechina kenneth mackenzie simon thompson phil trinder olivier boudeville csaba hoch amir ghaffari mario moro hernandez evaluating scalable distributed erlang scalability reliability ieee transactions parallel distributed systems natalia chechina mario moro hernandez phil trinder scalable reliable instant messenger using erlang libraries erlang pages new york usa acm koen claessen john hughes quickcheck lightweight tool random testing haskell programs proceedings acm sigplan international conference functional programming pages new york usa acm trinder chechina papaspyrou sagonas thompson scaling reliably crauwels potts van wassenhove local search heuristics single machine total weighted tardiness scheduling problem informs journal computing datasets paper included beasley orlib jeffrey dean sanjay ghemawat mapreduce simplified data processing large clusters commun acm david dewolfs jan broeckhove vaidy sunderam graham fagg ftmpi metacomputing generic name services case study pages berlin heidelberg alan donovan brian kernighan programming language professional marco dorigo thomas ant colony optimization bradford company scituate usa faith ellen yossi lev victor luchangco mark moir snzi scalable nonzero indicators proceedings annual acm symposium principles distributed computing pages new york usa acm jeff epstein andrew black simon towards haskell cloud haskell pages acm martin fowler refactoring improving design existing code longman publishing boston usa ana gainaru franck cappello errors faults techniques computing pages springer international publishing martin josef geiger new instances single machine total weighted tardiness problem technical report research report march datasets available http guillaume germain concurrency oriented programming termite scheme proceedings acm sigplan workshop erlang pages new york usa acm amir ghaffari benchmark tool distributed erlang https amir ghaffari investigating scalability limits distributed erlang proceedings acm sigplan workshop erlang pages acm amir ghaffari natalia chechina phip trinder jon meredith scalable persistent storage erlang theory practice proceedings acm sigplan workshop erlang pages new york usa acm seth gilbert nancy lynch brewer conjecture feasibility consistent available web services acm sigact news andrew grimshaw wulf legion team legion vision worldwide virtual computer communications acm carl hewitt actor model discretionary adaptive concurrency corr carl hewitt peter bishop richard steiger universal modular actor formalism artificial intelligence ijcai pages san francisco usa morgan kaufmann rich hickey clojure programming language dls pages new york usa acm trinder chechina papaspyrou sagonas thompson scaling reliably kozsik kitlei nagy melinda roland building refactoring tool erlang workshop advanced software development tools techniques wasdett laxmikant kale sanjeev krishnan portable concurrent object oriented system based acm sigplan notices volume pages acm david klaftenegger konstantinos sagonas kjell winblad scalability erlang term storage proceedings twelfth acm sigplan workshop erlang pages new york usa september acm david klaftenegger konstantinos sagonas kjell winblad delegation locking libraries improved performance multithreaded programs parallel volume lncs pages springer david klaftenegger konstantinos sagonas kjell winblad queue delegation locking ieee transactions parallel distributed systems appear rusty klophaus riak core building distributed applications without shared state acm sigplan commercial users functional programming cufp pages new york usa acm avinash lakshman prashant malik cassandra decentralized structured storage system sigops oper syst april lee python actor runtime library huiqing simon thompson automated api migration refactoring tool erlang programs tim menzies motoshi saeki editors automated software engineering ase ieee computer society huiqing simon thompson multicore profiling erlang programs using proceedings twelfth acm sigplan workshop erlang huiqing simon thompson improved semantics implementation testing quickcheck international workshop automation software test huiqing simon thompson safe concurrency introduction slicing pepm acm sigplan january huiqing simon thompson orosz melinda refactoring wrangler updated acm sigplan erlang workshop volume frank lubeck max neunhoffer enumerating large orbits direct condensation experimental mathematics andreea lutac natalia chechina gerardo phil trinder towards reliable scalable robot communication proceedings international workshop erlang erlang pages new york usa acm timer api january https kenneth mackenzie natalia chechina phil trinder performance portability placement distributed erlang proceedings acm sigplan workshop erlang pages acm jeff matocha tracy camp taxonomy distributed termination detection trinder chechina papaspyrou sagonas thompson scaling reliably algorithms journal systems software nicholas matsakis felix klock rust language acm sigada ada letters volume pages acm robert mcnaughton scheduling deadlines loss functions management science martin odersky scala programming language william opdyke refactoring objectoriented frameworks phd thesis university illinois nikolaos papaspyrou konstantinos sagonas preserving term sharing erlang virtual machine torben hoffman john hughes editors proceedings acm sigplan erlang workshop pages acm release project team framework project http konstantinos sagonas thanassis avgerinos automatic refactoring erlang programs principles practice declarative programming ppdp pages acm range updates using contention adapting search trees languages compilers parallel computing international workshop volume lncs pages springer marc snir steve otto walker jack dongarra steven husslederman mpi complete reference mit press cambridge usa sriram srinivasan alan mycroft kilim actors java ecoop pages berlin heidelberg syme adam granicz antonio cisternino expert springer simon thompson huiqing refactoring tools functional languages journal functional programming marcus linux timers may whatsapp https tom white hadoop definitive guide storage analysis internet scale revised updated reilly ulf wiger functional programming experiences ericsson project implementing functional languages ifl aachen germany september konstantinos sagonas kjell winblad scalable ordered set ets using adaptation proc acm sigplan workshop erlang pages acm september konstantinos sagonas kjell winblad contention adapting search trees proceedings international symposium parallel distributed computing pages ieee computing society konstantinos sagonas kjell winblad efficient support range queries
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top tips make research irreproducible neil chue tom ian lars kenji apr software sustainability institute university edinburgh department computing information systems cardiff metropolitan university school computer science university andrews insight centre data analytics university college cork ireland microsoft research cambridge http tcrick http http http http april noticed contributed number manifestos guides top tips make research reproducible however seen little published make research irreproducible irreproducibility default setting science irreproducible research particularly common across computational sciences study making work irreproducible without reviewers complaining much neglected area feel therefore encapsulating top irreproducibility filling gap domain literature following starter tips ensure work wrong nobody able check correct make everyone else disproportionately work build upon either case beneficiary unfortunate convention science research pretend reproducible top tips help salve conscience certain reviewers still bound fussy conventionality enabling enthusiastically recommend acceptance irreproducible work think big picture people interested science dull experimental setup describe necessary camouflage absence brief details insignificant aspects methodology abstract great way communicating ideas quickly clearly giving readers chance understand subtle implementation details particularly custom toolchains manual interventions actually make work short sweet limitations methods proofs obvious careful reader need waste space making however much work takes colleagues fill gaps still get credit say amazing experiments proofs pierre fermat cuius rei demonstrationem mirabilem sane detexi hanc marginis exiguitas non deficit model expert domain define algorithms data run experiments unhappy circumstance methods well means claiming exhaustive list making research irreproducible space saved way used cite critical papers field papers inflate well potential reviewers community curated benchmarks create bespoke benchmarks use preferably make available others share makes easier people scoop research ideas understand code actually instead say worst understand code actually work however important tip deceptively beautifully simple ensure work irreproducible make sure reproduce able reproduce would always danger somebody else able exactly much else follows example complete confidence inability reproduce work saves tedious time revising work advice reviewers unable browbeat editor accepting always resubmit elsewhere major advantage key insight strict discipline required ensure experience irreproducibility happily occur tiniest amount carelessness one number stages make simple conjecture experiment irreproducible exactly equivalent experiment never carried happy consequences conjecture experts irreproducibility published elsewhere extremely impressive experimental support close mantra scientists interested irreproducibility publishing research irreproducibility lets false observations obtain longevity references andreas james procter ten simple rules open development scientific software plos computational biology geir kjetil sandve anton nekrutenko james taylor eivind hovig ten simple rules reproducible computational research plos computational biology ian gent recomputation manifesto available http april lucas joppa greg mcinerny richard harper lara salido kenji takeda kenton hara david gavaghan stephen emmott troubling trends scientific software use science ian gent lars kotthoff experience first year lessons learned recomputability december greg wilson aruliah titus brown neil chue hong matt davis richard guy steven haddock kathryn huff ian mitchell mark plumbley ben waugh ethan white paul wilson best practices scientific computing plos biology carole goble better software better research ieee internet computing tom crick benjamin hall samin ishtiaq implement algorithm model reproducible research software proceedings international workshop sustainable software science practice experiences tom crick benjamin hall samin ishtiaq kenji takeda share enjoy publishing useful usable scientific models proceedings international workshop recomputability victoria stodden sheila miguez best practices computational science software infrastructure environments reproducible extensible research journal open research software exemplary example http
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restriction estimates free two step nilpotent group three generators apr valentina casarino paolo ciatti abstract let free two step nilpotent lie group three generators let sublaplacian compute spectral resolution prove operators arising decomposition enjoy type estimate introduction starting observation stein lebesgue exponent sufficiently close fourier transform function restricts sense may made precise compact hypersurface nonvanishing curvature various forms restriction theorems fourier transform became one main theme analysis satisfactory result obtained far theory theorem concerns restriction sense fourier transform function prn sphere result proved mid seventies last century important theorem great deal extensions applications different branches mathematics particular observed stein strichartz str estimates interpreted bounds concerning mapping properties lebesgue spaces operators arising spectral decomposition euclidean laplacian point view emphasised works sogge studied boundedness spectral projections operator spheres generally compact riemannian manifolds years later proved first time result sort subelliptic operator sublaplacian heisenberg group series papers authors article studied sort combination previous results considering estimates joint spectral projections operator sublaplacian years ago started investigation devoted extend result sublaplacians general two step nilpotent lie groups papers considered groups enjoying special nondegeneracy condition according quotient respect codimension one subspace center isomorphic heisenberg group paper instead concerned mapping properties operators arising spectral decomposition invariant sublaplacian free two step mathematics subject classification key words phrases free nilpotent lie groups restriction theorems research supported italian ministero dell istruzione dell della ricerca prin real complex manifolds geometry topology harmonic analysis gnampa project calcolo funzionale per operatori subellittici valentina casarino paolo ciatti nilpotent lie group three generators group quotient respect codimension one subspace center isomorphic direct product three dimensional heisenberg group real line therefore nondegeneracy property exploited previous works absent however analysis group made easier fact abelian component quotients respect planes center always one dimensional fact combined action rotation group two layers lie algebra gives rise family automorphisms conclude introduction short description next sections section describe features group concerned decomposing lie algebra direct sum center see nontrivial linear form fixed quotient respect null space direct product three dimensional heisenberg algebra times real line moreover two components heisenberg abelian one depend line spanned section derive spectral decomposition function respect sublaplacian decomposition expressed first terms fourier transform every dual function living transform followed section fourier transform direction radical finally decomposed eigenfunctions two dimensional twisted laplacian finally section prove operators arising spectral decomposition sublaplacian bounded nonisotropic lebesgue space lsz lpv integrations weighted exponents lsz basic ingredients estimates provided theorem fact dictates range exponents concerning integration estimates proved years ago koch ricci twisted laplacian range exponents estimates hold sharp examples analogue provided show generalities let free two step nilpotent lie group three generators assume connected simply connected lie algebra splits vector space direct sum centre three dimensional vector spaces proceed convenient introduce inner product respect orthogonal subspaces inner product induces norm norm space linear forms denote shall always identify lie algebra means exponential mapping use coordinates labeling points points vector fields restriction estimates free group left invariant satisfy fix point unit sphere denotes space linear forms call radical form space follows one dimensional satisfies thus fix two unit vectors satisfying subspace spanned lie algebra isomorphic three dimensional heisenberg algebra subspace orthogonal coincides kernel therefore quotient respect isomorphic lie algebra direct product equip algebra system coordinates decompose vector sum unit vector shall identify slight abuse notation linear form canonically associated writing fixing two orthonormal vectors obtain orthogonal basis vector uniquely written coordinates adapted transformation mapping rotation fixing hence pxq since universal property rotations extend automorphisms preserves maps sublaplacian vector fields generate entire lie algebra therefore theorem sublaplacian valentina casarino paolo ciatti hypoelliptic differential operator easy see hence homogeneous respect dilations defined lpf plf sublaplacian symmetric operator schwartz class extends selfadjoint operator pgq positive spectrum plan section consists deriving spectral decomposition construction largely inspired analogous derivations acds accomplish task given schwartz function take partial fourier transform central variables pxq zqdz introduce spherical coordinates writing belongs coordinates fourier inversion formula reads particular vector fields differential operators defined gpxq gpxq schwartz function explicitly given pza defininig differential operator gpxq gpxq find coordinates operator becomes restriction estimates free group two dimensional laplacian spectrum consists eigenvalues given write spectral decomposition respect gps gps maps onto eigenspace associated integral operators details see instance satisfying estimates proved koch ricci piecewise affine function defined order decompose eigenfunctions twisted laplacian define function inverse since recalling pxq pxq pxq makes sense decompose function eigenfuctions obtaining pxq pxq taking partial fourier transform follows pxq also pxq pxq valentina casarino paolo ciatti since operators acting different spaces namely commute thus inverting fourier transform obtain hence pxq pxq plugged yields pxq may deduce spectral decomposition respect replacing arguments sum integrals generalised eigenfunctions notice implies pxq pxq pxq pxq pxq restriction estimates free group pxq means integral must computed replacing last integral obtain pxq setting using fubini theorem deduce pxq pxq expression derived shows write pxq thus provides spectral resolution respect restriction theorem section show operators satisfy restriction estimates state prove result introduce nonisotropic norms defined lsz lpv obvious modifications equal shall prove following theorem theorem let schwartz function let lsz lsz lpv estimates false valentina casarino paolo ciatti proof sharpness range estimates hold may proved suitable easy modification example provided prove bound dependence right hand side dictated homogeneity therefore suffices discuss case reduce complexity notation make formulas readable consider tensor function schwartz functions reduces pxq denotes fourier transform consider another tensor function gpx lsz lqv compute zqgpx zqdzdx pxq pxq changing variables integral obtain since det restriction estimates free group setting write inequality yields set sup simplify notation remind according theorem inequality lsz holds lsz thus obtain lsz estimate integral last formula remind definition see also use inequality integral obtain valentina casarino paolo ciatti application inequality implies since plancherel theorem applied yields det reduces force estimates deduce applying inequality integration exponents deduce first integral finite since implies restriction estimates free group therefore may use minkowski integral inequality gives apply inequality inner integral replace coordinates deducing dxdydv lpv finally plugging obtain lsz lpv lsz lpv lsz lpv series fact converges since estimate asserted statement follows duality proving theorem valentina casarino paolo ciatti references acds str astengo cowling blasio sundari hardy uncertainty principle certain lie groups london math soc casarino norms complex harmonic projection operators canad casarino estimates joint spectral projections complex spheres math casarino ciatti transferring eigenfunction bounds studia casarino ciatti restriction theorem groups advances mathematics casarino ciatti restriction estimates full laplacian groups rend lincei mat appl casarino ciatti joint eigenfunction bounds quaternionic spheres fourier anal electronically published october doi http appear print koch ricci spectral projections twisted laplacian studia math martini spectral multipliers free group studia math restriction theorem heisenberg group ann math sogge oscillatory integrals spherical harmonics duke math sogge concerning norm spectral clusters elliptic operators compact manifolds funct strichartz harmonic analysis spectral theory laplacians funct thangavelu harmonic analysis heisenberg group progress mathematics vol boston boston degli studi padova stradella san nicola vicenza italy address degli studi padova via marzolo padova italy address
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dec matters programming moez abdelgawad college mathematics econometrics hunan university changsha hunan china informatics research institute new borg elarab alexandria egypt moez january abstract type systems recently model oop compared models oop combined comparisons provide clear technical account deep mathematical account relation nominal structural type systems presented help demonstrate key value nominal typing nominal subtyping developers language designers believe clearer understanding key semantic advantage pure nominal typing pure structural typing help remedy existing schism believe future foundational research relevance mainstream oop based less structural models oop nominal ones instead statements inheritance subtyping mainstream languages unnecessarily place restrictions inheritance rippled mantras research community years many mainstream developers language designers however accept statements languages developers language designers dearly familiar inheritance simply subtyping believe type inheritance inherently nominal notion structural one languages java scala among used programming languages today however value nominal typing mainstream developers means designing robust software seems wait full appreciation among perpetuating unnecessary schism many developers language designers many researchers side discounting even disregarding views essay strengthen complement earlier efforts demonstrate semantic value nominal typing presenting technical comparison nominal type systems structural introduction cook shocked programming languages research community declaring programming inheritance subtyping meaning correspondence type inheritance subtyping programming languages adding mainstream languages unnecessarily place strictions years statements rippled mantras research community day however many mainstream developers language designers digest accept identifying type inheritance subtyping simply statement cook true languages developers language designers dearly familiar see restriction type inheritance languages strongly justified structural view inheritance fact unjustified redefinition type inheritance view inherently nominal notion languages among top programming languages today examples languages include many mainstream programming languages java scala languages remained among top programming languages decade even conservative measures languages expected remain among top programming languages near future far one spite value nominal typing nominal subtyping mainstream developers means designing robust software readily understood maintained well value properties type systems depend nominality identification type inheritance subtyping seem yet fully appreciated among researchers led continuing tension schism two large significant communities many mainstream developers language designers one side many researchers side sides discounting even disregarding views opinions languages objects types nominal meaning objects types carry class names information part meaning objects meaning types class interface names trait names languages support used type names languages interface contracts typically written informally code documentation comments specifications behavioral design intentions software developers oop reference class interface trait name invariably considered reference associated contract given association type names corresponding behavioral contracts nominal typing allows associating types objects formal informal behavioral contracts using type names code developers using languages simple way refer corresponding referring richer specifications object state behavior checked statically used runtime readily access richer object expressed natural way using structural record types definition include class names nominally typed languages closer semantically typed languages structurally typed languages first mathematical models oop gain widespread recognition among programming languages researchers developed oop making first steps mainstream computer programming see section early models structural models oop developers models cardelli explained due influence functional programming research extant time models oop thus reflected view oop include class names information structural objects viewed models simply records functions object types accordance viewed record types type object specifies structure object meaning object types carry information names members objects fields methods inductively structural types members examples languages include lesserknown languages caml moby polytoil strongtalk pure languages class names information also called nominal information used part identity objects types neither static type checking runtime accordingly nominal information missing structural mathematical models oop main practical advantage structural typing nominal typing languages seems flexibility ability structurallytyped language supertypes get defined fact subtypes already defined light mainstream developers languages adopting structural typing inflexibility languages seems enough justification wider use structural typing particularly light advantages nominal typing discuss essay attempt thus essay close gap exists programming language researchers maintain view oop believe conclusions based view inheritance subtyping correspondence mainstream software developers language designers maintain view software accordingly reject conclusions based structural view giving precise technical account relation nominal structural type systems essay complements recent mathematical comparison nominal model oop structural models oop essay structured follows first section give details history modeling oop particularly details relevant realizing differences nominal typing structural typing development nominal structural models oop given structural typing less understood well among researchers section directly demonstrate value behavioral contracts nominal typing mainstream languages using comparison followed discussion comparison section first discuss detail value contracts value identifying inheritance subtyping mainstream software design using code examples section section compare type systems structurallytyped ones vividly illustrate main technical differences conclude section discussing nominal structural views type names importance recursive types mainstream oop conclude essay summarizing findings making final remarks section related work even though programming emerged got mature mainstream software development differences programming languages started getting discussed researchers spite early hint cardelli see value investigating nominal typing nominal subtyping value developers appreciated much decade later around year eighties mainstream oop early days cardelli built first denotational model oop cardelli work pioneering naturally given research modeling functional programming extant time cardelli heavily referred relied model cardelli constructed structural denotational model discussion versus languages including ducktyping merits demerits beyond scope essay interested reader check essay focus nominal structural languages quite significantly cardelli fact also hinted looking investigating nominal typing sadly cardelli hint went largely ignored years structural typing rather assumed superior nominal typing instead particularly late nineties cook colleagues worked improve cardelli model unlike cardelli cook emphasized work importance oop value level type level research led break identification inheritance subtyping bruce others presented discussion problem binary methods oop binary method method takes parameter type class method declared later bruce simons promoted structural view oop conclusions based number publications spite disagreement conclusions fundamental intuitions significant portion mainstream developers language designers pressure disagreement researchers started late stressing significance differences oop oop started acknowledging practical value nominal typing nominal subtyping accordingly attempts made develop languages complex type systems however eyes mainstream developers hybrid languages complex type systems languages either simply purely purely structurallytyped complexity typically results lesser productivity developers attempt use typing approaches software see also discussion end section operational models oop abadi cardelli first present model model structural view oop however operational models nominallytyped oop got later developed seminal work igarashi pierce wadler presented featherweight java operational model language even though first operational model nominallytyped oop see example widely known operational model tiny core subset mainstream language namely java development operational models nominallytyped oop motivated construction noop first model oop given different basis deriving data structuring functional programming based standard branches mathematics programming based biology featherweight java offers clear operational semantics tiny language worth mentioning models languages foundational models fewer assumptions operational ones provide denotational justification inclusion nominal information inclusion nominal information noop crucial proving identification inheritance subtyping oop identification inheritance subtyping taken assumption rather proven consequence nominality also models noop allows discussing issues oop type names binary methods foundational level provided operational models oop abstract description denotational models results conceptually clearer understanding programming notions described well relations also worth mentioning noop developed partially response technical challenge pierce presented lics lecture pierce looked precise relation structural nominal type systems notably development concluded implying question relation remained open question development purpose customary denotational models operational ones play complementary roles denotational models usually interest programming language designers operational ones usually interest programming language implementers fact seemingly nowadays forgotten researchers cardelli explicitly mentions publication cook bruce work later discuss discussion cardelli model cook model comparison noop nominal model oop versus structural models cardelli cook presented see later discussion binary methods mistakenly identified structural type systems binary methods semantics true binary methods uncover third problem pure languages call problem spurious binary conclude demonstration value nominal typing discussing depth nominal structural views type names importance recursive types mainstream oop section discussions comparisons section demonstrate nominal typing programming languages causes typing subtyping languages closer semantic typing subtyping respectively association nominal information class contracts closeness semantic typing simplicity resulting software design mental model importance recursive types mainstream developers language designers help explain practical value nominal typing mainstream developers language designers researchers also expressed dissatisfaction assuming views programming based researching functional programming including view assumes structural typing may apply without qualifications programming addition pierce earlier later researchers pointing importance distinguishing nominal typing structural typing macqueen example also noted many mismatches standard popular functional programming language classbased languages java later cook also pointed differences objects oop abstract data types adts common functional programming research results run similar vein since somewhat also point mismatches theory practice programming functional practice oop versus oop contracts nominality liskov substitution principle lsp contracts notions mainstream software development contract program similar contract real world specifies object expects client objects client objects expect members fields properties members form object interface outside world buttons front television set example interface electrical wiring side plastic casing one presses power button promised turn television common form interface group related methods together contract giving promises behavior methods similarly class contract agreement instances class expose present public interface api certain methods certain properties certain behaviors section first informally discuss importance contracts nominal typing nominal subtyping mainstream developers section discussing dbc design contract process discuss lsp liskov substitution principle expresses importance preserving contracts upon inheritance section section present code examples illustrate oop oop compare technical point view illustrate structural typing structural subtyping sometimes force breaking contracts comparison discuss two key problems structural type systems namely spurious subsumption converse missing section technical overview main typing notions explains technical jargon used essay sented provider client contract examples examples contracts software plenty examples familiar java developers example include contract comparable interface promising clients total ordering elements requiring classes implement interface adhere promise contract class object promising equals method hashcode method agreement requiring subclasses override one two methods override method accordingly also java class jcomponent contains default implementation methods accessible interface jcomponent actually declared implement interface contract associated accessible satisfied default implementation provided jcomponent example stresses association inherited contracts superclass names mainstream software class extends another class implements interface declaring inherits contract associated superclass superinterface maintain likewise class maintain contract associated another class interface declare extending implementing class interface discuss point section examples contracts may also include class implements tree layout algorithms contract class may require input graph tree may promise result input tree produce layout overlapping nodes edges labels general contracts whether written formally informally usually contain following pieces information side effects preconditions postconditions invariants sometimes even performance guarantees java class contracts set requirements promises usually stated javadoc comments requirements contract simply conditions use class example conditions argument values conditions order execution methods conditions execution parallel environment two rather artificial examples contracts expound show benefit output guaranteed comply postconditions need check output input guaranteed comply preconditions need check input obligation satisfy preconditions satisfy postconditions table design contract dbc benefits obligations source ferences nominal typing structural typing promise animal play another animal promise mathematical set contains repeated elements particular use two examples show structural subtyping lead breaking contracts associated presented examples easy see contract made two parts requirements upon caller client made class provider promises made class caller caller fulfills requirements class promises deliver service requirements may enforced throwing checked unchecked exceptions stated conditions violated promises enforced assertions end method according proponents design contract dbc classes software system communicate one another basis precisely defined benefits obligations preconditions obeyed client class method service provider deny service postcondition invariant violated uncovers problem service provider side benefits obligations clients providers along relative chronological order summarized table oop moving dbc design mainstream type systems languages ideally include behavioral contracts object types motivated dbc contracts used provider client mainstream developers constructing robust reliable reusable maintainable software since contracts promise specified properties objects example book titled effective java joshua bloch reflects use contracts mainstream oop asserts value contracts software design writing class island instances one class frequently passed another many classes depend objects passed obeying contracts associated superclasses violated contract simply know objects behave confronted obligation satisfy contract requirements satisfy contract promises table contracts oop benefits obligations world language tractable component contract enforced build time compiler java example class claims implement interface methods defined interface must appear source code class successfully compile run time assumed promises given interface maintained class happens class claims extend another class interface claims extend another interface inheritance requirements promises sometimes referred interface inheritance contract inheritance type using inheritance may necessary make changes superclass contract changes break caller determining change break caller professional developers use memorable phrase require promise less new specification require caller promise practice however inclusion behavioral contracts object types much ask type checker general problem able statically check contracts since behavioral contracts remarkably expressive solution language designers choose approximation association class names contracts type systems respecting nominal information typing subtyping decisions allows type system tractable approximation dbc hence language designers many mainstream languages use nominal typing languages typically require enforcement requirements promises contracts requirements promises rather assumed hold thereby encouraging requiring developers enforce contracts accurately reflect contracts used oop table modified table benefit output assumed comply contract promises need check output input assumed comply contract requirements need check input according programmers employ inheritance number different purposes provide subtyping reuse code allow subclasses customise superclasses behaviour categorise objects inheritance method classes share implementations code reuse technique limited notion inheritance unfortunately still entertained researchers code sharing part fuller picture inheritance means towards higher goal classes sharing contracts even architectures code inheritance contract sharing notion inheritance generally interested discuss essay inheritance subsumption contract preservation inheritance contracts mainstream oop implementing interface example allows class program become formal behavior promises provide interfaces form contract class outside liver less new specification compatible old break caller bloch hinting conventional wisdom among mainstream developers identifies type inheritance subtyping proceeds conclude based earlier observations contracts programming whenever principles violated program becomes difficult understand maintain example discussion contracts bloch gives examples demonstrating problems java libraries bloch coauthored resulted violating principles based discussion contracts inheritance two clear design principles among professional mainstream developers inheritance contracts appropriate circumstances subclass really subtype whenever contract obeyed inheriting class whenever type inheritance subsumption corresponding class types subtyping conversely bloch concludes responsibility subclass overriding methods superclass obey general contracts failure prevent classes depend contracts functioning properly conjunction whenever subsumption two class types whenever subtyping contracts superclass type obeyed subclass type type inheritance requirement subclasses maintain contracts superclasses expressed among professional developers stating software obey liskov substitution principle lsp according bloch given importance lsp expressing preservation contracts upon inheritance mainstream developers given contracts typically specifications object behavior easy conclude basing typing contracts make typing subtyping closer behavioral typing behavioral subtyping sometimes also called semantic typing semantic subtyping desirable property language illustrate importance identifying inheritance subtyping software developers following two sections present code examples point two problems structural subtyping exist oop first problem sometimes called problem spurious subsumption second problem inheritance implying subtyping call missing subsumption inheritance two classes imply subsumption two corresponding class types converse spurious subsumption problem lsp says important property type also hold subtypes method written type work equally well subtypes demonstrated bloch common knowledge among professional mainstream developers subsumption expressed lsp identification inheritance subsumption subtyping object types integral part mental model lsp third five design principles solid mainstream developers follow design robust software jargon developers code smells particular refused bequest class code obey lsp derived code breaks contract one base one superclasses lsp thus expresses class contracts preserved inheriting classes code examples skipped reader familiar two problems spurious subsumption problem two classes whose instances maintain contract considered subtypes according structural subtyping rules demonstrating example structural subtyping breaking lsp missing subsumption problem two classes whose instances maintain behavioral contract considered subtypes structural type system due structural type systems rebinding upon inheritance demonstrating example structural subtyping thus breaking identification inheritance subtyping due pure structural type systems always requiring rebinding upon inheritance also point problem pure structural subtyping call problem spurious binary note code class set class multiset two classes support precisely set four operations instances signatures four operations different contracts associated two classes specifying semantics behavior instances agree corresponding mathematical notions reflected different class names two classes structure two classes given oop respects class names thus associated class contracts oop ignores final assignment correctly disallowed oop wrongly allowed oop assignment allowed oop class multiset inherit class set allowed oop matching signatures operations supported two classes assignment allowed allowed demonstrated code allow repetition set elements value bound variable assignment given instance multiset variable declaration assumed set repeated elements assignment allowed thus break contract class set associated variable problem spurious subsumption similar problem accidentally mistaking values datatype another datatype think using floats modeling euros dollars mistaking euros dollars mistaking floats either similarly software developers think object context class hierarchy contracts associated class members key prescript oop class contracts inherited along class members maintained instances inheriting subclasses spurious subsumption oop multiset new multiset insert insert set allow assignment spurious subsumption oop subtyping imply inheritance may subtyping types corresponding two classes instances one used inheritance relation two classes illustrate let assume following definitions class set class multiset contract class set disallows repetition elements set agreement mathematical definition sets whereas contract class multiset allows repetition elements multiset agreement mathematical definition multisets sometimes also called bags class set boolean equals object void insert object void remove object boolean ismember object class multiset boolean equals object void insert object void remove object boolean ismember object lows unintended breaking rule since structural type checker fails reject program uses object one type behaviorally different structurally compatible type animal mate animal class cat extends animal behavior specific cats inheritance subtyping another problem structural subtyping converse spurious subsumption problem languages structural subtyping require subtyping types classes inheritance relation thus inherited contracts inheritance imply subtyping combination subtyping implying inheritance spurious subsumption structural subtyping thus totally separates notions inheritance subtyping based nonnominal view inheritance ignores inheritance class contracts associated class names illustrate inheritance implying subtyping oop assume following definitions class animal class cat class animal void move point void eat food void breathe void sleep time period generic animal behavior example mentioned earlier interface comparable java consisting single abstract method int compareto object public contract asserting compareto defines total ordering instances class inheriting comparable clients comparable thus depend property arbitrary class method int compareto object however generallyspeaking necessarily obey contract languages instances class bound spurious subsumption variables type comparable similar allowing binding instance first bound variable variable example spite fact developer asserts class implements comparable asserting compareto defines total ordering class author comparable conversely asserting class implement comparable interface instances bound variables type comparable since compareto method may unintentionally intentionally define total ordering instances generic animal behavior inherited class animal cats cat mate cat animal mate animal oop disagree signature method mate class cat oop assumes mate binary method see binary methods recognized problematic multiple approaches suggested dealing discuss technically section requires method natural signature cat mate cat expense making cats instances class cat animals instances class animal quite naturally oop hand assume mate method class animal binary method thus keeps using signature method upon inheritance class cat oop thus cats indeed animals meaning mainstream oop identify inheritance subtyping given oop oop differ whether inheritance implies subtyping languages structural type corresponding class cat subtype structural type corresponding class animal contravariance types method arguments unless unintuitive structural notion like matching expresses similarity recursive structure class cat class animal gain traction support mainstream oop added language used subtyping see also discussions section one may recall cardelli noting biological origin oop biological origin reason inheritance called inheritance first place assignment animal new cat correctly allowed oop wrongly disallowed oop summarize code examples presented demonstrate fundamental difference oop oop perspective developers oop class structure inherited class contracts multisets sets contrary mathematical definition cats animals contrary biological definition oop class contracts inherited via class names addition class structure multisets sets agreement mathematical definition cats animals agreement biological definition oop whether subtyping needed indicated presence absence explicit inheritance declarations accordingly code examples make clear generally oop sometimes forces subtyping unneeded sometimes bars needed oop forces subtyping explicitly needed bars omission explicitly unneeded conclusion demonstrates fundamental semantic practical value nominal information developers programming languages method method inside class takes argument type class method declared causes methods like playwith class animal whose semantics true binary methods mistaken ones thus subclass cat example playwith method restrictive signature void playwith cat allowing cats play cats animals call problem structural typing problem spurious binary methods false binary methods since method inadvertently considered binary method nominallytyped languages suffer problem languages treat method regular method thus signature method change upon inheritance nominal typing type names recursive types binary methods based discussion previous sections fundamental technical difference nominallytyped languages languages clearly lies two approaches typing languages differently view treat type names spurious binary methods languages type names viewed names type variables aba problem breviate type expressions shortcuts oop also related binary methods type names languages fact pure language one nominal typing features class oop generics used like animal method like say example define generic class enum java offers somewhat better also fully satisfacvoid playwith animal keeps signature subclass tory support true binary methods keeping identification inheritance subtyping based subclasses class animal pure preliminary research made expect future research oop always treats binary offer satisfactory alternative supporting true allow cats play dogs mouses mice example nary methods fully satisfactory alternative also see related discussion close end section useful even necessary defining recursive type expressions variable names however recursive type names languages name class used inside definition class gets interpreted get rebound different types upon inheritance get rebound types subtypes could break contravariant subtyping rule method parameter types thus break type safety languages languages resolve situation breaking correspondence inheritance subtyping demonstrated earlier code examples section section languages however nominality types means type names viewed part identity meaning type expressions given association type names public class contracts means class names treated variable names accordingly program type names fixed meanings change upon inheritance fixed type type name bound program break contravariant subtyping method parameters method type get inherited subclasses thus necessitating breaking identification inheritance subtyping demonstrated earlier code examples section section oop class directly refer using class names signature field method parameter return value kind reference called recursive reference sometimes circular reference also classes class refers indirectly via classes allowed oop kind reference inside class indirectly referencing called indirect reference reference pierce noted languages allow readily expressing circular class definitions since objects mainstream oop characterized selfreferential values autognostic cording cook since values typed using recursive types strong wide need circular class definitions oop direct indirect circular type references quite common mainstream oop ease recursive typing expressed languages one main advantages oop according pierce fact recursive types come essentially free nominal systems decided benefit languages demonstration influence views selfreferential classes properties type systems nominal structural models oop compared done brief detail easy see classes viewed differently nominal models oop structural models different views classes particular make nominal models oop lead simple mathematical proof identification type inheritance different conclusion one reached based structural models particular inclusion class signature constructs noop led simplicity mathematical proof identification see details aside theory difference nominal structural views type names oop demonstrates prominently practice different support different treatment provided languages languages binary methods mentioned section binary method method takes parameter type class method declared problem binary methods requiring supported languages main factor behind structural models oop leading identifying type inheritance subtyping languages given view type names type variable names get rebound require type argument method identified binary method upon inheritance method type corresponding subclass languages hand fixed interpretation type names offer somewhat solution totally avoiding binary methods overly embracing pure languages languages treat method taking argument class method declared like method needing special treatment oop thus quite support binary methods good reasons break identification inheritance contracts subtyping lose advantages nominal typing offers good approximation binary methods given meaning types names languages change upon inheritance languages provide methods whose type upon inheritance approximates true binary methods type input parameter method approximates binary method guaranteed supertype type true binary method given type parameter change subclasses degree approximation gets lesser deeper inheritance hierarchy method gets inherited light spurious binary methods problem uncovered structural type systems see section believe providing approximations binary methods smart design choice nominal type systems even likely avoiding spurious binary methods may consciously intended also noted problem spurious binary methods provides justification languages cautious fully embracing binary methods treating method looks like binary method indeed one also hinted earlier sections opinion structural typing arguably better support binary methods justify using structural typing since structural type systems problems support binary methods problem spurious binary methods conjecture generics may provide better solution binary methods possibly also oop require breaking identification inheritance subtyping thus require sacrificing closeness nominal semantic behavioral advantages nominal typing light spurious binary methods problem requiring explicit use generics opinion better approach towards supporting binary methods mainstream languages might allowing developers explicitly mark flag binary methods even precisely allow developers mark specific parameters methods parameters need treated binary methods hybrid languages add structural typing features vice versa conjecture useful part claimed flexibility structural typing may possible achieve languages supporting separate notion contract names thereby splitting class names contract names allowing classes additionally define satisfying supercontracts sub classes explored suggestion however since believe may complicate nominal type systems believe flexibility absolute necessity splitting class names contract names may suggestion worthy investigation believe suggestion viable reason agreement nominal spirit using multiple dispatch discussed possible solution problem binary methods also viable creating hybrid languages believe type system nominally structurally typed example implicit implicit type variables get included class signatures similar implicit included method signatures due problems supporting recursive types mentioned believe hybrid languages support recursive structural types example makes type system necessarily imply similarity behavvery complex probably even lends hybrid ior oop type inheritance implies features unusable refined contracts refined contracts imply subsumption class types vice versa oop subsumption class concluding remarks types implies refined contracts implying type inheritance essay added earlier efforts aimed putting facts together clear demonstrate semantic value nominal typing oop different class names informaparticularly association class names tion implies different different havioral class contracts making technical imply different class names informaparison nominal type systems tion identification types contracts tural type systems recently subtyping inheritance contracts makes model oop namely noop nominal typing nominal subtyping closer sealso compared models oop mantic typing semantic subtyping comparisons provide clear deep account essay thus stressed practical value relation nominal structural nominal typing mainstream oop particularly type systems due earlier lack showing theoretic model oop value nominal subtyping compile time presented help furand runtime respecting behavioral conther demonstrate researchers value tracts thus respecting design intents nominal typing nominal subtyping mainstream developers language designers value resulting identification instill deeper appreciation inheritance subtyping providing simpler essay particularly noted nominal conceptual model software softtyping prevents types structurally look ware components leading simpler design confused type since process software objects structure necessarily imply objects value making recursive types readily ior nominal typing identifies types expressible necessary static class names information nominal informatyping autognostic objects tion thus assert maintaining contract assert comparison also revealed problem spuristructural interface thus oop ous binary methods far unrecognized problem objects class type implies languages serting maintain contracts recent comparison nominal asserting maintain contracts implies structural denotational models oop demonstrates type different views fundamental notions similarly nominal subtyping allows subtyping mainstream oop namely objects type names lations decided based refinement types subtyping relation tracts maintained objects based subtyping type inheritance particthe refinements structure inclusion ular comparison demonstrates object contracts deciding subtyping relation nominal mainstream oop record tosubtyping thus also prevents types gether nominal information class types cially structurally similar confused record types whose elements inas subtypes since similarity structure stances additionally respect statements class tracts type inheritance correctly mainstream oop based less structural models oop nominal ones instead fied nominal subtyping table following page summarizes main differences nominal typing structural references typing pointed essay language specification version http hope development mathematical models oop comparisons presented essay elsewhere significant steps providing full account relation nominal structural type systems hope essay clearly explains rationale behind belief significant practical value significant semantic value nominal typing reasons mainstream software developers correctly choosing use languages believe clear view rationale behind many developers preference languages accurate technical mathematical view software present programming languages researchers better chances progressing mainstream languages making research relevant language designers mainstream software finally believe clearer understanding deeper appreciation key semantic advantage nominal typing structural typing help remedy existing schism researchers one hand developers language designers hand offering thereby better chances progressing mainstream languages particular believe future foundational research relevance generics example add expressiveness type systems programming languages hinted earlier particularly generics improve support binary methods languages maintaining identification inheritance subtyping believe building model generic oop along lines may offer better chances deeper understanding features generic mainstream languages java erasure variance annotations including notorious java wildcards polymorphic methods generic type inference programming languages scala programming language programming language tiobe index http martin abadi luca cardelli semantics object types proc lics martin abadi luca cardelli theory objects moez abdelgawad noop mathematical model programming phd thesis rice university moez abdelgawad overview nominaltyping versus objectoriented programming code examples technical report moez abdelgawad model programming journal electronic notes theoretical computer science entcs doi moez abdelgawad comparison noop structural models objectoriented programming preprint available http oop object interfaces nominal include class names contracts richer specifications object behavior types objects object type expressions included via class names meaning objects interfaces types class types oop structural include class names ignored meaning objects object interfaces object types record types class signatures record type expressions class signatures object types included object meanings class names associated public class contracts used type names fixed rebound upon inheritance includes inheritance class contracts respects contracts respects type inheritance objects carry types part object meanings type names synonyms type expressions inherent fixed meaning rebound upon inheritance type type names meaning type names type inheritance subtyping type inheritance versus subtyping software design mental model binary methods spurious missing subsumption spurious bin methods ignores behavioral class contracts ignores behavioral class contracts correspondence two relations independent simple inheritance hierarchy subtyping hierarchy supported approximations provided complex inheritance hierarchy separate independ subtyping hierarchy fully supported including false ones neither exist exist exist exist special constructs needed explicit expression typing subtyping recursive types readily naturally expressed typing subtyping closer typing subtyping table moez abdelgawad towards kim bruce luca cardelli giuseppe castagna ing generics technical report hopkins objects group gary leavens benjamin pierce binary methods theory practice object systems moez abdelgawad robert cartwright oop objects mere kim bruce foundations records inheritance subtyping submitted languages types semantics mit press journal publication jonathan aldrich power interoperabil luca cardelli semantics multiple inherity objects inevitable proceedings itance proc internat symp seof acm international symposium mantics data types volume pages new ideas new paradigms reflections programming software onward pages new york usa acm luca cardelli semantics multiple inheritance inform joseph bank barbara liskov andrew myers parameterized types java technical luca cardelli structural subtyping report notion power type acm proceedings popl joshua bloch effective java programming language guide sun microsystems mountain luca cardelli james donahue lucille glassview man mick jordan bill kalsow greg nel joshua bloch effective java prentice hall son report revised volume digptr ital systems research center john boyland giuseppe castagna parasitic robert cartwright moez abdelgawad methods implementation inheritance subtyping extended abstract java oopsla nordic workshop programming theory nwpt tallinn estonia bracha griswold strongtalk typechecking smalltalk production environment robert cartwright steele guy oopsla pages compatible genericity types gilad bracha martin odersky david java programming language craig stoutamire philip wadler making chambers editor acm symposium objectthe future safe past adding genericity oriented programming systems languages java programming language craig applications oopsla volume pages chambers editor acm symposium vancouver october acm oriented programming systems languages acm sigplan applications oopsla volume pages vancouver october acm chambers cecil ecoop acm sigplan bruce schuett van gent fiech clifton millstein leavens polytoil polymorphic objectc chambers multijava design rationale comoriented language acm transactions propiler implementation applications acm gramming languages systems transactions programming languages systems william cook denotational semantics inheritance phd thesis brown james gosling bill joy guy steele gilad bracha java language specification william cook understanding data abstraction revisited volume pages james gosling bill joy guy steele gilad bracha alex buckley java language acm specification william cook walter hill peter atsushi igarashi benjamin pierce philip canning inheritance subtyping wadler featherweight java minimal core popl proceedings calculus java acm transactions programming languages systems william cook jens palsberg may tional semantics inheritance correctness acm symposium reto kramer examples design contract programming systems languages applicain java design components object tions oopsla pages world berlin sophia drossopoulou susan eisenbach angelika langer java generics faq khurshid java type system sound tapos mohamed fayad douglas schmidt leroy doligez garrigue vouillon objective caml system application frameworks comable http mun acm october barbara liskov keynote abstrac robert bruce findler matthew flatt tion hierarchy acm sigplan notices volmatthias felleisen semantic casts contracts ume pages acm structural subtyping nominal world ecoop programming barbara liskov jeannette wing pages springer behavioral notion subtyping acm transactions programming languages systems fisher reppy design class toplas mechanism moby pldi david macqueen objectoriented formal aspects computing matthew flatt shriram krishnamurthi matthias felleisen classes mixins proceedings acm david macqueen gordon plotkin symposium principles programming lanr sethi ideal model recursive polymorguages pages acm phic types information control matthew flatt shriram krishnamurthi matthias felleisen programmer reduction boris magnusson code reuse considered harmsemantics classes mixins formal ful syntax semantics java pages donna malayeri jonathan aldrich inspringer tegrating nominal structural subtyping ecoop programming gil maman whiteoak introducing pages springer structural subtyping java oopsla donna malayeri jonathan aldrich ewan tempero hong yul yang james noble programmers inheritural subtyping useful empirical study tance java proceedings euesop ropean conference program erik meijer peter drayton static typing ming ecoop pages berlin heiwhere possible dynamic typing needed delberg end cold war programming kresten krab thorup mads torgersen unilanguages oopsla fying genericity ecoop programming pages springer bertrand meyer applying design contract computer yizhou zhang matthew loring guido salvaneschi barbara liskov andrew myers bertrand meyer software conlightweight flexible generics struction prentice hall proceedings acm sigplan conference programming language design milner tofte harper macplementation pldi pages new queen definition standard reyork usa acm vised mit press tobias nipkow david von oheimb javalight proceedings acm symposium principles programming languages pages acm martin odersky scala language specification http klaus ostermann nominal structural subtyping programming journal object technology benjamin pierce types programming languages mit press benjamin pierce types programming languages next generation lics harry porter iii separating subtype hierarchy inheritance implementation journal programming anthony simons theory classification part perspectives type compatibility journal object technology mayjune
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jan stillman question exterior algebras herzog conjecture betti numbers syzygy modules jason mccullough abstract let field characteristic consider exterior algebras finite dimensional spaces short paper exhibit principal quadric ideals family whose regularity unbounded negatively answers analogue stillman question exterior algebras posed peeva show examples dual modules polynomial rings yield counterexamples conjecture herzog betti numbers linear strand syzygy modules introduction let field let symmetric algebra finite dimensional space stillman problem posed following question projective dimension pds homogeneous ideal bounded purely terms number degrees minimal generators caviglia showed question equivalent parallel question one replaces projective dimension regularity theorem recently gave positive answer question full generality let positively graded exterior algebra finite dimensional kvector space resolutions need finite regularity rege finitely generated finite since koszul algebra rege defined rege sup tore irena peeva posed following variant stillman question joint introductory workshop msri fall question peeva regularity rege homogeneous ideal bounded purely terms number degrees generators surprisingly contrary symmetric algebra case answer question section present family principal quadric ideals exterior algebras arbitrary field whose regularity unbounded let finitely generated graded let denote minimal degree generator consider betti numbers linear strand mathematics subject classification primary secondary jason mccullough dimk torsi length linear strand max herzog proposed following lower bound betti numbers linear strand kth syzygy modules conjecture herzog graded kth syzygy module linear strand length lin conjecture proved following cases herzog proved case herzog motivated result green contained case see also similar results reiner welker proved case monomial ideal proved following cases lin proved conjecture full generality modifying recent construction use bernsteingel fand bgg correspondence produce counterexamples virtually cases herzog conjecture precisely section construct finitely generated graded nth sygyzy module linear resolution length graded betti numbers lin note lin contradicting conjecture examples bgg dual principal quadric mentioned note different construction also gives family counterexamples conjecture although aim specifically show theorem ideal cyclic module graded betti numbers otherwise particular setting syzn gives nth syzygy module graded betti numbers degree shift stillman question herzog conjecture bgg correspondence briefly recall fand bgg correspondence refer reader details let field characteristic fix positive integer let denote standard graded polynomial ring set let homk denote vectorspace dual let exterior algebra let dual basis thus also view positively graded ring deg keep notation paper consistent viewed negatively graded ring let denote functor category graded category linear free complexes defined follows given graded emodule define viewed complex graded free differential induced sxj extend functor complexes taking total complex resulting double complex adjoint function complexes complexes defined similar way creates equivalence bounded derived categories graded graded important property equivalence purpose paper equivalence functor identifies finitely generated linear free counterexamples herzog conjecture first recall fact exterior algebras theorem theorem let generic element linear transformation injective surjective remarked suffices pick come first main result convention set theorem fix field characteristic let standard graded polynomial ring exists finitely generated graded smodule generated degree elements reg graded betti numbers otherwise particular set syzn contradicting conjecture proof let general element consider graded complex free jason mccullough let since usual koszul complex shift homological degree cone complex fits short exact sequence since follows otherwise theorem every possible cancelation occur occur thus minimal subcomplex graded betti table note bgg correspondence first linear strand corresponds anne second linear strand particular since set truncation minimal free resolution shift graded degrees ensure generated degree particular graded betti table form setting syzn see nth syzygy linear free resolution graded betti numbers prescribed question exterior algebras let exterior algebra finite dimensional space set exterior variables degree rather previous section resolutions need finite since koszul algebra regularity rege finitely generated finite rege defined rege sup tore irena peeva posed following variant stillman question joint introductory workshop msri fall following theorem gives promised negative answer question theorem let field characteristic fix positive integer let space basis consider exterior algebra set rege proof consider minimal free resolution note may identify anne theorem anne elements degree thus rege hand proof stillman question herzog conjecture theorem see linear free dictionary theorem theorem homk anne linear free therefore rege remark previous theorem also holds field positive characteristic longer true ann linear free resolution since particular example finally show one construct nth syzygy modules linear free resolutions length graded betti numbers smaller predicted herzog conjecture make use bgg package written abo decker eisenbud schreyer smith stillman demonstrate case true exterior algebra variables polynomial ring variables ideal sum list ideal ideal betti res resolution reg total bettitally ann ideal ideal product flatten entries vars jason mccullough loadpackage bgg bgg package bgg matrix coker betti res linear strand resolution total bettitally bgg comodule matrix coker nth syzygy module shift grading betti res resolution bgg correspondence total bettitally clear extreme counterexample herzog conjecture note module rank syzygy module theorem bruns corollary exists free submodule rank still syzygy module module would betti table stillman question herzog conjecture would interesting know general minimal betti numbers nth syzygy module linear free resolution acknowledgements author thanks srikanth iyengar irena peeva mark walker useful conversations concerning paper also jerzy weyman pointing reference references tigran ananyan melvin hochster small subalgebras polynomial rings stillman conjecture arxiv aldo conca herbig srikanth iyengar koszul property moment map classical representations arxiv david eisenbud geometry syzygies graduate texts mathematics vol springerverlag new york second course commutative algebra algebraic geometry david eisenbud gunnar schreyer sheaf cohomology free resolutions exterior algebras trans amer math soc david eisenbud jee koh linear syzygy conjectures adv math graham evans phillip griffith syzygies london mathematical society lecture note series vol cambridge university press cambridge mark green koszul cohomology geometry projective varieties differential geom herzog linear strand graded free resolution unpublished notes srikanth iyengar mark walker examples finite free complexes small rank homology arxiv jason mccullough alexandra seceleanu bounding projective dimension commutative algebra expository papers dedicated david eisenbud occasion birthday peeva london london guillermo jan snellman conjectures hilbert series generic ideals exterior algebra homology homotopy appl part roos festschrift volume irena peeva mike stillman open problems syzygies hilbert functions commut algebra reiner welker linear syzygies ideals math scand tim bounds betti numbers algebra address jmccullo iowa state university department mathematics ames
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recruitment market trend analysis sequential latent variable models chen hengshu hui pengliang fang baidu baidu data lab rutgers university zhuhengshu dingpengliang xiefang hxiong dec abstract recruitment market analysis provides valuable understanding economic growth plays important role employers job seekers rapid development online recruitment services massive recruitment data accumulated enable new paradigm recruitment market analysis however traditional methods recruitment market analysis largely rely knowledge domain experts classic statistical models usually general model dynamic recruitment data difficulties capture market trends end paper propose new research paradigm recruitment market analysis leveraging unsupervised learning techniques automatically discovering recruitment market trends based recruitment data specifically develop novel sequential latent variable model named mtlvm designed capturing sequential dependencies corporate recruitment states able automatically learn latent recruitment topics within bayesian generative framework particular capture variability recruitment topics time design hierarchical dirichlet processes mtlvm processes allow dynamically generate evolving recruitment topics finally implement prototype system empirically evaluate approach based recruitment data china indeed visualizing results mtlvm successfully reveal many interesting findings popularity lbs related jobs reached peak half decreased keywords trend analysis recruitment market latent variable model introduction scarcity skilled talents stimulated global recruitment industry past years article corresponding author permission make digital hard copies part work personal classroom use granted without fee provided copies made distributed profit commercial advantage copies bear notice full citation first page copyrights components work owned others acm must honored abstracting credit permitted copy otherwise republish post servers redistribute lists requires prior specific permission fee request permissions permissions kdd august san francisco usa acm isbn doi http forbes reported corporations spend nearly billion year variety recruiting services worldwide number likely three times bigger along line growing challenge provide effective trend analysis recruitment market forecasting recruitment demand predicting market status employers job seekers benefit study recruitment market trends moreover macro level analysis also provide valuable understanding economic growth business analysts rapid development online recruitment services linkedin dice lagou massive recruitment posting data accumulated example end million job positions companies across industries mobile internet cloud computing available lagou chinese tech hiring service website huge data enable new paradigm studying recruitment market trends holistic manner recruitment market analysis classic topic human capital economics recruitment market either treated factor macro economic phenomenons analysis focused advising people make best job decisions general economic framework however previous studies rely largely knowledge domain experts classic statistical models thus usually general capture high variability recruitment market evolution recruitment topics also studies limited efforts understanding market trends forecasting recruitment situation specific company next months therefore appealing design new research paradigm recruitment market analysis analysis massive recruitment data along line major challenges first model intrinsically sequential dependency recruitment states hiring freeze forming market trend second model semantic relationship market trend job postings different companies finally model variability recruitment postings long time period tackle challenges propose unsupervised learning approach recruitment market trend analysis automatically discern underlying trend recruitment market first develop novel sequential market trend latent variable model mtlvm designed capturing temporal dependencies recruitment states companies able automatically learn figure trend number job postings related different skill requirements two years indicates demands recruitment market latent recruitment topics demands recruitment data within bayesian generative framework specific assume current recruitment state specific company influenced state previous epoch impel company review appropriate recruitment demands recruitment different recruitment demands generate different recruitment topics experienced algorithm engineer finally generate recruitment postings particular capture variability recruitment topics time design hierarchical dirichlet processes mtlvm dynamically generate recruitment topics finally implement intelligent prototype system empirically evaluate approach based recruitment data set collected china time period indeed visualizing results mtlvm successfully observe many interesting discoveries popularity lbs related jobs reaches peak half decreases generally contributions paper summarized follows best knowledge paper first attempt leverage unsupervised learning approach automatically modeling trend recruit market work provides new research paradigm recruitment market analysis propose sequential latent variable model named mtlvm learning latent recruitment states demands topics simultaneously particularly mtlvm dynamically generate recruitment topics integrating hierarchical dirichlet processes develop prototype system empirically evaluate approach indeed visualizing results obtained mtlvm successfully observe many interesting useful findings overview section first introduce preliminaries recruitment market modeling formally present overview model mtlvm jan jul jan jul figure word cloud representation job postings related mobile software engineer respect different epochs data set size keyword proportional frequency preliminaries recent years witnessed rapid development online recruitment services already become important venue talent seeking especially hightech companies therefore job posting data services help researchers better understand trend recruitment market perspectives individual company also whole industry intuitively different job postings indicate different recruitment demands companies related positions usually change different epochs example figure demonstrates trend number job postings related different skill requirements january november based data set observe skill information retrieval becomes less popular compared skills data mining machine learning meanwhile spark emerging technique big data processing attracted attention years indeed evolution recruitment demands inherently determined change latent recruitment states companies different epochs strong sequential dependency example alibaba one largest companies china hugely enlarged recruitment followed recruitment state hiring freeze result recruitment demands related ecommerce largely shrank capture change recruitment states model semantic relationship recruitment demand state model mtlvm follows beysian latent variable model markov assumption current state determined state previous epoch indeed analyzing descriptions job postings observe detail similar recruitment demands recruiting mobile software engineer influenced corresponding recruitment states thus corresponding demands usually high variability generate different recruitment topics example unique recruitment states specifically first epoch corresponding companies sampled uniform distribution following epochs company releases job postings previous epoch namely exists current recruitment state sampled multinomial distribution determined previous state otherwise drawn overall recruitment state epoch average company market state epoch addition name chains consisting neighbouring belong company data chain therefore company occasionally releases jobs may one data chain according formulation furthermore define generative process job posting company epoch follows first recruitment demand generated latent factor sampled dirichlet process determined current recruitment state sample recruitment topic demand observation finally observation generated multinomial distribution determined corresponding topic specifically figure shows graphical representation mtlvm figure graphical representation mtlvm figure shows word cloud representation job postings related mobile software engineer respect different epochs observe mobile game development hot topic second half android based web technology becomes popular first half model semantic relationships among recruitment states recruitment demands recruitment topics model mtlvm follows idea hierarchical dirichlet processes infinity version topic modeling model job postings therefore topic number automatically determined section introduce technical details model mtlvm illustrate important mathematical notations table modeling trend recruitment market model inference according introduction section summarize parameterizations mtlvm follows diri overview mtlvm formally regard job posting company epoch bag words basic observation job postings keywords job description modeling trend recruitment market first divide job postings different data units respect companies timestamps contain job postings company epoch introduced assume current recruitment state specific company influenced state previous epoch impel company review appropriate recruitment demands different recruitment demands generate different recruitment topics finally generate recruitment postings therefore define parameter represent recruitment state company epoch number words originally chinese automatically translated commercial translation tool multi gce multi following parameterizations get joint probability distribution set including specifically default initial recruitment state fixed indeed equation divided two parts follows multinomial distribution multi follows dirichlet processes respectively specifically computed table mathematical notations gce symbol description tokens job posting company epoch job posting company epoch observation unit containing job postings company epoch entire data set job postings recruitment state company epoch hyperparameter dirichlet prior transition matrix recruitment state probability measure representing recruitment strategy state probability measure representing recruitment demand recruitment topic tokens job posting company epoch base measure dirichlet process generating concentration parameter dirichlet process generating concentration parameter dirichlet process generating concentration parameter dirichlet process generating base measure dirichlet process generating set hyperparameters including default initial recruitment state number tokens job posting company epoch number job postings company epoch number unique recruitment states therefore objective learning mtlvm find set optimal parameters gce maximize probability equation paper propose framework learn model gibbs sampling method first step introduce optimize transition matrix constituted given depending equation could get conditional distribution since given challenge calculate follow inference give directly means number recruitment states application mtlvm learning stage mtlvm used predicting future trend recruitment market recruitment states demands topics basic observations specifically given company estimate current recruitment state arg max pearing except means number pair appearing except second step introduce compute parameters related dirichlet process equation indeed task regarded analog chinese restaurant process crp metaphor explained follows cuisine styles recruitment state franchise company restaurants job postings everyday franchise change cuisine style according cuisine styles last day particular menus different restaurants may different even share cuisine style table restaurant dish topic determined first customer basic observation job postings sitting shared among customers sit table new customer enters restaurant sit occupied table new table chooses new table order new dish menu according metaphor crp easily obtained gibbs sampling scheme posterior sampling given detailed definition inference found appendix transition matrix therefore probability recruitment state next epoch obtained multi furthermore recruitment topics obtained way introduced section thus probability basic observation keywords job description company appearing epoch computed obtained equation experiments section study performance model mtlvm huge data set collected major online recruitment website china furthermore developed prototype system empirically evaluate model system visualize results model provide analysis recruitment market analysis help people understand high variability recruitment market figure shows screenshot prototype system system show trend analysis entire recruitment market detail evolution recruitment state companies table average comparison mtlvm mtlvm mtlvm lda learning recruitment states recruitment topics specially following set symmetric dirichlet distribution parameters prior topic distributions simplicity set directly another hyperparameter also set empirically figure screenshot system recruitment market analysis following visualization results section obtained prototype system table statistics data set raw data filtered data job postings unique companies data units data chains data set experimental setup data set used experiments collected major online recruitment website china contains job postings companies released january november specially figure demonstrate statistics data set mentioned data unit figure means job posting set released company epoch data chain means chains consisting neighbouring belong company statistics observe companies randomly release job postings therefore represent trend recruitment market avoid bias conserve companies released job postings table shows detailed statistics raw data set filtered data set filtered original data number companies declined however average number job postings per company increases average length chain also increases make reasonable training mtlvm particular job posting keywords job description job responsibility skill requirements treated basic observations stop words removed guarantee modeling performance note model trained original chinese words experimental results translated english commercial translation tool facilitating demonstration following subsections comprehensively study performance mtlvm term trend analysis evaluation recruitment topics quantitatively evaluate performance latent variable models always open problem although perplexity likelihood common measures evaluating prediction results demonstrate coherent meaningful latent factors recruitment states topics therefore paper follow measures introduced inspired evaluating mtlvm specifically picked top keywords learned recruitment topic asked senior experts human resource evaluate value experts first required judge whether topic valuable topic valuable need continue judge many words relevant top keyword list based manually labeled results metrics validity measure coherence measure defined relevant words valid topics topics words valid topics besides evaluate number recruitment states affects performance train mtlvm different settings state number respectively furthermore select widely used topic model latent dirichlet allocation lda baseline convergence numbers topic models automatically determined therefore set lda table shows average results observe terms mtlvm mtlvm outperform lda lot mtlvm best performance terms performance mtlvm best mtlvm worse lda may many states make model relatively sparse thus make relevant words scattered different topics particular performance mtlvm worst may states accurately describe market trend well overall since mtlvm balanced results set state number following experiments evaluation recruitment states empirically evaluate learned recruitment states several aspects figure shows trend popularity recruitment states discovered mtlvm time obvious figure distribution number companies release job posting different epochs number companies respect number job postings number data units respect number contained job postings number data chains respect length table probabilities top recruitment topics selected recruitment states corresponding word cloud representations topics shown figure state state state state top top top top figure trend popularity recruitment states discovered model time figure transition matrix recruitment states element means transition probability state state deeper color means higher probability states change time dramatically specifically state kepng relative high popularity long period popularity always low meanwhile popularity state rising dramatically since february several states state represent totally opposite trends furthermore figure shows transition matrix recruitment states element row column represents transition probability state state observe states highest transition probabilities due momentum recruitment market also color columns relatively deeper indicates importance states results show model mtlvm ability capture high variability recruitment market discovering latent recruitment states recruitment state inspection test whether recruitment states discovered model comprehensible solve problem select representative recruitment states according analysis show top topics figure word cloud representations larger words higher generative probabilities meanwhile table shows corresponding generative probabilities topics find topic programming always high probability every state particular top topics state state state means demands related positions always exuberant since words linux mysql directly indicate fundamental skill requirements actually salary software engineer kept rising long time support discovery model states also show work games also popular consistent observations next illustrated top figure stateinspect itstates obvious topic top topic containing data analysis research algorithm indicates demand recruiting senior researchers algorithmic engineers top topic may propaganda public relationship contains several paperwork advertising related words compose edit wechat blog social network based advertising besides makes state different top topic contains human resource office assist administrative management obviously actually model topic exists state since research development propagandism administrative management essential companies conclude state covers fundamental talent demands companies state state state state figure word cloud representations top topics selected recruitment states size keyword proportional generative provability generative probabilities recruitment topics shown table figure also indicates state popular state state closely related state find top topics relative normal top topic contains data programming algorithm apparently however makes topic different word lbs service lbs map navigation map localization cornerstone kind business model uses online mobile drive offline local sales become popular since figure notice popularity topic increased declined may industry doorsill field related high large companies capability get field actually largest companies china baidu tencent sogou provide service state recruitment state top topic find word well merchant business actually proposal internet plus thousands companies focusing sprung across country meanwhile others topics state related normal terms technology top topics popular programming language database may indicate business concept rather technology concept state shown figure state exploded since february word cloud representations topics top related meaningful high frequency words baidu data mining large distributed indicate big data related topic machine learning data mining trend directly reveals companies baidu paid attention big data related fields visualization trend companies evaluate model checking trend several representative companies important fields baidu famous company alibaba largest company china visualize evolution recruitment states companies figure figure first observe state common state among companies consistent analysis topics state besides find state relevant lbs appears relatively frequently among baidu sogou tencent actually companies provide map service china especially baidu baidu map popular navigation tool china many companies use lbs api provided baidu improve services baidu remarkable strengths lbs paid much attention indeed furthermore tencent one largest companies china business scattered covers many fields game social network media entertainment kind business strategy directly reflected figure recruitment state tencent changes frequently meanwhile baidu alibaba sogou another search engine prefer state relevant big data machine learning data mining considering core business search engine ecommerce several practical applications advertising recommender system preference also reasonable addition happened find interesting company zuora whose state almost state actually zuora enterprise software company aim automate billing commerce finance operations subscription business model business model naturally related big data processing thus need senior talents furthermore observe state related appears company tuniu qunar feiniu frequently indeed companies aim connect offline merchants online customers high consistent tuniu qunar aim provide travel booking service hotel ticket car rental feiniu wesite invested large retail corporation goal establishment fuse traditional offline service channel online sale channel evaluation model application evaluate proposed model predicting basic observations future introduced section make prediction calculating probability basic observations equation experiment compare likelihood overall observations test set prove performance model test set built following approach extract recruitment postings november companies also released jobs october means companies test set recruitment states previous time span end test set contains companies recruitment postings besides recruitment postings january october treated train data select two model baselines one dynamic topic model dtm dtm classic topic model analyzing evolution topics assumes topics evolve smoothly respect time thus chains topics adjacent epochs state space models performance proved predicting next year science given articles previous years obvious dtm used prediction directly due assumption topics extracted model future data better static topic models cases experiment follow method prove assumption dtm solid problem indicate necessity model code dtm got number topics set parameters set default values addition developed simple sequence approach called byesian multivariate hidden markov model baseline compared mtlvm baseline associates states words directly joint probability distribution follows first term equation shown second term assume given recruitment state table prediction performance mtlvm baseline methods terms log likelihood log likelihood mtlvm dtm observations conditionally independent ulti baseline aim prove latent hierarchical structure model meaningful modeling recruitment markets similarity mtlvm details inference follow mtlvm omitted table shows log likelihood prediction respect different models larger number means better performance mtlvm outperform dtm largely indicates reasonable employ latent recruitment states model trend recruitment markets performance mtlvm also better bmhmm clearly validate effectiveness proposed model related work generally related works paper grouped two categories namely recruitment market analysis sequential latent variable model recruitment market analysis traditionally recruitment market analysis regarded classic topic human capital economics attracts generations researchers contribute ever since adam smith macro perspective labor always crucial element studying gross economy growth money market exchange market equilibrium ever since solow proposed growth model therefore economists usually study topics example demographic structure participation rate labor relation inflation unemployment rate labor contributes gross productivity expenditures etc another micro perspective relevant paper studies set basic market cleaning framework employees choose best balance leisure work employers hire budget constrain consequently wage derived marginal labor cost later researches improve understandings releasing constraints acknowledging market detailed investigations forming better utility functions employees studying actions employees game theory recently several researchers computer science try employ data mining technology solve problems offer categorization job skill analysis however previous research economics efforts relies largely knowledge domain experts classic statistical models thus usually general capture high variability recruitment market neglect finegrained market trend hand recent research computer science still focuses traditional human resource problems therefore paper figure visualization change recruitment states several companies different colors representing different states pose new research paradigm recruitment market analysis leveraging unsupervised learning approach sequential latent variable model indeed novel sequential latent variable model mtlvm regarded combination hidden markov model hmm hierarchical dirichlet processes hdp within bayesian generative framework intrinsically capture sequential dependency variability latent variable recruitment states topics specially hmm based sequential latent variable models successfully applied problems variety fields signal processing speech recognition biometrics genetics economics mobile internet mining much history hmms implemented using recursive algorithms developed parameter estimation viewed black boxes many statisticians recent years researchers proposed use bayesian methods simulate hmm parameters posterior distribution provide scalable stable process parameter estimation hmm compared traditional estimation mle based hmm learning solution bayesian methods directly maximize probability hidden variables given observed data integrating possible parameter values rather searching optimal set parameter values end model proposed paper also follows bayesian generative framework latent dirichlet allocation lda based latent variable models become one powerful tools mining textual data however topic models need predefined parameter indicate number topics thus fail capture variability topics end hierarchical dirichlet processes hdp proposed infinity version topic model automatically learn number topics therefore paper propose ingrate hdp mtlvm capturing variability latent recruitment topics conclusion paper provided data driven analysis recruitment market trends specifically developed novel sequential latent variable model named mtlvm designed capturing temporal dependencies corporate recruitment states able automatically learn latent recruitment topics within bayesian generative framework moreover capture variability recruitment topics time designed hierarchical dirichlet processes mtlvm processes allow dynamically generate recruitment topics finally implemented prototype system empirically evaluate approach based recruitment data results showed approach could effectively discover recruitment market trends provide guidances job recruiters job seekers appendix describe analog chinese restaurant process crp corresponding inference detail specifically first define variables sampled gce job posting linked index words besides let denote number linked according crp integrate get conditional distribution follows probability measure concentrated new sampled indicates second term side equation chosen need add sample new equation allocate sampled value existed need allocate second define variables sampled recruitment state linked index words besides let denote number linked similarly integrate get conditional distribution follows psc psc psc sampling process similar third let denote variables sampled linked index also let denote number write conditional distribution directly next describe gibbs sampling method yielded specifically follow inference method sample rather dealing directly sampling relying equation easily compute conditional distribution new used except variable likelihood simply given variables prior probability samples existed proportional prior probability new proportional process sampling similar sampling sampling given mutually independent conditional distribution related linked follows density measure parameter references baidu translation http code dynamic topic model https dice http forbes article http internet plus https plus lagou http linkedin http aldous exchangeability related topics springer bao cao chen tian xiong unsupervised approach modeling personalized contexts mobile users knowledge information systems baum petrie soules weiss maximization technique occurring statistical analysis probabilistic functions markov chains annals mathematical statistics blei lafferty dynamic topic models proceedings international conference machine learning pages acm blei jordan latent dirichlet allocation journal machine learning research chang gerrish wang blei reading tea leaves humans interpret topic models advances neural information processing systems pages churchill stochastic models heterogeneous dna sequences bulletin mathematical biology fredkin rice bayesian restoration patch clamp recordings biometrics pages goldwater griffiths fully bayesian approach unsupervised tagging annual computational linguistics volume page citeseer guha neuberg bayesian hidden markov modeling array cgh data journal american statistical association hamilton new approach economic analysis nonstationary time series business cycle econometrica journal econometric society pages hayashi sakai nash implementation competitive equilibria market international journal game theory huai chen zhu xiong bao liu tian toward personalized context recognition mobile users semisupervised bayesian hmm approach acm transactions knowledge discovery data tkdd juang rabiner hidden markov models speech recognition technometrics litecky igou aken skills management oriented enterprise system job markets proceedings annual conference computers people research pages acm liu neuwald lawrence markovian structures biological sequence alignments journal american statistical association malherbe cataldi ballatore bringing order job market efficient job offer categorization proceedings international acm sigir conference research development information retrieval pages acm rabiner tutorial hidden markov models selected applications speech recognition proceedings ieee romer chow advanced macroeconomic theory shapiro stiglitz equilibrium unemployment worker discipline device american economic review teh jordan beal blei hierarchical dirichlet processes journal american statistical association varian repcheck intermediate microeconomics modern approach volume norton company new york xie xing integrating document clustering topic modeling arxiv preprint zhang kim xing dynamic topic modeling monitoring market competition online text image data proceedings acm sigkdd international conference knowledge discovery data mining pages acm zhu zhu chen liu xiong tracking evolution social emotions topic models knowledge information systems pages zhu liu xiong chen popularity modeling mobile apps sequential approach cybernetics ieee transactions
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matrix completion related problems via strong duality yingyu david hongyang jan abstract work studies strong duality matrix factorization problems show certain dual conditions problems dual optimum well understood convex optimization little known problems propose novel analytical framework show certain dual conditions optimal solution matrix factorization program thus global optimality program achieved solving convex dual conditions satisfied wide class matrix factorization problems although matrix factorization problems hard solve full generality analytical framework may independent interest optimization broadly apply framework two prototypical matrix factorization problems matrix completion robust principal component analysis pca examples efficiently recovering hidden matrix given limited reliable observations framework shows exact recoverability strong duality hold sample complexity guarantees matrix completion robust pca carnegie mellon university email ninamf university email yliang carnegie mellon university email dwoodruf corresponding author carnegie mellon university email hongyanz introduction matrix factorization problems emerging object study theoretical computer science optimization machine learning many domains theoretical computer science optimization study models led significant advances provable algorithms converge local minima linear time machine learning matrix factorization serves building block prediction recommendation systems winning submission netflix prize two prototypical examples matrix completion robust principal component analysis pca work develops novel framework analyze class matrix factorization problems strong duality leads exact recoverability matrix completion robust principal component analysis pca via solution convex problem matrix factorization problems stated finding target matrix form minimizing objective function known value min kabkf factor matrices function characterizes desired properties work motivated several promising areas analytical framework matrix factorizations applicable first area matrix completion shown matrix exactly recovered finding solution form consistent observed entries assuming incoherent problem received tremendous amount attention due important role optimization wide applicability many areas quantum information theory collaborative filtering second area robust pca fundamental problem interest data processing aims recovering sparse components exactly superposition component corresponds product sparse component captured proper choice function norm believe analytical framework potentially applied problems broadly matrix sensing dictionary learning weighted approximation deep linear neural network may independent interest without assumptions structure objective function direct formulations matrix factorization problems optimize general standard assumptions structure problem sufficiently many samples optimization problems solved efficiently convex relaxation methods run local search algorithms given initialization close enough global solution basin attraction however methods sample complexity significantly larger lower bound see table comparison problem becomes challenging number samples small enough initialization far desired solution case algorithm run local minimum saddle point another line work focused studying loss surface matrix factorization problems providing positive results approximately achieving global optimality one nice property line research spurious local minima specific applications matrix completion matrix sensing dictionary learning phase retrieval linear deep neural networks etc however results based concrete forms objective functions also even local minimum guaranteed globally optimal general remains escape saddle points additional arguments needed show achievement local minimum importantly existing results rely strong assumptions sample size strong duality primal problem problem surface surface theorem dual certificate common optimal solution figure strong duality matrix factorizations results work studies exact recoverability problem variety matrix factorization problems goal provide unified framework analyze large class matrix factorization problems achieve efficient algorithms main results show although matrix factorization problems hard optimize general certain dual conditions duality gap zero thus problem converted equivalent convex program main theorem framework following theorems strong duality informal certain dual conditions strong duality holds optimization problem argmin convex closed function closed means set closed set words problem problem argmin convex function defined exactly optimal solutions sense maxm kmkr kmkr sum first largest squared singular values theorem connects program convex counterpart via strong duality see figure mention strong duality rarely happens optimization region matrix approximation quadratic optimization two quadratic constraints among paradigms enjoy nice property given strong duality computational issues original problem overcome solving convex problem positive result framework complemented lower bound formalize hardness problem general assuming random problem hard see conjecture give strong negative result deterministic algorithms also bpp see section discussion conclusion holds randomized algorithms succeeding probability least theorem hardness statement informal assuming random hard average problem form deterministic algorithm achieving opt objective function value requires time opt optimum absolute constant bpp conclusion holds randomized algorithms succeeding probability least framework requires dual conditions theorem verified show two prototypical problems matrix completion robust pca obey conditions belong linear inverse problems form proper choice function aim exactly recovering hidden matrix rank given limited number linear observations matrix completion linear measurements form support set uniformly distributed among subsets cardinality strong table comparison matrix completion methods condition number obeys max accuracy output min first line upper bound second line approach work sample complexity log log condition log condition max max log lower max log log condition log log log max log condition log log log log condition condition similar conditions condition condition condition condition duality either study exact recoverability primal problem investigate validity convex dual problem study former tools geometric functional analysis recall analysis matrix completion one typically requires condition given matrix skinny svd vectors entry equal entries equal incoherence condition claims information spreads throughout left right singular vectors quite standard matrix completion literature standard condition following results theorems matrix completion informal unique matrix rank consistent measurements high probability provided log satisfies incoherence addition exists convex optimization matrix completion form exactly recovers high probability provided log condition number best knowledge result first connect convex matrix completion matrix completion two parallel lines research received significant attention past years table compares result prior results robust pca instead studying exact recoverability problem matrix completion investigate problem directly robust pca problem decompose given matrix sum component sparse component obtain following theorem robust pca lower bound theorems robust pca informal exists convex optimization formulation robust pca high form problem exactly recovers incoherent matrix min probability even rank max size support support set uniformly distributedq among sets cardinality incoherence parameter satisfies constraints bounds theorem match best known results robust pca literature supports uniformly sampled assumption arguably intuitive see section note results hold even close full rank constant fraction entries noise independently work developed framework analyze loss surface problems applied framework matrix completion robust pca bounds matrix completion sample complexity log pca outlier entries deterministic number method tolerate zhang also studied robust pca problem using optimization outlier entries deterministic number outliers algorithm tolerate strong duality approach unique work techniques reduction approximation results inspired approximation problem min know local solutions globally optimal see lemma strong duality holds extend property general problem main given matrix insight reduce problem form using term prior work attempted apply similar reduction conclusions either depended unrealistic conditions local solutions local solutions conclusions relied strong assumptions objective functions objective functions concrete instead general results formulate strong duality via existence dual certificate applications existence dual certificate converted mild assumptions number measurements sufficiently large positions measurements randomly distributed illustrate importance randomness may exist deterministic world blessing randomness desired dual certificate hardness result shows problem weighted approximation cast form without randomization measurements made underlying low rank matrix achieve good objective value mention achieve strong duality similar phenomenon observed deterministic matrix completion thus utilize randomness analyze existence dual certificate matrix completion assumption measurements random standard angle space space matrices consistent observations space space matrices small high probability namely almost unique matrix consistent measurements thus dual certificate represented another form convergent neumann series concerning projection operators spaces remainder proof show construction obeys dual conditions prove dual conditions matrix completion use fact subspace complement space almost orthogonal sample size sufficiently large implies projection dual certificate space small norm exactly matches dual conditions geometric analysis strong duality implies primal problem problem exactly solutions sense thus show exact recoverability linear inverse problems matrix completion robust pca suffices study either primal problem convex counterpart former analysis matrix completion mention traditional techniques convex optimization break problem since subgradient objective function may even exist instead apply tools geometric functional analysis analyze descent cone geometry problem geometric analysis stark contrast prior techniques convex geometric analysis figure feasibility convex combinations constraints used define minkowski functional definition atomic norm method uses constraint null matrix completion problem two hard constraints rank output matrix larger implied form output matrix consistent sampled measurements study feasibility condition problem geometric unique feasible solution problem starting perspective rank increases directions constraint set requirement feasibility condition geometrically interpreted descent cone rank constraint set must intersect uniquely see figure means unique matrix satisfies constraints shown following tangent cone argument let set matrices rank around underlying matrix tangent cone argument definition subset tangent cone latter cone interest nice form namely space mentioned space matrices leverage results prior work imply large enough sample size namely among matrices form matrix rank consistent observations using argument show sample size needed exact recovery matrix completion matches known lower bound constant factor putting things together summarize new analytical framework following figure geometric analysis exact recovery problem optimal sample complexity problem exact recovery convex problem general randomness construction dual certificate reduction strong duality approximation techniques alternative method investigate exact recoverability problem via standard convex analysis find induced function similar nuclear norm observation prove validity robust pca form combining property standard techniques preliminaries use calligraphy represent set bold capital letters represent matrix bold letters represent vector letters represent scalars specifically denote underlying matrix use indicate column row entry row column represented xij condition number let max min function input matrix conjugate function defined maxm furthermore let denote conjugate function frequently use rank constrain rank equivalently represented qpby restricting number columns rows norms denote kxkf xij frobenius norm matrix let singular values nuclear norm trace norm defined operator norm kxk denote maxij two matrices equal dimensions denote aij bij denote function evaluated define indicator function convex set set denote otherwise cone denote set indices observed entries complement without confusion also indicates linear subspace formed matrices entries denote orthogonal projector subspace consider single norm operators namely operator norm denoted kak defined kak supkxkf orthogonal projection operator subspace know kpt whenever dim distributions denote standard gaussian random variable uniform uniform distribution cardinality ber bernoulli distribution success probability matrix factorizations new analytical framework section develop novel framework analyze general class matrix factorization problems framework applied different specific problems leads nearly optimal sample complexity guarantees particular study matrix factorization problem convex closed min show suitable conditions duality gap dual problem zero problem converted equivalent convex problem strong duality abi fixed leading objective first consider easy case kyk function abkf case establish following lemma globally optimal local minimum lemma given matrix objective function around saddle point negative given svdr directional curvature moreover local proof lemma basically calculate gradient let equal zero see appendix details given lemma reduce form plus extra term max abi max max define abkf lagrangian problem second equality holds closed convex argument fixed value lemma local minimum globally optimal minimizing equivalent minimizing fixed saddle remaining part analysis choose proper point mina problem optimal solution introduce following condition later show condition holds high probability problem exists condition solution explanation condition note abbt fixed particular set condition implies either saddle point local minimizer function fixed following lemma states local minimizer strong duality holds global minimizer exists dual certificate lemma dual certificate let local minimizer fixed strong satisfying condition pair svdr duality holds moreover relation saddle proof sketch assumption lemma show point lagrangian see appendix show strong duality fact inequality holds saddle point mina mina one hand mina hand weak duality mina mina therefore mina mina strong duality holds therefore argminab argminab abi argminab desired svdr lemma leads following theorem full rank prior work studying loss surface matrix approximation assumes matrix singular values work generalize result removing two assumptions one easily check minm lagrangian constraint optimization problem mina little abuse notation call lagrangian unconstrained problem well small large dual certificate figure geometry dual condition general matrix factorization problems optimal solution problem define matrix space theorem denote strong duality holds problem provided kpt conditions lemma hold proof proof idea construct dual certificate satisfy following condition condition svdr local minimizer assumption lemma kpt kmk turns matrix fact frequently use sequel denote left singular space right singular space linear space equivalently represented therefore imply note null col row vice versa svdr implies orthogonal decomposition svdr kek conversely kek condition imply therefore dual conditions equivalent kpt show dual condition theorem intuitively need show angle subspace small see figure specific function following see section demonstrate applications randomness obey dual condition high probability matrix completion matrix completion hidden matrix rank given measurements uniform sampled uniformly random subsets cardinality goal exactly recover high probability apply unified framework section matrix completion setting quantity governing difficulties matrix completion incoherence parameter intuitively matrix completion possible information spreads evenly throughout matrix intuition captured incoherence conditions formally denote skinny svd fixed matrix rank introduced condition matrix conditions shown condition holds many random matrices incoherence parameter log two positive results first result upper bound standard incoherence condition unique matrix rank consistent observations proof deferred appendix theorem upper bound let uniform support set uniformly distributed among sets cardinality suppose log absolute constant unique matrix rank condition probability least proof sketch consider feasibility matrix completion problem find matrix rank proof first identifies feasibility condition problem shows matrix obeys feasibility condition sample size large enough specifically note obeys conditions problem therefore matrix obeys condition follow condition descent cone matrices note descent cone contained subspace tool geometry functional analysis thus fact sample size large proof completed describe simple inefficient algorithm given theorem section positive result matches lower bound prior work claims sample complexity theorem optimal theorem lower bound theorem denote uniform support set uniformly distributed among sets cardinality suppose log absolute constant exist infinitely many matrices rank obeying probability least second positive result converts feasibility problem theorem convex optimization problem efficiently solved theorem efficient matrix completion let uniform support set uniformly distributed among sets cardinality suppose condition number absolute constants probability least output convex problem argmin provided log obeys unique exact argmina proof sketch shown theorem problem optimal sample complexity strong duality exactly recovers holds optimization problem equivalently converted convex program theorem straightforward strong duality suffices apply unified framework section prove strong duality show dual condition theorem holds high probability following arguments let global solution problem need show third equality holds since kpt interesting see dual condition satisfied angle subspace subspace small see figure sample size becomes larger larger angle becomes smaller smaller angle zero show sample size log sufficient condition condition hold robust principal component analysis section develop theory robust pca based framework problem robust pca given observed matrix form matrix corruption matrix sparse goal recover hidden matrices observation set make information spreads evenly throughout matrix matrix one entry whose absolute value significantly larger entries robust pca problem introduced extra incoherence condition recall skinny svd kuv work make following incoherence assumption robust pca instead note condition similar incoherence condition robust pca problem two notions incomparable note condition intuitive explanation namely entries must scatter almost uniformly across matrix following results robust pca theorem robust pca suppose matrix rank obeys incoherence assume support set uniformly distributed among sets cardinality probability least output optimization problem argmin provided rank exact namely positive absolute constants function given bounds rank sparsity theorem match best known results robust pca prior work assume support set sampled uniformly computational aspects computational efficiency discuss computational efficiency given strong duality note dual primal problem given see appendix dual max min maxhm problems solved efficiently due convexity particular grussler provided computationally efficient algorithm compute proximal operators functions hence algorithm find global minimum error function value time poly computational lower bounds unfortunately strong duality always hold general problems present strong lower bound based random hypothesis fairly standard conjecture complexity theory gives constant factor inapproximability problem deterministic algorithms even running exponential time additionally assume bpp bpp class problems solved probabilistic polynomial time class problems solved deterministic polynomial time conclusion holds randomized algorithms also standard conjecture complexity theory implied existence certain strong pseudorandom generators problem deterministic exponential time exponential size circuits therefore subexponential time algorithm achieving sufficiently small constant factor approximation problem general would imply major breakthrough complexity theory lower bound proved reduction maximum edge biclique problem details presented appendix theorem computational lower bound assume conjecture hardness random exists absolute constant deterministic algorithm achieving opt objective function value problem requires time opt optimum addition bpp conclusion holds randomized algorithms succeeding probability least proof sketch theorem proved using hypothesis random hard show hardness maximum edge biclique problem deterministic algorithms reduction maximum edge biclique problem problem complete proofs related work found appendices acknowledgments thank rong jason lee zhouchen lin tengyu benjamin recht tuo zhao useful discussions work supported part nsf grants nsf nsf nsf nsf nsf sloan research fellowship microsoft research faculty fellowship simons investigator award simons collaboration grant would like thank rina foygel finding bug proof theorem previous version related work matrix factorization popular topic studied theoretical computer science machine learning optimization review several lines research studying global optimality optimization problems global optimality matrix factorization lots matrix factorization problems shown spurious local minima either require additional conditions local minima based particular forms objective function specifically burer monteiro showed one minimize aat convex function solving directly without introducing local minima provided rank output larger rank true minimizer xtrue however condition often impossible check rank xtrue typically unknown priori resolve issue bach proved aat global minimizer local minimizer aat twice differentiable convex function haeffele vidal extended result allowing general form objective function twice differentiable convex function compact level set proper convex function lower however major drawback line research result fails local minimizer full rank matrix completion matrix completion prototypical example matrix factorization one line work matrix completion builds convex relaxation achieve optimal sample complexity recently showed matrix completion spurious local optimum sufficiently large matrix incoherent result positive matrices sample complexity optimal another line work built upon good initialization global convergence recent attempts showed one first compute form initialization singular value decomposition close global minimizer use approaches reach global optimality alternating minimization block coordinate descent gradient descent result contrast reformulate matrix completion problems equivalent convex programs guarantees global convergence initialization robust pca robust pca also prototypical example matrix factorization goal recover sparse components exactly superposition widely applied various tasks video denoising background modeling image alignment photometric stereo texture representation subspace clustering spectral clustering typically two settings robust pca literature support set sparse matrix uniformly sampled support set sparse matrix deterministic entries row column matrix large work discuss first case framework provides results match best known work setting matrix factorization problems matrix sensing another typical matrix factorization problem bhojanapalli showed matrix recovery model mina achieves optimality every local minimum operator satisfies restricted isometry property gave lower bound showed unstructured operator may easily lead local minimum globally optimal matrix factorization problems also shown nice geometric properties property local minima global minima examples include dictionary learning phase retrieval linear deep neural networks linear neural networks goal learn projection represents weight matrix connects hidden units layers study linear models central theoretical understanding loss surface deep neural networks activation functions dictionary learning aim recover complete square invertible dictionary matrix given signal form provided representation coefficient sufficiently sparse problem centers around solving matrix factorization problem sparsity constraint representation coefficient examples matrix factorization models range classic unsupervised learning problems like pca independent component analysis clustering recent problems matrix factorization weighted matrix approximation sparse coding tensor decomposition subspace clustering etc applying framework problems left future work atomic norms atomic norm recently proposed function linear inverse problems many norms norm nuclear norm serve special cases atomic norms widely applied problems compressed sensing matrix recovery blind deconvolution etc norm defined minkowski functional associated convex hull set kxka inf particular set convex hull infinite set matrices equals nuclear norm mention objective term kabkf problem similar atomic norm slight differences unlike atomic norm set infinite set matrices rank achieve better sample complexity guarantees based methods proof lemma local minimum globally lemma restated given matrix optimal objective function around saddle point negative directional curvature moreover local maximum proof critical point equivalently abbt note fixed matrix resp function convex coefficients resp prove desired lemma following claim claim two matrices define critical point global mapping form satisfying proof define critical point holds general solution satisfies property moorefor matrix penrose also abbt equivalently mmt rewritten plugging relation symmetric thus note matrix desired prove lemma also need following claim claim denote ordered set ordered largest distinct values let smallest eigenvalues uir denote matrix formed orthonormal eigenvectors associated ordered eigenvalues whose multiplicities two matrices define critical point exists ordered set invertible matrix matrix matrix block equal orthogonal projector dimension critical point real symmetric covariance matrix always represented proof note orthonormal matrix consisting eigenvectors diagonal matrix eigenvalues satisfy abbt define critical point converse notice put equivalently uput thus yields uput uput equivalently put notice diagonal matrix distinct eigenvalues put matrix blocks orthogonal projector dimension corresponding eigenvalues therefore exists index set put matrix follows uput since column space coincides column space form given thus local minimizer given globally optimal according show index set corresponding largest eigenvalues consists combinations indices eigenvalues corresponding pair given strict saddle point claim pair given strict saddle point proof let denote enough slightly perturb column space towards direction eigenvector precisely fix two indices notice thus largest index let etr let direct calculation shows hence lim note critical points form pair given strict saddle point pair given local minimum conclude local maximum proof completed proof lemma global minimizer exists dual certificate lemma restated let condition pair local minimizer fixed strong svdr duality holds moreover relation local minimizer proof assumption lemma function independent according lemma namely globally minimizes fixed furthermore argmina implies convexity function meaning due concavity variable thus saddle point prove strong duality fact saddle point one hand inequality holds min max min min max hand weak duality min max max min therefore mina mina strong duality holds hence argmin argmin argmin svdr desired proof theorem theorem upper bound restated let uniform support set uniformly distributed among sets cardinality suppose log absolute constant unique matrix rank probability least proof note sampling model uniform equivalent sampling model ber frequently use sequel see appendix consider feasibility matrix completion problem find matrix rank proof first identifies feasibility condition problem shows matrix obeys feasibility condition sample size large enough denote rank define rank following proposition feasibility problem proposition feasibility condition unique feasible solution problem proof notice problem equivalent another feasibility problem find matrix rank suppose since rank implies note means unique feasible solution problem remainder proof show proceed note escaping mesh techniques matrix sensing work matrix completion since drawn grassmanian according haar measure address issue instead need following lemmas first lemma claims tangent cone set evaluated slightly larger cone cone lemma theorem let subset real normed space follows cone tangent cone set point defined second lemma states tangent cone set evaluated represented closed form lemma theorem let skinny svd matrix tangent cone set rank linear subspace given ult mvt ready prove theorem lemma cone first equality holds definition meaning unique feasible solution problem thus rest proof find sufficient condition following lemma lemma assume ber incoherence condition holds probability least provided log absolute constant proof ber theorem high probability kpt provided log note however since therefore triangle inequality kpt since kpt proof completed note implies proof completed proof theorem argmina shown theorem problem optimal sample complexity strong duality holds exactly recovers optimization problem equivalently converted convex program theorem straightforward strong duality suffices apply unified framework section prove strong duality show global solution problem dual condition theorem holds high probability let need show third equality holds since kpt following lemma lemma construct kpt kpt eqn holds probability least construct proof prove lemma first claim following theorem theorem theorem assume sampled according bernoulli model success probability incoherence condition holds absolute constant log log probability least provided perturbation matrix suppose condition holds let exists setting prove condition eqn observe kpt ykf kpt kpt kpt kpt need show kpt proceeding begin introducing normalized version ppt note operator whenever kpt according theorem operator represented convergent neumann series kpt also note log sufficiently large absolute constant thus kpt kpt kpt kpt kpt kpt high probability proof completed thus suffices construct dual certificate conditions hold end partition partitions size assumption may choose log sufficiently large constant let ber denote set indices corresponding jeb set partitions define theorem kwk follows kwb following lemma together implies strong duality straightforwardly lemma assumptions theorem dual certification obeys dual condition probability least proof well known matrix completion uniform model uniform equivalent bernoulli model ber element included probability independently see section brief justification equivalence suppose ber prove lemma preliminary need following lemmas lemma lemma suppose fixed matrix suppose ber high probability log log absolute constant max max zib max zaj lemma lemma suppose ber fixed matrix high probability provided log absolute constant lemma lemma suppose fixed matrix ber log sufficiently large high probability observe lemma kwj lemma kwj kwj therefore kpt kpt let denote lemma kpt log log log log log log log note facts assume wlog setting max kei max kei etj etj maxheti etj substituting max keti ketj log max ketj proof completed obtain kpt proof theorem theorem robust pca restated suppose obeys incoherence assume support set uniformly distributed among sets cardinality probability least output optimization problem argmin provided rank exact namely positive absolute constants function given subgradient function lemma let skinny svd matrix rank subdifferential evaluated given kwk proof note fixed function set optimal solutions problem form subdifferential conjugate function evaluated set notice function unitarily invariant von neumann trace inequality optimal solutions problem given diag value larger given optimal solution problem max solution unique proof complete dual certificates lemma assume unique solution problem exists pair sign kwk kkf proof let optimal solution problem definition subgradient inequality follows kpt kpt kpt hkf note hkf hkf hkf khkf kpt hkf hkf hkf kpt hkf implies hkf hkf kpt hkf kpt therefore kpt kpt second inequality holds optimal thus note implies completes proof according lemma show exact recoverability problem sufficient find appropriate dual certification least squares golfing scheme remainder proof construct dual condition holds true introducing construction assume ber equivalently ber allowed large absolute constant note distribution drawn independently replacement ber dlog obeys log implies construct based distribution construction separates two terms construct apply golfing scheme introduced specifically constructed inductive procedure construct apply method least squares sign note thus neumann series observe sign prove dual condition suffices show kwl kws proof dual conditions since constructed dual certificate remainder show obeys dual conditions high probability following lemma assume ber assumptions theorem given obeys dual condition proof let set log proof holds kpt kpt log log lemma note lemma lemma therefore kwl log log log log log since log incoherence used fact proof follows theorem log proof definition know since shown suffices prove lemma incoherence log log choose absolute constant true constant sufficiently small prove given obeys dual condition following lemma assume ber assumptions theorem given obeys dual condition proof according standard argument equivalent studying case signs independently distributed proof recall sign sign sign bound first term ksign sign bound second term let denote sizes respectively know sign yxt sign sup yxt sign sup consider random variable yxt sign zero expectation hoeffding inequality exp exp xyt therefore union bound sign exp note conditioned event kgk sign exp following lemma guarantees event holds high probability small absolute constant lemma cor suppose ber incoherence holds probability least provided log absolute constant setting completes proof proof recall sign sign sign wij hws etj etj let etj hoeffding inequality union bound sup exp note conditioned event unconditionally sup etj kpt etj uut vvt exp lemma setting proof completed proof theorem computational lower bound problem assumes hardness random conjecture random let constant consider random formula variables clause literals clauses picked independently probability algorithm always outputs random formula satisfiable outputs probability least random formula unsatisfiable must run time input absolute constant based conjecture following computational lower bound problem show problem general hard deterministic algorithms additionally assume bpp conclusion holds randomized algorithms high probability theorem computational lower bound restated assume conjecture exists absolute constant algorithm achieves opt objective function value problem constant probability requires time opt optimum addition bpp conclusion holds randomized algorithms succeeding probability least proof theorem proved using hypothesis random hard show hardness maximum edge biclique problem deterministic algorithms definition maximum edge biclique problem input bipartite graph output complete bipartite subgraph maximized showed random assumption exist two constants efficient deterministic algorithm able distinguish bipartite graphs clique size bipartite cliques size reduction uses bipartite graph least edges large probability constant given given bipartite graph define follows define matrix yij edge yij edge wij edge wij poly edge choose large enough constant let wij exists biclique least edges number remaining edges solution min cost hand exist biclique edges number remaining edges least solution min cost least choose large enough combined result completes proof deterministic algorithms rule randomized algorithms running time function observe define new problem problem except input description padded string length string irrelevant solving problem changes input size poly argument previous paragraph deterministic algorithm still requires time solve problem new input size however randomized algorithm solve time runs poly time contradicts assumption bpp completes proof matrix completion upper bound theorem formulates matrix completion feasibility problem however priori unclear algorithm finding log sample complexity incoherence via solving feasibility problem answer question mention matrix completion solved finite time minimum assumptions namely note feasibility problem equivalent finding zero polynomial eti abej unknowns since assumed orthogonal entries written poly bits kbkf exp poly means one rounds entries nearest additive grid multiple exp poly get matrix entry represents true entry optimal additive exp poly error course one write cases entries irrational found exp time gives exponential time algorithm solve feasibility problem theorem matrix completion dual problems section derive dual problems program according primal problem equivalent min max therefore dual problem given max min max max problem derived min min max max min maxx convex function holds definition conjugate function problems solved efficiently due convexity particular provided computationally efficient algorithm compute proximal operators functions hence algorithm find global minimum error function value time poly equivalence bernoulli uniform models lemma let number bernoulli trials suppose ber probability least provided log absolute constant proof scalar chernoff bound exp exp taking log appropriate absolute constant exp taking log appropriate absolute constant exp given conclude probability least provided log absolute constant references naman agarwal zeyuan brian bullins elad hazan tengyu finding approximate local minima nonconvex optimization linear time arxiv preprint sanjeev arora aditya bhaskara rong tengyu algorithms provable dictionary learning arxiv preprint pranjal awasthi balcan nika haghtalab hongyang zhang learning compressed sensing asymmetric noise annual conference learning theory pages anima anandkumar rong efficient approaches escaping higher order saddle points optimization arxiv preprint animashree anandkumar rong daniel hsu sham kakade matus telgarsky tensor decompositions learning latent variable models journal machine learning research christoph monaldo mastrolilli ola svensson inapproximability results maximum edge biclique minimum linear arrangement sparsest cut siam journal computing alekh agarwal sahand negahban martin wainwright noisy matrix decomposition via convex relaxation optimal rates high dimensions annals statistics pages ali ahmed benjamin recht justin romberg blind deconvolution using convex programming ieee transactions information theory zeyuan katyusha first direct acceleration stochastic gradient methods arxiv preprint aditya bhaskara moses charikar ankur moitra aravindan vijayaraghavan smoothed analysis tensor decompositions acm symposium theory computing pages amir beck yonina eldar strong duality nonconvex quadratic optimization two quadratic constraints siam journal optimization pierre baldi kurt hornik neural networks principal component analysis learning examples without local minima neural networks srinadh bhojanapalli anastasios kyrillidis sujay sanghavi dropping convexity faster optimization annual conference learning theory pages samuel burer renato monteiro local minima convergence semidefinite programming mathematical programming francis bach julien mairal jean ponce convex sparse matrix factorizations arxiv preprint srinadh bhojanapalli behnam neyshabur nati srebro global optimality local search low rank matrix recovery advances neural information processing systems pages saugata basu richard pollack marie francoise roy combinatorial algebraic complexity quantifier elimination acm stephen boyd lieven vandenberghe convex optimization cambridge university press balcan hongyang zhang matrix completion via adaptive sampling advances neural information processing systems pages yudong chen matrix completion ieee transactions information theory anna choromanska mikael henaff michael mathieu ben arous yann lecun loss surfaces multilayer networks international conference artificial intelligence statistics emmanuel xiaodong john wright robust principal component analysis journal acm emmanuel ben recht exact matrix completion via convex optimization foundations computational mathematics emmanuel benjamin recht simple bounds recovering models mathematical programming pages venkat chandrasekaran benjamin recht pablo parrilo alan willsky convex geometry linear inverse problems foundations computational mathematics emmanuel terence tao power convex relaxation matrix completion ieee transactions information theory yudong chen martin wainwright fast estimation projected gradient descent general statistical algorithmic guarantees arxiv preprint uriel feige relations average case complexity approximation complexity proceedings annual ieee conference computational complexity canada may page rong furong huang chi jin yang yuan escaping saddle points online stochastic gradient tensor decomposition annual conference learning theory pages rong chi jin zheng spurious local minima nonconvex low rank problems unified geometric analysis arxiv preprint andreas goerdt lanka approximation hardness result bipartite clique electronic colloquium computational complexity report volume rong jason lee tengyu matrix completion spurious local minimum advances neural information processing systems pages david gamarnik quan hongyi zhang matrix completion samples linear time arxiv preprint christian grussler anders rantzer pontus giselsson optimization convex constraints arxiv preprint gross recovering matrices coefficients basis ieee transactions information theory quanquan zhaoran wang han liu sparse structure pursuit via alternating minimization international conference artificial intelligence statistics pages moritz hardt understanding alternating minimization matrix completion ieee symposium foundations computer science pages moritz hardt ankur moitra algorithms hardness robust subspace recovery arxiv preprint moritz hardt raghu meka prasad raghavendra benjamin weitz computational limits matrix completion annual conference learning theory pages benjamin haeffele vidal global optimality tensor factorization deep learning beyond arxiv preprint bingsheng xiaoming yuan convergence rate alternating direction method siam journal numerical analysis benjamin haeffele eric young rene vidal structured matrix factorization optimality algorithm applications image processing international conference machine learning pages russell impagliazzo avi wigderson bpp requires exponential circuits derandomizing xor lemma acm symposium theory computing pages johannes jahn introduction theory nonlinear optimization springer berlin heidelberg michel francis bach absil rodolphe sepulchre optimization cone positive semidefinite matrices siam journal optimization chi jin rong praneeth netrapalli sham kakade michael jordan escape saddle points efficiently arxiv preprint prateek jain raghu meka inderjit dhillon guaranteed rank minimization via singular value projection advances neural information processing systems pages prateek jain praneeth netrapalli sujay sanghavi matrix completion using alternating minimization acm symposium theory computing pages kenji kawaguchi deep learning without poor local minima arxiv preprint yehuda koren robert bell chris volinsky matrix factorization techniques recommender systems ieee computer raghunandan hulikal keshavan efficient algorithms collaborative filtering phd thesis stanford university raghunandan keshavan andrea montanari sewoong matrix completion entries ieee transactions information theory raghunandan keshavan andrea montanari sewoong matrix completion noisy entries journal machine learning research michel ledoux concentration measure phenomenon number american mathematical society yuanzhi yingyu liang andrej risteski recovery guarantee weighted approximation via alternating minimization international conference machine learning pages praneeth netrapalli niranjan sujay sanghavi animashree anandkumar prateek jain robust pca advances neural information processing systems pages sahand negahban martin wainwright restricted strong convexity weighted matrix completion optimal bounds noise journal machine learning research michael overton robert womersley sum largest eigenvalues symmetric matrix siam journal matrix analysis applications benjamin recht simpler approach matrix completion journal machine learning research james renegar computational complexity geometry theory reals part introduction preliminaries geometry sets decision problem existential theory reals symb james renegar computational complexity geometry theory reals part general decision problem preliminaries quantifier elimination symb ilya razenshteyn zhao song david woodruff weighted low rank approximations provable guarantees acm symposium theory computing pages ruoyu sun luo guaranteed matrix completion via nonconvex factorization ieee symposium foundations computer science pages sun qing john wright geometric analysis phase retrieval ieee international symposium information theory pages sun qing john wright complete dictionary recovery sphere overview geometric picture ieee transactions information theory sun qing john wright complete dictionary recovery sphere recovery riemannian method ieee transactions information theory nathan srebro adi shraibman rank international conference computational learning theory pages springer reinhold schneider uschmajew convergence results projected methods varieties matrices via lojasiewicz inequality siam journal optimization yuan shen zaiwen wen yin zhang augmented lagrangian alternating direction method matrix separation based factorization optimization methods software gongguo tang badri narayan bhaskar parikshit shah benjamin recht compressed sensing grid ieee transactions information theory stephen ross boczar mahdi soltanolkotabi benjamin recht solutions linear matrix equations via procrustes flow arxiv preprint roman vershynin lectures geometric functional analysis pages roman vershynin introduction analysis random matrices arxiv preprint roman vershynin estimation high dimensions geometric perspective sampling theory renaissance pages springer wang huan stability matrix factorization collaborative filtering international conference machine learning pages zaiwen wen wotao yin yin zhang solving factorization model matrix completion nonlinear successive algorithm mathematical programming computation xinyang dohyung park yudong chen constantine caramanis fast algorithms robust pca via gradient descent advances neural information processing systems pages qinqing zheng john lafferty convergent gradient descent algorithm rank minimization semidefinite programming random linear measurements advances neural information processing systems pages qinqing zheng john lafferty convergence analysis rectangular matrix completion using factorization gradient descent arxiv preprint hongyang zhang zhouchen lin chao zhang counterexample validity using nuclear norm convex surrogate rank european conference machine learning principles practice knowledge discovery databases volume pages hongyang zhang zhouchen lin chao zhang completing matrices corrupted samples coefficients general basis ieee transactions information theory hongyang zhang zhouchen lin chao zhang edward chang exact recoverability robust pca via outlier pursuit tight recovery bounds aaai conference artificial intelligence pages hongyang zhang zhouchen lin chao zhang junbin gao robust latent low rank representation subspace clustering neurocomputing hongyang zhang zhouchen lin chao zhang junbin gao relations among low rank subspace recovery models neural computation xiao zhang lingxiao wang quanquan nonconvex free lunch plus sparse matrix recovery arxiv preprint tuo zhao zhaoran wang han liu nonconvex optimization framework low rank matrix estimation advances neural information processing systems pages
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nov heat kernel based community detection kyle kloster david gleich purdue university west lafayette purdue university west lafayette kkloste dgleich abstract heat kernel type graph diffusion like personalized pagerank diffusion useful identifying community nearby starting seed node present first deterministic local algorithm compute diffusion use algorithm study communities produces algorithm formally relaxation method solving linear system estimate matrix exponential norm prove algorithm stays localized large graph constant runtime depends parameters diffusion size graph large graphs experiments indicate communities produced method better conductance produced pagerank although take slightly longer compute community identification task heat kernel communities perform better pagerank diffusion categories subject descriptors discrete mathematics graph algorithms pattern recognition general terms algorithms theory keywords heat kernel local clustering introduction community detection problem identify set nodes graph internally cohesive also separated remainder network one popular way capture idea conductance measure set treat idea formally next section permission make digital hard copies part work personal classroom use granted without fee provided copies made distributed profit commercial advantage copies bear notice full citation first page copyrights components work owned others author must honored abstracting credit permitted copy otherwise republish post servers redistribute lists requires prior specific permission fee request permissions permissions kdd august new york usa copyright held publication rights licensed acm acm http informally conductance set ratio number edges leaving set number edges touched set vertices value small indicates set many internal edges edges leaving many surveys empirical studies conductance measure surfaces one reliable measures community although measure critized producing communities empirical properties communities correlate highly sets produced algorithms optimize conductance furthermore methods identifying overlapping sets communities use conductance find realworld communities better alternative virtually rigorous algorithms identify sets small conductance based eigenvector computations local graph diffusions one notable exception graclus method uses relationship kernel variant conductance paper study new algorithm uses heat kernel diffusion identify communities network heat kernel discussed formally section although properties diffusion analyzed theory chung work work provide efficient algorithm compute diffusion recently chung simpson stated randomized monte carlo method estimate diffusion paper introduces efficient deterministic method estimate diffusion use study properties small conductance sets identified method communities use deterministic approach critical need differentiate subtle properties diffusion primary point comparison personalized pagerank diffusion used establish new insights properties communities large scale networks thus wish understand communities produced heat kernel compare produced personalized pagerank basic operation algorithm coordinate relaxation step used efficiently compute personalized pagerank known push operation graph term coordinate relaxation classical name operation dates back method distinguishes approach prior work use coordinate relaxation implicitly constructed linear system estimate heat kernel diffusion formally exponential random walk transition matrix began looking problem recently workshop paper showed style algorithm successfully estimates related quantity paper tackles fundamentally new direction although similar techniques required entirely new analysis particular able estimate diffusion constant time norm depends parameters diffusion size graph python implementation algorithm accomplish task presented figure remainder paper first review topics formally sections present algorithm section discuss related work section show new approach differs improves upon personalized pagerank diffusion synthetic problems summary contributions propose first local deterministic method accurately compute heat kernel diffusion graph code simple scalable graph access inexpensive method always localized even massive graphs provably constant runtime degreeweighted norm compare new heat kernel diffusion method venerable pagerank diffusion synthetic small large networks billion edges demonstrate differences large networks twitter method tends produce smaller tighter communities also accurate detecting communities make experimental codes available spirit reproducible research https preliminaries begin fixing notation let simple undirected graph vertices fix ordering vertices refer vertex numeric vertex denote degree node let associated adjacency matrix undirected symmetric furthermore let diagonal matrix degrees dii random walk transition matrix finally denote vector appropriate length zeros single entry vector conductance given subset denote vol sum degrees vertices boundary number edges one endpoint inside one endpoint outside notation conductance set given min vol vol conceptually probability random walk length one land outside given start node chosen uniformly random inside matrix exponential heat kernel graph involves matrix exponential wish briefly mention facts operation see higham treatment consider general matrix exponential function matrix exponential applied rather given substituting matrix taylor series expansion exponential function exp said exponential diagonal matrix exponential function applied diagonal entries phenomenon occurs powers diagonal matrix simply diagonal elements raised given power matrix ggt generalization notion symmetric matrix called normal matrix eigenvalue decomposition simple albeit inefficient means computing exponential exp exp computing matrix exponential even product exponential vector exp still active area research finding small conductance communities diffusions graph diffusion sum following form stochastic vector sums one intuitively diffusion captures quantity material node flows graph terms provide decaying weight ensures diffusion eventually dissipates context paper interested diffusions single nodes neighborhood sets single vertex cases small set call origins diffusion seeds given estimate diffusion seed produce small conductance community using sweep procedure involves computing sorting nodes descending order magnitude vector computing conductance prefix sorted list due properties conductance efficient means computing conductances return set smallest conductance community around seeds personalized pagerank diffusion one instances framework personalized pagerank diffusion fix defined properties diffusion studied extensively particular andersen establish local cheeger inequality using particular algorithm called push locally distributes mass local cheeger inequality informally states seed nearby set small conductance result sweep procedure set related conductance moreover show push algorithm estimates error weighted norm looking edges heat kernel diffusion another instance framework heat kernel diffusion ply replaces weights exp satisfying degree weighted bound exp known estimating gave rise similar type local cheeger inequality chung simpson monte carlo approach methods known estimate quantity efficiently new algorithm deterministic approach suitable comparing properties diffusions terminates exploring edges theorem parameter grows heat kernels compared pagerank different sets coefficients simply assign different levels importance walks varying lengths heat kernel coefficients decay much quickly heat kerk nel heavily weights shorter walks overall sum depicted figure property turn important consequences study methods large graphs section weight using standard properties matrix exponential factor exp exp scale problem equivalent computing satisfying exp characterization must satisfy similar weighted objective used push algorithm pagerank outline algorithm accomplish first approximate exp degree taylor polynomial compute use large implicitly formed linear system avoid explicitly evaluating taylor polynomial linear system state relaxation method spirit pagerank push algorithm order compute accurate approximation taylor polynomial exp determining exponential matrix sensitive computation rich history general matrix approximation via taylor polynomial exp length figure curve represents coefficients sum walks dotted blue lines give red give indicated values inaccurate kgk large mixed signs large powers contain large oppositely signed numbers cancel properly exact arithmetic however intend compute exp nonnegative calculation rely delicate cancellations furthermore approximation need highly precise therefore use polynomial exp approximation details choosing see section assume chosen exp algorithm overall idea local clustering algorithm approximate heat kernel vector form exp perform sweep describe method call approximating algorithm rooted recent work computing accurate column exp heavily tuned objective thus overall strategy classical pagerank push method classic relaxation method simplifications efficient implementation entirely novel particular new objective paper enables get constant runtime bound independent property graph differs markedly previous methods objective recall final step finding small conductance community involves dividing degree node thus goal compute since original publication work proven upperbound terms way compute satisfying triangle inequality exp objective error weights using degree taylor polynomial ultimately approximates approximating term sum polynomial times vector total error computed solution weighted sum errors individual term show lemma weights given polynomials define fixed degree taylor polynomial exponential define polynomials closely related functions central exponential integrators odes note guarantee total error satisfies criterion enough show error taylor term satisfies inequality analogous discussed detail section deriving linear system define basic step algorithm show influence total error rearrange taylor polynomial computation linear system denote kth term vector sum note pvk identity implies terms exactly satisfy linear system let approximate solution would block components desired approximation practice update single length solution vector adding updates vector instead maintaining different block vectors part furthermore block matrix side never formed explicitly block system place describe algorithm steps graph dictionary sets seed array seeds eps psis precomputed store dictionaries initialize residual collections deque initialize queue seed len seed append len popleft rvj perform relax step rvj mass rvj float len neighbors next next block last step add soln rvj len continue next next thresh math exp eps len thresh thresh psis next thresh next mass thresh append next add queue next next mass figure algorithm working python code graph stored dictionary sets statement returns set neighbors associated vertex solution vector indexed vertices residual vector indexed tuples pairs vertices steps fully working demo may downloaded github https algorithm given random walk transition matrix scalar seed vector inputs solve linear system follows denote initial solution vector initial residual denote entry corresponding node residual block idea iteratively remove entries satisfy organize process begin placing nonzero entries queue place updated entries satisfy proceeds follows step pop top entry call subtract entry making add add pei residual block entry updated add entry back satisfies entries satisfy removed resulting solution vector satisfy prove section along bound work required achieve present working code method figure shows optimize computation using sparse data structures make highly efficient practice choosing last detail algorithm need discuss pick want guarantee accuracy using exp exp tpt tpt get new upperbound exp noting exp tpt tpt exp tpt tpt since stochastic know norm exp tpt tpt bounded ktpt guarantee enough choose implies determined effi ciently simply iteratively computing terms taylor polynomial error less desired error practice required choice greater log think made rigorous outline convergence result proof proceeds follows first relate error vector taylor approximation error vector solving linear system described section second express error vector terms residual blocks linear system involve writing sum residual blocks weights third use previous results upperbound use show guaranteed stopping criterion finally prove performing steps stopping criterion attained requires work bounded related work wish highlight ideas recently emerged literature clarify method differs discuss terms community detection matrix exponential fast diffusion methods relaxation methods community detection conductance conductance often appears community detection known one important measures community personalized pagerank method one scalable methods find sets small conductance although recent work opened new possibility localized maxflow algorithms pagerank algorithm use point comparison zhu recently provided improved bound performance algorithm finding sets high internal conductance internal conductance set minimum conductance subgraph induced set would expect realworld communities large internal conductance due similarity algorithm personalized pagerank diffusion believe similar result likely holds well matrix exponential network analysis recently matrix exponential frequently appeared tool network analysis literature used estimate node centrality graph kernels already mentioned clustering community detection many studies involve fast ways approximate entire matrix exponential instead single column study instance sui describe decomposition network useful estimating katz scores matrix exponential orecchia mahoney show heat kernel diffusion implicitly approximates diffusion operator using particular type generalized entropy provides principled rationale use fast methods diffusions perhaps related work recent monte carlo algorithm chung simpson estimate heat kernel diffusion via random walk sampling scheme approach involves directly simulating random walk transition probabilities mirror taylor series expansion exponential comparison approach entirely deterministic thus useful compare diffusions eliminates algorithmic variance endemic monte carlo simulations similar idea used borgs achieve randomized sublinear time algorithm estimate largest pagerank entries fact monto carlo methods frequently feature pagerank computations due relationship diffusion random walk deterministic approaches matrix exponential involve least one product relaxation methods algorithm use coordinate relaxation method similar gausssouthwell applied symmetric positive definite matrix would coordinate descent method proposed pagerank difference cases type relaxation method also used estimate katz diffusion recently used estimate column matrix exponential exp strict error able prove sublinear convergence bound assuming slowly growing maximum degree degree distribution paper comparision treats scaled exponential exp norm also shows constant runtime independent network property experimental results compare local clustering algorithm pprpush algorithms accept inputs symmetric graph seed set parameters required pprpush algorithms compute respective diffusion ranks starting seed set perform sweepcut resulting ranks difference algorithms lies solely diffusion used particular parameters conducted timing experiments dual cpu system intel xeon processor ghz cores cores total ram none experiments needed anywhere near memory parallelism implementation uses matlab sparse matrix data structure mex interface uses unordered maps store sparse vectors equivalent code figure synthetic results section study behavior pagerank heat kernel diffusions symbolic image graph chaotic function graphs result loosely reminiscent social network pockets structure like communities also chaotic behaviour results like property symbolic image function graph node represents region space edges represent action function region space consider functions unit square associate node pixel image illustrate vectors images figure left illustrate graph construction works remaining examples let map results chaotic nonlinear dynamical system use construction sample points region space symmetrize result discard weights discretize space grid results node graph edges figure right also show pagerank vector uniform teleportation image illustrate structure function next figure compare vectors sets identified diffusions starting single seed node chose parameters two methods perform amount work results would expected figure many remaining experiments show pagerank diffuses larger region graph whereas remains focused pagerank finds large community nodes whereas heat kernel finds small community around nodes slightly worse conductance experiment suggests results also hold networks communities often small heat kernel diffusion produce accurate communities networks ppr vector ppr set size figure left illustration symbolic image function graph large blue node represents region space thick blue edges represent thin red values function behave region right global pagerank vector image illustrates features chaotic map shows pockets structure vector set size figure compare pagerank diffusions symbolic image chirikov map see figure pprgrow finds larger set slightly better conductance whereas hkgrow finds tighter set conductance networks smaller sets like communities runtime conductance next compare runtime conductance algorithms suite social networks pprpush fix compute pagerank multiple values output set best conductance obtained matches way method commonly used past work compute heat kernel rank four different parameter sets output set best conductance among also include early termination criterion case degrees nodes relaxed sum dil exceeds however even smaller input graphs condition likely met smaller value appear reached threshold furthermore main theorem paper imp plies quantity dil exceed datasets use summarized table datasets modified undirected single connected component datasets originally presented following papers compare runtimes two algorithms display figure graph percentiles runtimes trials performed given graph trial consisted choosing node graph uniformly random seed calling pagerank heat kernel algorithms larger datasets much broader spectrum node degrees therefore greater variance instead performed trials additionally display percentiles conductances achieved exact set trials trendlines figures omit trials order better show trends median results plotted open circles figures show small graphs faster pprpush gets larger worse conductance picture reverses large graphs slower finds smaller better conductance sets cluster size conductance highlight individual results previous experiment symmetrized twitter network find finds sets better conductance pprpush sizes communities network see figure clusters produced conclude evaluation identifying groundtruth communities datasets experiment dataset first located known communities dataset size greater thank amirmahdi ahmadinejad pointing small mistake original implementation experiment though reported table original publication carried experiment commmunities size greater ahmadinejad noted practice used communities size less results shown table corrected experiment fact even favorable algorithm original reported results hkgrow pprgrow runtime friendster conductances ppr hkgrow pprgrow table result evaluating heat kernel pagerank finding communities heat kernel finds smaller accurate sets slightly worse conductance data conductance set size amazon dblp youtube orkut friendster given one community using every single node individual seed looked sets returned pprpush using standard procedure picked set seed highest measure recall measure harmonic mean precision recall report mean measure conductance set size average taken trials table results show produces slightly inferior conductance scores using much smaller sets substantially better measures suggests better captures properties communities pagerank diffusion sense tighter sets produced heat kernel better focused around communities larger sets produced pagerank diffusion conclusions results suggest algorithm viable companion celebrated pagerank push algorithm may even worthy competitor tasks require accurate communities large graphs furthermore conductances graph runtime ppr table datasets figure top figure runtimes shown percentile trendlines select set experiments bottom conductances shown way suspect method useful myriad uses diffusions linkprediction even logic programming future plan explore method directed networks well better methods selecting parameters diffusion also possible new ideas may translate faster methods diffusions katz modularity methods plan explore diffusions well acknowledgements work supported nsf career award references abrahao soundarajan hopcroft kleinberg separability structural classes communities kdd pages higham computing action matrix exponential application exponential integrators siam sci march alberich marvel universe looks almost like real social network arxiv preprint alon milman isoperimetric inequalities graphs superconcentrators comb theory series conductance ppr density clustersize ppr conductance figure top figure shows scatter plot conductance community size twitter graph two community detection methods bottom figure shows kernel density estimate conductances achieved method shows likely return set lower conductance andersen chung lang local graph partitioning using pagerank vectors focs andersen lang algorithm improving graph partitions soda pages january avrachenkov litvak nemirovsky osipova monte carlo methods pagerank computation one iteration sufficient siam numer february bahmani chakrabarti xin fast personalized pagerank mapreduce sigmod pages new york usa acm bahmani chowdhury goel fast incremental personalized pagerank proc vldb arenas models social networks based social distance attachment phys rev nov boldi rosa santini vigna layered label propagation multiresolution ordering compressing social networks www pages march bonchi esfandiar gleich greif lakshmanan fast matrix computations pairwise columnwise commute times katz scores internet mathematics borgs brautbar chayes teng matrix sampling pagerank computation internet mathematics online chung heat kernel pagerank graph pnas chung local graph partitioning algorithm using heat kernel pagerank internet mathematics chung simpson solving linear systems boundary conditions using heat kernel pagerank algorithms models web graph pages springer dhillon guan kulis weighted graph cuts without eigenvectors multilevel approach ieee trans pattern anal mach november estrada characterization molecular structure chemical physics letters estrada higham network properties revealed matrix functions siam review farahat lofaro miller rae ward authority rankings hits pagerank salsa existence uniqueness effect initialization siam journal scientific computing fiedler algebraic connectivity graphs czechoslovak mathematical journal higham functions matrices theory computation siam jeh widom scaling personalized web search www pages acm kang faloutsos beyond caveman communities hubs spokes graph compression mining icdm pages washington usa ieee computer society katz new status index derived sociometric analysis psychometrika march kloster graph diffusions matrix functions fast algorithms localization results phd thesis purdue university kloster gleich fast relaxation method computing column matrix exponential stochastic matrices large sparse networks arxiv kloster gleich method approximating column matrix exponential matrices large sparse networks algorithms models web graph page press kloster gleich heat kernel based community detection proceedings acm sigkdd international conference knowledge discovery data mining pages acm kondor lafferty diffusion kernels graphs discrete input spaces icml pages san francisco usa morgan kaufmann publishers kunegis lommatzsch learning spectral graph transformations link prediction icml pages kwak lee park moon twitter social network news media www pages leskovec huttenlocher kleinberg signed networks social media chi pages leskovec kleinberg faloutsos graph evolution densification shrinking diameters acm trans knowl discov data march leskovec lang dasgupta mahoney community structure large networks natural cluster sizes absence large clusters internet mathematics september liou novel method evaluating transient response appendix proceedings ieee convergence theory luo tseng convergence coordinate descent method convex differentiable state main result bounding work required minimization optim theory approximate heat kernel accuracy scribed mcsherry uniform approach accelerated pagerank theorem let section steps computation www pages performed entries residual satisfy minchev wright norges produces approximation universitet mislove marcon gummadi druschel exp satisfying bhattacharjee measurement analysis online social networks proceedings acm sigcomm exp conference internet measurement pages acm amount work required bounded moler van loan nineteen dubious ways work compute exponential matrix siam review moler van loan nineteen dubious ways producing satisfying compute exponential matrix years exp later siam review newman network datasets equivalent producing satisfying http exp newman structure scientific collaboration networks pnas show error vector steps newman modularity community structure satisfies networks pnas following lemma expresses error vector orecchia mahoney implementing weighted sum residual blocks linear system regularization implicitly via approximate eigenvector shows polynomials weights computation icml pages orecchia sachdeva vishnoi lemma let defined section approximating exponential lanczos method notation section express error vector hkan otilde spectral algorithm balanced relax terms residual blocks follows separator stoc pages orecchia zhu algorithms local graph clustering soda pages schaeffer graph clustering computer science review proof consider recall shepelyansky zhirov google matrix let matrix first subdiagonal dynamical attractors ulam networks phys rev equal rewrite linear system march conveniently sui lee whang savas jain pingali dhillon parallel clustered approximation graphs application link prediction let true solution vectors approximating kasahara kimura editors languages showed section error compilers parallel computing volume lecture fact sum errors express notes computer science pages springer berlin terms residual partitions heidelberg given step pre cooperative association internet data multiplying yields analyais network datasets exactly definition note http error vector linear accessed system explicit computation inverse wang mazaitis cohen see details yields programming personalized pagerank locally groundable probabilistic logic proceedings acm international conference conference information knowledge management cikm pages new york usa acm whang gleich dhillon overlapping purposes full block vectors individual community detection using seed set expansion cikm partitions unimportant want sum pages new york usa acm previously discussed yang leskovec defining evaluating network communities based proceedings next use express hence acm sigkdd workshop mining data semantics mds pages new york usa acm zhu lattanzi mirrokni local algorithm terms residual blocks accomplish examfor finding clusters icml pages ining coefficients arbitrary block turn requires analyzing powers fix residual block consider product single term since multiplies term want know blocks look like know otherwise means contains single nonzero block hence summing blocks power yields bounded desired shown upperbound first show prove claim recall since polynomial equals tpt taking infinity norm applying triangle inequality definition gives equals returning original objective proving claim allows continue stopping criterion requires every entry residual satisfies satisfying equivalent condition guarantees finally implies exp completing proof first part theorem thus matrix coefficient side expression substituting side yields bounding work remains bound work required perform steps hkrelax stopping criterion achieved stopping criterion requires residual entry satisfy know step must operate entry larger threshold step relax entry satisfying consider solution vector note entry really sum values deleted always contains nonnegative values seed vector nonnegative step involves setting entry adding scaled column nonnegative hence equals sum added finally since entries nondecreasing change add positive values know ktn implies using fact values relax must satisfy dil simplifying yields dil lemma know giving lowerbound dil finally note dominating suboperation step consists relaxing spreading heat kernel rank neighbors node block since node neighbors step consists adds work performed dil exactly quantity bounded figure available png format http figure available png format http figure available png format http figure available png format http figure available png format http
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modeling intensity function point process via recurrent neural networks shuai junchi stephen xiaokang hongyuan may shanghai jiao tong university east china normal university ibm research china georgia tech benjaminforever xkyang jcyan schu zha abstract event sequence asynchronously generated random timestamp ubiquitous among applications precise arbitrary timestamp carry important clues underlying dynamics lent event data fundamentally different whereby series indexed fixed equal time interval one expressive mathematical tool modeling event point process intensity functions many point processes involve two components background effect history due inherent spontaneousness background treated time series need handle history events paper model background recurrent neural network rnn units aligned time series indexes history effect modeled another rnn whose units aligned asynchronous events capture dynamics whole model event type timestamp prediction output layers trained approach takes rnn perspective point process models background history effect utility method allows treatment modeling intensity often parametric form point processes meanwhile training opens venue reusing existing rich techniques deep network point process modeling apply model predictive maintenance problem using log dataset atms global bank headquartered north america introduction event sequence becoming increasingly available variety applications transactions social network activities conflicts equipment failures etc event data carry rich information event attribute type participator also timestamp indicating event occurs treated random variable event stochastically generated asynchronous manner timestamp makes correspondence author junchi yan research partially supported national key research development program china nsfc stcsm china postdoctoral science foundation funded project program nsf copyright association advancement artificial intelligence rights reserved event sequence fundamentally different time series montgomery jennings kulahci equal fixed time interval whereby time point serves role index major line research aalen borgan gjessing devoted study event sequence especially exploring timestamp information model underlying dynamics system whereby point process snyder miller powerful compact framework direction recently many machine learning based models scalable point process modeling attribute progressions direction part smart mathematical reformulations optimization techniques lewis mohler zhou zha song zhou zha song well novel parametric forms conditional intensity function shen ertekin rudin mccormick carefully designed researchers prior knowledge capture character dataset study however one major limitation parametric forms point process due specialized restricted expression capability arbitrary distributed event data trends oversimplified even infeasible capturing problem complexity real applications moreover runs risk model underfitting due misjudgement model choice recent works zhou zha song start turn form fit structure point process method hawkes process formulation formulation runs risk model mischoice paper view conditional intensity point process nonlinear mapping predicted transient occurrence intensity events different types model input information event participators event profile system history nonlinear mapping expected complex flexible enough model various characters real event data application utility fact deep learning models convolutional neural networks cnns lecun recurrent neural networks rnns pascanu mikolov bengio attracted wide attention recent vision speech language communities many dominated competing results perceptual benchmark tasks russakovsky particular turn rnns figure time series event sequence synergically modeled former used timely capture recent window features latter capture dependency time note dependency event sequence easily captured event sequence lstm less steps takes much steps using time series fixed time interval days figure note many unit steps top time series omitted figure clarity natural way encode nonlinear dynamic mapping effort modeling nonlinear intensity mapping without prior knowledge key idea highlights model interprets conditional intensity function point process nonlinear mapping synergetically established composite neural network two rnns building blocks illustrated time series top row event sequence bottom row distinct time series suitable carry synchronously fixed pace regularly updated constant profile features event sequence compactly catch event driven abrupt information affect condition intensity function longer time period specifically highlights paper first make observation many conditional intensity functions viewed integration two effects spontaneous background component inherently affected internal attributes individual event type effects history events meanwhile information real world also covered continuously updated features like age temperature asynchronous event data clinical records failures motivates devise general approach use rnn whose units aligned time points time series units rnn whose units aligned events time series rnn timely track spontaneous background event sequence rnn used efficiently capture dependency history arbitrary time intervals allows fit arbitrary dynamics point process otherwise difficult often impossible specified parameterized model certain assumptions best knowledge first work fully terpret instantiate conditional intensity function fused time series event sequence rnns opens room connecting neural network techniques traditional point process emphasizes specific model driven domain knowledge importantly introduction full rnn treatment lessen efforts design parametric point process model complex learning algorithms often call special tricks prohibiting wide use practitioners contrast neural networks specifically rnn becoming tools getting widely used recently model simple general trained target predictive maintenance problem data global bank headquartered north america consisting decades thousands event logs large number automated teller machines atms performance failure type timestamp prediction corroborates suitability applications related work motivation view related concepts work section mainly focused recurrent neural networks rnns applications time series sequences data respectively give point view existing point process methods connection rnns observations indeed motivate work paper recurrent neural network building blocks model recurrent neural networks rnns elman pascanu mikolov bengio modern variant long memory lstm units hochreiter schmidhuber graves rnns dynamical systems whose next state output depend present network state input general models networks rnns long explored perceptual applications many decades however difficult train rnns learn dynamics perhaps part due vanishing exploding gradients problem lstms provide solution incorporating memory units allow network learn forget previous hidden states update hidden states given new information recently rnns lstms successfully applied vision gregor speech graves rahman mohamed hinton language sutskever vinyals problems rnns series data application perspective view rnns works two scenarios particularly considered paper rnns synchronized series evenly spaced interval time series indexed sequence pure order information language asynchronous sequence random timestamp event data synchronized series rnns long time natural tool standard time series modeling prediction connor martin atlas han chandra zhang chen chang chang whereby indexed series data point fed input unfold rnn broader sense video frames also treated time series rnn widely used recent visual analytics works jain tripathi speech graves rahman mohamed hinton rnns also intensively adopted sequence modeling tasks chung bengio order information considered asynchronous event contrast event sequence timestamp occurrence asynchronously randomly distributed continuous time space another typical input type rnns choi esteban che despite title time series one key differentiation first scenario timestamp time duration events together features taken input rnns event dependency effectively encoded point process point process principled framework modeling event data aalen borgan gjessing dynamics point process well captured conditional intensity function whose definition briefly reviewed short time window represents rate occurrence new event conditioned history lim expectation number events happened interval given historical observations conditional intensity function played central role point processes many popular processes vary parameterized poisson process kingman homogeneous poisson process simple form intensity function poisson process generalization assumed independent history reinforced poisson processes pemantle shen model captures mechanism characterized compact intensity function recently used popularity prediction shen hawkes process hawkes hawkes process received wide attention recently social network analysis zhou zha song viral diffusion yang zha criminology lewis etc explicitly uses triggering term model excitation effect history events originally motivated analyze earthquake aftershocks ogata reactive point process ertekin rudin mccormick regarded generalization hawkes process adding term account inhibiting effects history events process isham westcott background part increases steadily decreased constant every time new event appears reformulate intensity functions general form table tries separate spontaneous background component history event effect explicitly predictive maintenance predictive maintenance mobley sound testbed model refers practice involves equipment risk prediction allow proactive scheduling corrective maintenance early identification potential concerns helps deploy limited resources cost effectively reduce operations costs table conditional intensity functions point processes model poisson process reinforced poisson process hawkes process reactive point process process background history event effect exp note dirac function kernel constant function maximize equipment uptime grall predictive maintenance adopted wide variety applications fire inspection madaio data center sirbu babaoglu electrical grid ertekin rudin mccormick management practical importance different scenarios relative rich event data modeling target model dataset automated teller machines atms global bank headquartered north america network structure learning taking sequence input rnn generates hidden states outputs sequence elman pascanu mikolov bengio specifically implement rnn long short term memory lstm hochreiter schmidhuber graves popularity capability efficient longrange dependency learning fact rnn variant gated recurrent units gru chung also alternative choice reiterate formulation lstm tanh tanh denotes multiplication recurrent activation logistic sigmod function system reduced lstm equation lstm consider two types input continuously evenly distributed data temperature event data whose occurrence time interval random network comprised two rnns using evenly spaced time series model background intensity events occurrence event sequence capture event dependency result hyt cyt lstmy hzt czt lstmz tanh hyt hzt softmax softmax denotes main type subtype events respectively timestamp associated event total loss sum time prediction loss loss event type log utj wuj log ujt log sjt number training samples indexed sjt timestamp coming event history information underlying rationale third term encourage correct classification coming event type also reinforce corresponding timestamp event shall close ground truth adopt gaussian penalty function fixed sjt exp figure network trained time series event sequence fed two rnns lstm connected embedding mapping layer fuses information two lstms three prediction layers used output predicted main type subtype events associated timestamp time penalty loss square loss respectively used event type timestamp prediction total max min mean std training set table statistics event count per atm timestamp interval days atms brackets ticket error prt cng idc comm lmtp misc testing set output timestamp prediction layer fed classification loss layer compute penalty given actual timestamp sjt sample following importance weighting methodology skewed data model training rosenberg weight parameters subtype set inverse sample number ratio type total size samples order weight classes fewer training samples loss independent subtype prediction shown set weight parameter zero respectively adopt rmsprop gradients dauphin shown work well training deep networks learn parameters data ticket error prt cng idc comm lmtp misc experiments data use failure prediction predictive atms maintenance typical example event based point process modeling prior knowledge dynamics complex system task involve arbitrarily working schedules heterogeneous mix conditions takes much cost even impractical devise specialized models problem real data description maintenance support services device fails equipment owner raises maintenance service ticket technician assigned repair failure fact history log relevant profile information equipment indicative signals coming failures studied dataset comprised event logs involving error reporting failure tickets originally collected large number atms owned anonymous global bank headquartered north america bank also customer technical support service department fortune company atm models training data consists atms testing data atms total wincor atms cover atm machine models procash type numbers bracket indicate number machines training testing event type two main types ticket error mar statistics presented table moreover error divided subtypes regarding component error occurs printer prt cash dispenser module cng internet data center idc communication part comm printer monitor lmtp miscellaneous hip card module usb misc features input features two rnns time series rnn length days time series rnn extract features including inventory information atm models age location etc event statistics including tickets events maintenance records errors system log occurrence frequencies used features concatenation two categories features serves features time series point event sequence rnn event type time interval two events model setting use single layer lstm size sigmoid gate activations tanh activation hidden representation embedding layer fully connected uses tanh activation outputs dimensional vector embedding used event type representation large number types embedding representation compact efficient time series rnn set length evenly spaced time interval days number way observation length days time series length event sequence arbitrarily long take also test degraded versions model follows time series rnn input event sequence right half yellow part removed note design spirit similar many lstm models jain tripathi used video analytics whereby frame sequence treated time series input lstms event sequence rnn rnn whose input time series left half yellow part removed intensity rnn two rnn fused shown three methods output layer directly subtype events hierarchical structure shown top left part dark green also term three hierarchical versions whose two hierarchical prediction layers used time series hrnn event sequence hrnn intensity hrnn addition compare three major peer methods logistic model input concatenation feature vectors active time series rnn set paper rmtpp hawkes process train model event sequences associated information fact rmtpp process event data similar input information event rnn logistic model use logistic regression event timestamp prediction use another independent logistic classification model event type prediction recurrent marked temporal point processes rmtpp uses neural network model event dependency flexibly method sample transient time series features event happens use partially parametric form base intensity hawkes process enable event prediction use hawkes process similar zhou zha song also add sparsity regularization term mutual infection matrix lowrank assumption removed subtypes evaluation metrics use several popular prediction metrics performance evaluation coming event type prediction adopt precision recall score confusion matrix main types error ticket well figure hierarchical layer flat independent layer confusion matrix subtypes error note metrics computed type averaged types event time prediction use mean absolute error mae measures absolute difference predicted time point actual one settings similar evaluate type timestamp prediction jointly devise two strict metrics type prediction narrow test samples whose timestamp prediction error mae days compute new timestamp recompute new samples whose coming event correctly predicted platform code based theano running linux server memory cpus cores intel xeon cpu also use gpu geforce gtx titan acculturation results discussion evaluations performed testing dataset distinctive training set whose statistics shown table averaged performance table shows averaged performance among various types events shown test two architectures event type prediction layer hierarchical predictor flat independent predictors main type includes ticket error subtype include ticket six subtypes error describe earlier paper confusion matrix confusion matrix six subtypes error event well two main types ticket error shown various methods make observations analysis based results shown flat architecture directly predicts main types outperforms hierarchical one different settings input rnn well varying evaluation metrics explained loss function focuses misclassification subtype prediction hierarchical layer performs better since fuses output prediction layer embedding layer shown surprisingly event type timestamp prediction main approach intensity rnn fuses two rnns outperforms counterparts time series rnn event sequence rnn notable margin event rnn also often performs better time series counterpart suggests least studied dataset history event effects important future event occurrence main method intensity rnn almost always superior methods except table ablation test method peer methods hawkes process recurrent hawkes process logistic classification type regression event timestamp numbers averaged types subtype precision time series rnn event sequence rnn intensity rnn hawkes process logistic prediction rmtpp recall time series rnn event sequence rnn intensity rnn hawkes process logistic prediction rmtpp score time series rnn event sequence rnn intensity rnn hawkes process logistic prediction rmtpp mae days time series rnn event sequence rnn intensity rnn hawkes process logistic prediction rmtpp hierarchical output subtype time series rnn event sequence rnn intensity rnn hawkes process logistic prediction rmtpp model time series rnn event sequence rnn intensity rnn hawkes process logistic prediction rmtpp diction task whereby logistic classification model performs better however challenging tasks subtype prediction event timestamp prediction method significantly outperforms especially subtype prediction task interestingly point process based models obtain better results task suggests point process models promising compared classical classification models indeed methodology provides learning mechanism without modeling point process empirical results realworld tasks suggest efficacy approach conclusion use conclude position model development implicit explicite modeling intensity function point process fact hawkes process uses full explicit parametric model rmtpp misses dense time series features model base intensity assumes partially parametric form make step full implicit mapping model model intensity hrnn time series hrnn event hrnn intensity rnn time series rnn event rnn hawkes process logistic logistic hawkes rmtpp rmtpp figure confusion matrixes type three methods top middle row use hierarchical flat structure respectively zoom better view figure evolving point process modeling see simple general learned standard backpropagation opens new possibilities borrowing advances neural network learning area point process modeling applications representative study paper clearly suggests high potential problems even domain knowledge problem hand contrast existing point process models assumption dynamics often need specified beforehand references aalen borgan gjessing aalen borgan gjessing survival event history analysis process point view springer science business media bengio bengio vinyals jaitly shazeer scheduled sampling sequence prediction recurrent neural networks nips chandra zhang chandra zhang cooperative coevolution elman recurrent neural networks chaotic time series prediction neurocomputing che che purushotham cho sontag liu recurrent neural networks multivariate time series missing values chen chang chang chen chang chang reinforced recurrent neural networks flood forecasts journal hydrology choi choi bahadori schuetz stewart sun doctor predicting clinical events via recurrent neural networks chung chung gulcehre cho bengio empirical evaluation gated recurrent neural networks sequence modeling connor martin atlas connor martin atlas recurrent neural networks robust time series prediction ieee transactions neural networks dauphin dauphin vries chung bengio rmsprop equilibrated adaptive learning rates optimization dai trivedi upadhyay gomezrodriguez song recurrent marked temporal point processes embedding event history vectore kdd elman elman finding structure time cognitive science ertekin rudin mccormick ertekin rudin mccormick reactive point processes new approach predicting power failures underground electrical systems annals applied statistics esteban esteban staeck yang tresp predicting clinical events combining static dynamic information using recurrent neural networks grall grall dieulle berenguer roussignol scheduling deteriorating system ieee transactions reliability graves rahman mohamed hinton graves rahman mohamed hinton towards speech recognition recurrent neural networks icml graves graves generating sequences recurrent neural networks gregor gregor danihelka graves rezende wierstra draw recurrent neural network image generation icml han han yin prediction chaotic time series based recurrent predictor neural network ieee transactions signal processing hawkes hawkes spectra mutually exciting point processes biometrika hochreiter schmidhuber hochreiter schmidhuber long memory neural computation isham westcott isham westcott pint process advances applied probability jain jain singh koppula soh saxena recurrent neural networks driver activity anticipation via architecture icra kingman kingman poisson processes volume oxford university press lecun lecun bottou bengio haffner learning applied document recognition proceedings ieee lewis mohler lewis mohler nonparametric algorithm multiscale hawkes processes journal nonparametric statistics lewis lewis mohler brantingham bertozzi point process models insurgency iraq ucla cam reports madaio madaio chen haimson zhang cheng firebird predicting fire risk prioritizing fire inspections atlanta kdd mobley mobley introduction predictive maintenance montgomery jennings kulahci montgomery jennings kulahci introduction time series analysis forecasting john wiley sons ogata ogata statistical models earthquake occurrences residual analysis point processes amer statist assoc pascanu mikolov bengio pascanu mikolov bengio difficulty training recurrent neural networks icml pemantle pemantle survey random processes reinforcement probability survey rosenberg rosenberg classifying skewed data importance weighting optimize average recall interspeech russakovsky russakovsky deng krause satheesh huang karpathy khosla bernstein berg imagenet large scale visual recognition challenge international journal computer vision shen shen wang song modeling predicting popularity dynamics via reinforced poisson processes aaai sirbu babaoglu sirbu babaoglu holistic approach log data analysis computing systems case ibm blue parallel processing workshopss snyder miller snyder miller random point processes time space springer science business media sutskever vinyals sutskever vinyals sequence sequence learning neural networks nips tripathi tripathi lipton belongie nguyen context matters refining object detection video recurrent neural networks bmvc nemati zha patient flow prediction via discriminative learning mutuallycorrecting processes ieee transactions knowledge data engineering yang zha yang zha mixture mutually exciting processes viral diffusion icml zhou zha song zhou zha song learning social infectivity sparse networks using hawkes processe aistats zhou zha song zhou zha song learning triggering kernels hawkes processes icml
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oct annals applied probability vol doi institute mathematical statistics ergodic control queues regime ari anup guodong university texas pennsylvania state study dynamic scheduling problem queueing network large pool statistically identical servers arrival processes poisson service times patience times assumed exponentially distributed class dependent optimization criterion expected long time average ergodic general nonlinear running cost function queue lengths consider control problem qed regime number servers total offered load scale like constant problem proposed ann appl probab section optimal solution control problem approximated corresponding ergodic diffusion control problem limit introduce broad class ergodic control problems controlled diffusions includes large class queueing models diffusion approximation establish complete characterization optimality via study associated hjb equation also prove asymptotic convergence values queueing control problem value associated ergodic diffusion control problem proof relies approximation method spatial truncation ergodic control diffusion processes markov policies follow fixed priority policy outside fixed compact set received april revised november supported part office naval research grant supported part award simons foundation university texas austin part office naval research electric ship research development consortium supported part marcus endowment grant harold inge marcus department industrial manufacturing engineering penn state ams subject classifications primary secondary key words phrases markovian queues qed regime diffusion scaling long control ergodic control stable markov optimal control spatial truncation asymptotic optimality electronic reprint original article published institute mathematical statistics annals applied probability vol reprint differs original pagination typographic detail arapostathis biswas pang contents introduction contributions comparisons organization notation controlled system regime markovian model ergodic control problem regime broad class ergodic control problems diffusions controlled diffusion model structural assumptions piecewise linear controlled diffusions existence optimal controls hjb equation technical proofs approximation via spatial truncations asymptotic convergence lower bound upper bound conclusion acknowledgements references introduction one classical problems queueing theory schedule network optimal way problems known scheduling problems arise wide variety applications particular whenever different customer classes present network competing resources optimal scheduling problem long history literature one appealing scheduling rules rule static priority policy assumed customer marginal delay cost average service time classes prioritized decreasing order static priority rule proven asymptotically optimal many settings markov modulated queueing network considered averaged shown asymptotically optimal discounted control problem important aspect queueing networks may choose leave system queue service therefore important include customer abandonment modeling queueing systems atar considered queueing network customer abandonment proved modified priority policy referred rule asymptotically optimal long run average cost fluid scale dai tezcan showed asymptotic optimality static priority policy finite ergodic control regime time interval parallel server model assumed conditions ordering abandonment rates running costs although static priority policies easy implement may optimal control problems many queueing systems queueing network discounted cost control problems studied asymptotically optimal controls problems constructed minimizer hjb equation associated controlled diffusions regime article interested ergodic control problem queueing network regime network consists single pool statistically identical servers buffer infinite capacity customer classes arrivals independent poisson processes parameters service rate customers customers may renege queue started receive service patience times customers renege queue rates scheduling policies server stays idle queues nonempty assume system operates regime arrival rates number servers scaled appropriately manner traffic intensity system satisfies regime system operations achieve high quality high server levels high efficiency high servers utilization hence also referred qed regime see example regimes consider ergodic cost function given lim sup running cost nonnegative convex function polynomial growth queue length process worth mentioning addition running cost based add cost provided polynomial growth running cost structure analysis goes control allocation servers different classes customers service completion times value function defined infimum cost admissible controls among scheduling policies article interested existence uniqueness asymptotically optimal stable arapostathis biswas pang stationary markov controls ergodic control problem asymptotic behavior value functions tends infinity section stated analysis type problems important modeling call centers contributions comparisons usual methodology studying problems consider associated continuum model controlled diffusion limit regime study ergodic control problem controlled diffusion ergodic control problems governed controlled diffusions well studied literature models fall two categories running cost defined requirement value outside compact set exceeds optimal average cost thus penalizing unstable behavior see assumption details controlled diffusion uniformly stable every stationary markov control stable collection invariant probability measures corresponding stationary markov controls tight however ergodic control problem hand fall frameworks first running cost consider total queue length total number customers system hand clear controlled diffusion uniformly stable unless one imposes nontrivial hypotheses parameters remains open problem one main contributions article solve ergodic control problem broad class nondegenerate controlled diffusions certain way viewed mixture two categories mentioned show section stability diffusion optimal stationary markov control occurs due certain interplay drift running cost model studied section far general queueing problem described thus separate interest ergodic control present comprehensive study broad class ergodic control problems includes existence solution ergodic hjb equation stochastic representation verification optimality theorem uniqueness solution certain class theorem convergence vanishing discount method theorem results extend results running costs assumptions theorems verified multiclass queueing model corresponding characterization optimality obtained corollary includes growth estimates solution hjb also introduce new approximation technique spatial truncation controlled diffusion processes see section shown freeze markov controls fixed stable markov control outside compact set still obtain nearly optimal controls class markov ergodic control regime controls large compact sets keep mind property true general method also thought approximation class controlled diffusions uniformly stable remark fixed control controlled diffusions queueing model regarded special case piecewise linear diffusions considered shown diffusions stable constant markov controls proof via suitable lyapunov function conjecture uniform stability holds controlled diffusions associated queueing model markovian model gamarnik stolyar show stationary distributions queue lengths tight policy theorem also wish remark allow negative assuming abandonment rates strictly positive abandonment rates zero another important contribution work convergence value functions associated sequence queueing models value ergodic control problem say corresponding controlled diffusion model obvious one asymptotic optimality existence optimal stable controls hjb equations controlled diffusions fact relatively straightforward cost consideration discounted situation tightness paths finite time horizon sufficient prove asymptotic optimality situation finite time behavior stochastic process plays role cost particular need establish convergence controlled steady states although uniform stability stationary distributions queueing model case abandonment rates zero established obvious stochastic model considered property uniform stability therefore use different method establish asymptotic optimality first show value functions asymptotically bounded study upper bound construct sequence markov scheduling policies uniformly stable see lemma key idea used establishing stability results spatial truncation technique markov policies follow fixed priority policy outside given compact set believe techniques also used study ergodic control problems queueing models scheduling policies consider paper allow preemption customer service interrupted server serve customer different class service resumed later fact asymptotic optimality shown within class preemptive policies preemptive nonpreemptive policies studied nonpreemptive scheduling control policy constructed hjb equation associated preemptive policies thus shown arapostathis biswas pang asymptotically optimal however far know optimal nonpreemptive scheduling problem ergodic cost remains open similar line work uncontrolled settings refer reader admission control single class model ergodic cost criterion regime studied controlled problems finite server models asymptotic optimality obtained conventional regime main advantage uniform exponential stability stochastic processes obtained using properties skorohod reflection map recent work studying ergodic control queueing network summarize main contributions paper introduce new class ergodic control problems framework solve establish approximation technique spatial truncation provide best knowledge first treatment ergodic control problems diffusion scale many server models establish asymptotic optimality results organization section summarize notation used paper section introduce many server queueing model describe regime ergodic control problem setting introduced section main results asymptotic convergence stated theorems section introduces class controlled diffusions associated ergodic control problems contains queueing models diffusion scale key structural assumptions section verified generic class queueing models section characterized piecewise linear controlled diffusions section concerns existence optimal controls general hypotheses section contains comprehensive study hjb equation section devoted proofs results section spatial truncation technique introduced studied section finally section prove results asymptotic optimality notation standard euclidean norm denoted set nonnegative real numbers denoted stands set natural numbers denotes indicator function denote set nonnegative integers closure boundary complement set denoted respectively open ball radius around denoted given two real numbers minimum maximum denoted respectively ergodic control regime define integer part real number denoted use notation denote vector ith entry equal entries equal also let given two vectors inner product denoted denote dirac mass function domain define oscillation follows osc sup nonnegative function let denote space banach space functions satisfying norm sup also let denote subspace consisting functions satisfying lim sup slight abuse notation also denote generic member spaces two nonnegative functions use notation indicate denote lploc set functions locally loc set functions lloc whose ith weak derivatives lloc set bounded continuous functions denoted cloc denote set functions continuously differentiable whose kth derivatives locally continuous exponent define cbk set functions whose ith derivatives continuous bounded denote cck subset cbk compact support path use notation denote jump time given polish space denote set probability measures endow prokhorov metric borel measurable map often use abbreviated notation quadratic variation square integrable martingale denoted optional quadratic variation presentation purposes use time variable subscript diffusion processes also used generic constants whose values might vary place place arapostathis biswas pang fig schematic model system controlled system regime markovian model let given complete probability space stochastic variables introduced defined expectation denoted consider markovian queueing system consists customer classes parallel servers capable serving customers see figure system buffer assumed infinite capacity customers class arrive according poisson process rate customers enter queue respective classes upon arrival processed customers class served fcfs service discipline waiting queue customers abandon system service times patience times customers classdependent assumed exponentially distributed class customers served rate renege rate assume customer arrivals service abandonment classes mutually independent regime study queueing model whitt regime qed regime consider sequence systems indexed arrival rates number servers increase appropriately let rni mean offered load classpi customers traffic intensity nth system given rni regime ergodic control regime parameters assumed satisfy following rni implies scaling common models note make assumption sign state descriptors let xin xin total number class customers system qni qni number class customers queue zin zin number class customers service following basic relationships hold processes xin qni zin qni zin describe processes using collection ani sin rin independent poisson processes define ani sin zin dynamics take form xin xin scheduling control following consider policies nonanticipative allow preemption server becomes free customers waiting queue server stays idle customers multiple classes waiting queue arapostathis biswas pang server make decision customer class serve service preemption allowed service customer class interrupted time serve class customers original service resumed later time scheduling control policy determines processes must satisfy constraints constraint define action set thus write also assume controls nonanticipative define ftn gtn rin qni collection sets filtration ftn represents information available time gtn contains information future increments processes say control policy admissible adapted ftn ftn independent gtn time iii process agrees law sin process agrees law rin denote set admissible control policies ergodic control problem regime define processes ergodic control regime xin qni zin express defined ani square integrable martingales filtration ftn note define define adapted define thus represents fraction customers queue total queue size positive show later convenient view control note controls nonanticipative preemption allowed arapostathis biswas pang cost minimization problem next introduce running cost function control problem let given function satisfying positive constants also assume locally lipschitz assumption includes linear convex running cost functions example let holding cost rate class customers typical running cost functions following running cost functions evidently satisfy condition given initial state scheduling policy define cost function lim sup running cost function satisfies note running cost defined using scaled version associated cost minimization problem becomes ninf refer value function given initial state nth system easy see redefining rewrite control problem inf lim sup infimum taken admissible pairs satisfying simplicity assume initial condition deterministic ergodic control regime limiting controlled diffusion process one formally deduces provided exists limit every finite time interval limit process diffusion process independent components dxt dwt initial condition drift takes form diag diag control lives nonanticipative standard wiener process independent initial condition covariance matrix given diag formal derivation drift obtained detailed description equation related results given section let set admissible controls diffusion model definition see section ergodic control problem diffusion scale define function analogy define ergodic cost associated controlled diffusion process running cost function lim sup consider ergodic control problem inf call optimal value initial state controlled diffusion process shown later independent detailed treatment related results corresponding ergodic control problem given section next state main results section proof found section arapostathis biswas pang theorem let also assume hold lim inf given theorem suppose assumptions theorem hold addition assume convex lim sup thus conclude convex running cost function theorems establish asymptotic convergence ergodic control problem queueing model broad class ergodic control problems diffusions controlled diffusion model dynamics modeled controlled diffusion process taking values euclidean space governed stochastic differential equation dxt dwt random processes live complete probability space process standard wiener process independent initial condition control process takes values compact metrizable set jointly measurable moreover nonanticipative independent completion relative process called admissible control let denote set admissible controls impose following standard assumptions drift diffusion matrix guarantee existence uniqueness solutions equation local lipschitz continuity functions locally lipschitz lipschitz constant depending words also assume continuous ergodic control regime affine growth condition satisfy global growth condition form trace local nondegeneracy holds aij integral form written dws third term side stochastic integral say process solution continuous defined satisfies well known admissible control exists unique solution theorem define family operators plays role parameter aij refer controlled extended generator diffusion elsewhere paper adopted notation also use standard summation rule repeated subscripts superscripts summed words side stands fundamental importance study functionals formula defined holds lus dws arapostathis biswas pang local martingale krylov extension formula page extends functions local sobolev space loc recall control called markov measurable map called stationary markov depend correspondingly said strong solution given wiener process complete probability space exists process continuous satisfies strong solution called unique two solutions agree viewed elements well known assumptions markov control unique strong solution let usm denote set stationary markov controls usm process strong markov denote transition function pvt also follows work usm transition probabilities densities locally continuous thus defined aij usm generator strong feller let pvx denote probability measure evx expectation operator canonical space process control usm conditioned process starting need following definition definition function called set set compact empty property holds simply say recall control usm called stable associated diffusion positive recurrent denote set controls ussm let denote unique invariant probability measure diffusion control ussm also let ussm recall ussm exists function bounded domain constant satisfying denote first exit time process set defined inf ergodic control regime open ball radius centered origin denoted let assume running cost function nonnegative continuous locally lipschitz first argument uniformly without loss generality let lipschitz constant summary assume continuous satisfies constant general may convex set therefore often useful enlarge control set redefine drift running cost easy see drift running cost defined satisfy aforementioned conditions follows assume controls take values controls generally referred relaxed controls endow set relaxed stationary markov controls following topology usm usm compact metric space topology section refer topology topology markov controls control said precise takes value easy see precise control also understood relaxed control abusing notation denote drift running cost respectively action relaxed control understood structural assumptions assumptions described effect throughout analysis unless otherwise stated assumption open set following hold running cost exist functions arapostathis biswas pang without loss generality assume nonnegative remark statement assumption refrain using constants interest notational economy loss generality since functions always scaled eliminate unnecessary constants next assumption structural one rather necessary requirement value ergodic control problem finite otherwise problem vacuous define lim sup assumption exists assumption alone imply ussm however combined assumption case following lemma asserts lemma let assumptions hold exists ussm moreover exists nonnegative function positive constant conversely holds assumption holds proof first part result follows theorem whereas converse part follows lemma proofs stated later paper remark loss generality using constant assumption since always scaled achieve also observe problem reduces ergodic control problem cost obtain ergodic control problem uniformly stable controlled diffusion piecewise linear controlled diffusions controlled diffusion process belongs large class controlled diffusion processes called piecewise linear controlled diffusions describe class controlled diffusions show satisfies assumptions section ergodic control regime definition square matrix said written nonnegative matrix property denotes spectral radius let given matrix whose diagonal elements positive remaining elements note queueing model positive diagonal matrix results hold general let nonsingular define assume consider following controlled diffusion dxt dwt constant matrix invertible easy see satisfies conditions analysis types diffusion approximations established tradition queueing systems often easy deal limiting object also helps obtain information behavior actual queueing model next introduce running cost function let locally lipschitz polynomial growth positive constants depend typical examples running costs positive vector remark controlled dynamics running cost clearly general model described section denotes diffusion approximation number customers system regime ith component denotes diffusion approximation number class customers therefore denotes total number customers queue diagonal diagonal entries denote service arapostathis biswas pang abandonment rates respectively customer classes ith coordinate denotes fraction customers waiting queue therefore process denotes diffusion approximation numbers customers service different customer classes proposition let given respectively satisfies assumptions appropriate positive constants proof recall nonsingular exists positive definite matrix strictly positive definite therefore positive constant holds set chosen later open convex cone running cost function let nonnegative function constant let positive constant choose estimate note globally bounded follows smallest largest eigenvalues respectively use young inequality ergodic control regime obtain bound constant similar calculation shows constant holds also note select large enough hence exists ikc evident scale multiplying constant takes form ikc running cost follows satisfied next show satisfies assumption let write dxt dwt shown solution positive recurrent therefore stable markov control done finding suitable lyapunov function particular theorem shown exists positive definite matrix define suitably chosen constant smooth function given suitable constant holds arapostathis biswas pang large enough positive constant define determined later note constant hence straightforward calculation shows choose small enough constant holds large enough since large enough see satisfies control hence assumption holds lemma existence optimal controls definition recall definition define also let usm inf inf inf given well known set ergodic occupation measures controlled process closed convex subset lemmas use notation want indicate ergodic occupation measure associated control ussm words lemma holds therefore proof let solution recall first exit time formula ergodic control regime therefore letting using fatou lemma obtain bound min thus lim sup analysis use function roughly speaking order lies multiple assumption existence function guaranteed assumption following lemma shows lemma define open set assumption exists function locally lipschitz first argument uniformly second argument satisfies ihc positive constant moreover ihc function assumption proof denotes side ihc ikc since therefore assumption set bounded every hence exists increasing sequence open balls centered dnc let nonnegative smooth function clearly continuous locally lipschitz first argument satisfies holds clear fact arapostathis biswas pang remark clear proof lemma could fix value constant say however keep variable provides flexibility choice also order able trace along different calculations remark note satisfies note also view proposition queueing model ikc ikc ihc therefore satisfies remark often use fact bounded map lower easily follows two facts expressed increasing limit bounded continuous functions bounded continuous continuous theorem let lim sup ussm set invariant probability measures corresponding controls satisfies ergodic control regime particular tight exists min proof using formula follows ihc since together implies lim sup lim sup part follows fix inequality defined implies set mean empirical measures borel sets tight case limit point mean empirical measures ergodic occupation measure lemma view remark obtain lim sup ergodic occupation measure therefore disintegrating measure obtain associated control ussm ergodic theory also know lim sup evx almost every follows since clear equality must hold among three quantities implies holds therefore parts follow arapostathis biswas pang existence satisfying follows assumption theorem follows stochastic representation lemma proves existence stationary markov control optimal asserted following theorem theorem let denote set ergodic occupation measures corresponding controls ussm corresponding controls set compact exists usm proof theorem set tight let convergent sequence denote limit since closed since map lower follows therefore closed hence compact since equality follows also obtained disintegrating remark reader might noticed point assumption may weakened significantly really required existence open set functions satisfying inf use convention inf empty set also note equivalent statement bounded met define following proof lemma assert existence satisfying fact throughout rest paper assumption really invoked use function satisfying naturally assumption hjb equation let theorem bound ergodic control regime therefore since lim inf min inf exists arg let proof theorem relative running cost imply follows theorem tight since lower limit point taking limits obtain lim sup since closed implies therefore equality must hold words optimal ergodic occupation measure theorem exists unique function bounded solves hjb min inf words optimal value ergodic control problem running cost also stationary markov control optimal ergodic control problem relative satisfies min moreover every exists osc arapostathis biswas pang measurable selector minimizer hamiltonian satisfies inf stationary control ussm hitting time ball theorem following hold let theorem function converges uniformly compact sets satisfies min also limit point topology markov controls set satisfies stationary markov control optimal ergodic control problem relative satisfies min moreover optimal usm lim function stochastic representation lim inf lim usm usm satisfies ergodic control regime convex set strictly convex whenever constant strictly convex measurable minimizer converges pointwise thus usm minimizer theorem guarantees existence optimal stable control made precise ergodic diffusion control problem running cost function moreover convexity property part optimal stable control obtained pointwise limit minimizing selector instance let choosing proposition see approximate value function approximate control converge desired optimal value function optimal control respectively concerning uniqueness solution hjb equation recall case existing uniqueness results follows exists unique solution pair class functions bounded moreover satisfies restriction removed general multiple solutions since model function general bounded however show later lemma negative part grows slower holds defined section therefore second part theorem follows may viewed extension uniqueness results apply ergodic control problems running cost third part theorem resembles hypotheses uniqueness apply problems blanket stability hypothesis theorem let solution min following hold measurable selector minimizer associated hamiltonian ussm necessarily arapostathis biswas pang applying results queueing diffusion model following corollary corollary queueing diffusion model controlled dynamics given drift given running cost exists unique solution satisfying associated hjb class functions whose negative part solution agrees theorem proof existence solution follows theorem select proof proposition solution stated class follows lemma corollary appear later sections respectively proof proposition follows therefore uniqueness follows theorem also obtain hjb equation via traditional vanishing discount approach following theorem asserts similar results shown degenerate ergodic diffusion control problem certain ergodic diffusion control problems allowing degeneracy spatial periodicity theorem let theorem define inf function converges uniformly compact subsets moreover proofs theorems given section following result follows directly provides way find controls proposition let minimizing selector theorem corresponding invariant probability measures almost surely lim ergodic control regime technical proofs recall lemma need following lemma define inf set clearly quote following result theorem remark lemma provided defined minimal nonnegative solution min hjb lemma similar equation theorem concerns characterization discounted control problem lemma let precise markov control corresponding generator let nonnegative solution loc let nondecreasing function exists constant depends osc inf proof define solves also inf hence theorem exists positive constant sup inf arapostathis biswas pang implying sup inf next consider solution inf applying maximum principle theorem follows sup attains minimum boundary theorem therefore nonnegative function hence harnack inequality exists constant thus combining display obtain osc sup sup completes proof lemma let lemma exists constant osc proof recall stationary probability distribution process control ussm lemma since criterion nonnegative ball using fubini theorem obtain inf ergodic control regime last inequality used lemma theorem therefore estimate inf result follows lemma continue proof theorem proof theorem consider function view lemma lemma see locally bounded uniformly therefore standard elliptic theory partial derivatives uniformly bounded bounded ball constant depending theorem page therefore extract subsequence along converges result follows theorems lemma theorem proof follows lemma lemma remark proof following lemma elsewhere paper use fact ussm set controls corresponding set invariant probability measures tight map closure continuous map fact latter continuous total variation norm topology lemma also recall closed convex subsets therefore compact note also since compact tightness set invariant probability measures equivalent tightness corresponding set ergodic occupation measures lemma collection measurable selectors minimizer corresponding invariant probability measures tight moreover along subsequence following hold stable markov control proof set ergodic occupation measures corresponding tight remark applies set also part holds part follows equivalence existence invariant probability measure arapostathis biswas pang controlled diffusion stability associated stationary markov control see theorem part follows since equality holds continue following lemma asserts continuity mean hitting time ball respect stable markov controls lemma let collection markov controls topology markov controls let invariant probability measures corresponding controls respectively holds evxn proof define easy see infcompact locally lipschitz therefore theorem sup since also lemma obtain sup evxn let positive number greater exists positive evx evx assertion see sup therefore order prove lemma enough show follows lemma iii ergodic control regime lemma let theorem satisfy exists subsequence converges uniformly compact sets satisfies min also limit point topology markov controls set satisfies moreover admits stochastic representation inf usm follows unique limit point proof see family uniformly locally bounded hence applying theory elliptic pde follows uniformly bounded loc consequently uniformly bounded cloc therefore along subsequence uniformly compact sets also lemma therefore passing limit obtain hjb equation straightforward verify holds lemma theorem taking limits obtain evx inf usm also theorem bound inf using lemma taking limits obtain lower bound inf lemma theorem ussm define arapostathis biswas pang therefore dominated convergence theorem fact obtain thus imply lim inf thus lim inf applying formula obtain taking limits using dominated convergence theorem obtain recall definition section need following lemma lemma let lemma holds proof let lemma applying formula obtain ihc therefore adding term ergodic control regime sides using stochastic representation obtain inf stochastic representation inf also straightforward show eexx therefore since follows map implies hand obtain implies restriction support follows next prove theorem proof theorem part contained lemma prove part let control satisfying lemma map theorem satisfies ihc implies ussm applying formula obtain lim sup therefore since implies lim sup since follows lim sup arapostathis biswas pang therefore formula deduce lim sup hand since limit point mean empirical measures view remark obtain proves equality holds lim inf lim sup may replaced conversely suppose usm optimal satisfy exists nontrivial nonnegative ibr converges weakly along subsequence applying formula obtain define evx since bounded theorem invoking corollary obtain lim lim evx evx therefore taking limits first obtain taking limits since strictly positive density obtain contradiction completes proof part ergodic control regime first equality follows lemma taking limits show second equality holds optimal control suppose satisfies sup follows see lemma lim sup since must lim sup first equality obtain defined replaced thus analogy obtain lim inf rest follows proof lemma via next prove part assume convex set strictly convex identically constant fixed fix point define easy see depend also easy check closed set let limit solution corresponding limit already shown stable markov control next show fact precise markov control assumption unique minimizing selector moreover continuous definition clear restriction depend let using strict convexity property easy verify converges unique minimizer fact since open sequence holds follows definition minimizer uniform convergence therefore see precise markov control pointwise also easy check pointwise convergence implies convergence topology markov controls embark proof theorem arapostathis biswas pang proof theorem hypothesis implies map also satisfies ihc therefore follows proves part since implies lim sup since follows lim sup therefore formula deduce lim sup implies since hypothesis must sup follows lemma lim sup since must lim sup using following steps proof second equality obtain inf sup inf ergodic control regime taking limits since must proof part complete prove part note part haver therefore theorem implies hypothesis therefore converges proposition course implies tends similarly deduce applying formula obtain another application results therefore result follows part finish section proof theorem proof theorem first show let applying formula obtain follows ihc ihc taking limits first evaluating optimal control relative obtain estimate using also follows arapostathis biswas pang multiplying taking limits obtain lim sup lim sup inequalities hold lim therefore let lim note similar result lemma holds satisfies lim usm obtained without assumption running cost see example lemma lemma follows hand since must strong maximum principle approximation via spatial truncations introduce approximation technique turn used prove asymptotic convergence results section let ussm control fix control complement ball leave parameter free inside words define otherwise otherwise consider family controlled diffusions parameterized given dxt dwt associated running costs denote usm subset usm consisting controls agree let well known exists nonnegative solution loc poisson equation see lemma satisfies ergodic control regime recall lyapunov function assumption following theorem theorem let assumptions hold exists solution loc hjb equation min lul lul elliptic differential operator corresponding diffusion moreover following hold iii nonincreasing exists constant independent uniformly restriction proof earlier show inf minimal nonnegative solution min lul loc moreover measurable selector minimizer optimal control similar estimate lemma holds therefore exists subsequence along converges loc satisfies see also lemma show minimizing selector use following argument since claim exists nonnegative function indeed true since integrability uniform integrability function given measure equivalent see also proof lemma since every control usm agrees blc map constant usm equivalence iii lemma implies sup arapostathis biswas pang thus uniformly integrable respect family usm follows theorem inf yields part moreover view lemmas deduce holds constant independent also evident decreasing fix since nonnegative obtain using analogous argument one used proof lemma usm thus since choice holds obtain proves part fix let minimizing selector note usm therefore stable markov control let topology markov controls along subsequence evident usm also lemma evx using lemma obtain lower bound evx theorem see also holds evx evx blc ergodic control regime therefore using preceding inequality obtain evx fact nonnegative ihc evx evx combining obtain evx evx earlier using property fact bounded choose large enough since part iii follows part clear regularity theory elliptic pde theorem page similar theorem show oscillations uniformly bounded compacts therefore let obtain hjb equation min theorem bound arapostathis biswas pang positive constant course implies moreover straightforward show ussm lim sup evx therefore addition lim sup evx follows theorem iii lim sup theorem suppose assumptions theorem moreover proof let sequence measurable selectors minimizer corresponding sequence ergodic occupation measures since theorem tight remark limit point subsequence also denote corresponding limit point therefore lower lim also holds lim inf hence applying rule obtain hand optimal stationary markov control hypothesis fact proposition deduce converges course together implies tends therefore evaluating applying rule obtain combining two estimates thus equality must hold used fact exists optimal markov control theorem ergodic control regime next use stochastic representation fix since compact follows map defined continuous therefore map lower follows lim hand since follows uniformly integrable respect measures therefore also shown lemma lim since uniformly compact sets follows therefore theorem obtain taking limits using fact obtain since must theorem remark seen proof theorem assumption replaced weaker hypothesis easy see one replaces otherwise remark arapostathis biswas pang positive valued continuous function conclusion theorem holds consider controlled dynamics given running cost exists function satisfying fact proved proposition also exists lyapunov function satisfying assumption theorem relative control selected remark indeed order construct recall function let function complement unit ball centered origin observe positive constants holds straightforward calculation shows holds choice stochastic representation follows proved following corollary corollary queueing diffusion model controlled dynamics given running cost given exists solution additive constant associated hjb class functions whose positive part grows faster whose negative part conclude section following remark remark comparing approximation technique introduced section section see spatial truncation technique relies restrictive assumption lyapunov function running cost function theorem fact growth also restricts growth therefore class ergodic diffusion control problems considered section restrictive example running cost satisfies obvious one obtain lyapunov function growth order instance drift strictly growth expected lyapunov function growth larger therefore class problems considered section larger considered section asymptotic convergence section prove value ergodic control problem corresponding queueing network asymptotically converges value ergodic control controlled diffusion ergodic control regime recall processes defined defined square integrable martingales filtration ftn quadratic variations lower bound section prove theorem proof theorem recall definition consider sequence supn let function satisfying defined section denotes jump process time applying formula see theorem obtain arapostathis biswas pang since bounded sequence easy show exist positive constants independent provided next compute terms corresponding jumps first see jump size order also find positive constant sup using taylor approximation obtain inequality sup hence combining facts obtain suitable positive constants independent second inequality use fact optional martingale sum squares jumps martingale therefore positive constants holds ergodic control regime combined assumption supn implies sup lim sup turn obtain sup lim sup repeating argument coordinates obtain sup lim sup introduce process otherwise since follows takes values uin represents fraction class customers queue define mean empirical measures borel sets see family tight hence sequence exists subsequence also denoted evident tight let along subsequence therefore hard show lim defined earlier complete proof theorem need show ergodic occupation measure diffusion consider recall arapostathis biswas pang lemmas therefore using formula definition obtain ani fxi bin fxi fxi fxi ani bin first bound last term using taylor formula see fxi kkf positive constant use fact jump size hence using fact independent poisson processes simultaneous jumps using identity obtain fxi ergodic control regime kkf therefore first letting using see expectation side bounded therefore side tends thus fact compactly supported obtain therefore upper bound proof upper bound theorem little involved lower bound generally helpful one uniform stability across see uniform stability obtained reflected dynamics skorohod mapping however establish asymptotic upper bound using technique spatial truncation introduced section let precise continuous control ussm satisfying first construct admissible policy see define measurable map follows let note define sup define policy follows zin otherwise arapostathis biswas pang therefore whenever state system acn system works fixed priority policy least priority given jobs first show policy large enough show xin must xin xin since obtain xin hold large hence policy well defined large policy defined markov process generator given zin qni easy see lemma let markov process corresponding control let even positive integer exists sup lim sup process corresponding process defined proof proof technique inspired lemma define ergodic control regime positive constants determined later first show suitable choice exist constants independent choose large enough policy well defined define yin note also zin qyin qni qyin qyin qni qyin qni yin qyin qni last inequality use fact qni let last estimate due assumptions concerning parameters regime qyin qni qyin qni arapostathis biswas pang large qni let acn use fact holds also thus obtain maps qni acn hence obtain qyin qni yin used fact yjn observe exists independent due min result obtain qyin qni ergodic control regime yin ynj next estimate last term side let using young inequality obtain estisupn mate therefore yin yjn yjn choose define min choice follows yin ynj using preceding inequality obtain qyin qni arapostathis biswas pang combining obtain young inequality bounds thus choosing properly obtain proceed complete proof lemma applying first observe finite quantity dominated poisson arrival process therefore see implies hence proof follows dividing sides letting proof theorem let given running cost polynomial growth exponent let recall convex satisfies exponent choose ussm continuous precise control invariant probability measure also want control property outside large ball obtain see theorems ergodic control regime remark find ball large ussm continuous invariant probability measure corresponding note might continuous let sequence functions vanishes takes value define sequence convergence uniform complement neighborhood also proposition corresponding invariant probability measures exponentially tight thus combining two expressions easily find satisfies construct scheduling policy lemma lemma see constant holds sup lim sup define since follows large provided define arapostathis biswas pang define mean empirical measure family tight next show lim lim sup select sequence tkn along lim sup attained tightness exists limit point since support discrete lattice therefore lim sup lim sup family tight hence limit definition thus using continuity property follows along subsequence therefore order complete proof need show lim sup since policies observe therefore positive constants ergodic control regime given choose lim sup use observe thus holds order complete proof need show invariant probability measure corresponding shown using convergence generators proof theorem conclusion answered interesting questions ergodic control problem markovian queueing model current study raised questions future research one interesting questions consider nonpreemptive policies try establish asymptotic optimality class nonpreemptive admissible polices also interesting study similar control problem system multiple heterogeneous agent pools routing observed customers service requirements patience times nonexponential situations therefore important interesting address similar control problems general assumptions service patience time distributions acknowledgements thank anonymous referee many helpful comments led significant improvements paper ari arapostathis acknowledges hospitality department industrial manufacturing engineering penn state visiting early stages work guodong pang acknowledges hospitality department electrical computer engineering university texas austin visiting work part work done anup biswas visiting department industrial manufacturing engineering penn state hospitality department acknowledged references arapostathis borkar ghosh ergodic control diffusion processes encyclopedia mathematics applications cambridge univ press cambridge arisawa lions ergodic stochastic control comm partial differential equations atar scheduling control queueing systems many servers asymptotic optimality heavy traffic ann appl probab atar biswas control multiclass queue moderate deviation regime ann appl probab atar giat shimkin rule queues abandonment oper res arapostathis biswas pang atar giat shimkin asymptotic optimality rule ergodic cost queueing syst atar mandelbaum reiman scheduling multi class queue many exponential servers asymptotic optimality heavy traffic ann appl probab bogachev krylov regularity transition probabilities invariant measures singular diffusions minimal conditions comm partial differential equations borkar optimal control diffusion processes pitman research notes mathematics series longman harlow brown gans mandelbaum sakov shen zeltyn zhao statistical analysis telephone call center queueingscience perspective amer statist assoc budhiraja ghosh liu scheduling control markovmodulated multiclass queueing systems heavy traffic queueing syst budhiraja ghosh lee ergodic rate control problem single class queueing networks siam control optim dai tezcan optimal control parallel server systems many servers heavy traffic queueing syst dieker gao positive recurrence piecewise uhlenbeck processes common quadratic lyapunov functions ann appl probab gamarnik stolyar multiclass multiserver queueing system heavy traffic regime asymptotics stationary distribution queueing syst gamarnik zeevi validity heavy traffic approximation generalized jackson networks ann appl probab garnett mandelbaum reiman designing call center impatient customers manuf serv oper manag gilbarg trudinger elliptic partial differential equations second order grundlehren der mathematischen wissenschaften fundamental principles mathematical sciences springer berlin gurvich diffusion models approximations exponentially ergodic markovian queues ann appl probab krylov existence strong solutions stochastic equations via approximations probab theory related fields halfin whitt limits queues many exponential servers oper res harrison zeevi dynamic scheduling multiclass queue heavy traffic regime oper res ichihara sheu large time behavior solutions equations quadratic nonlinearity gradients siam math anal kallenberg foundations modern probability springer new york ergodic control regime kim ward dynamic scheduling queue multiple customer classes queueing syst ward admission control queue abandonment queueing syst krylov controlled diffusion processes applications mathematics springer new york translated russian aries mandelbaum stolyar scheduling flexible servers convex delay costs optimality generalized oper res ocone weerasinghe degenerate variance control onedimensional stationary case electron probab electronic pang talreja whitt martingale proofs limits markovian queues probab surv stannat nonsymmetric dirichlet operators existence uniqueness associated markov processes ann scuola norm sup pisa sci van mieghem dynamic scheduling convex delay costs generalized rule ann appl probab yosida functional analysis grundlehren der mathematischen wissenschaften fundamental principles mathematical sciences springer berlin arapostathis biswas department electrical computer engineering university texas austin guadalupe uta austin texas usa ari anupbiswas pang harold inge marcus department industrial manufacturing engineering college engineering pennsylvania state university university park pennsylvania usa
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role playing learning socially concomitant mobile robot navigation may mingming rui jiang shuzhi sam fellow ieee tong heng lee member ieee paper present role playing learning rpl scheme mobile robot navigate socially human companion populated environments neural networks constructed parameterize stochastic policy directly maps sensory data collected robot velocity outputs respecting set social norms efficient simulative learning environment built maps pedestrians trajectories collected number crowd data sets learning iteration robot equipped policy created virtually learning environment play companied pedestrian navigate towards goal socially concomitant manner thus call process role playing learning formulated reinforcement learning framework policy optimized using trust region policy optimization trpo consideration imperfectness robot sensor measurements simulative experimental results provided demonstrate efficacy superiority method index concomitant navigation mobile robot neural network reinforcement learning ntroduction capability navigate densely populated dynamic environments one important features enable deployment mobile robots unstructured environment schools shopping malls transportation hubs key difference problem navigating among humans traditional path planning obstacle avoidance problems humans tend smoothly evade interactively cooperatively rather remaining static maintaining indifferent trajectory dynamics words social norms need understood complied achieve maximum comfort involved pedestrians navigation refer problem social navigation aims model social norms develop robotic navigation policy socially acceptable pedestrians around social navigation traditional approaches based dynamic window approach dwa potential fields usually limited efficacy pedestrians simply regarded uncooperative obstacles illustrative example freezing robot problem frp mobile robot stuck narrow corridor facing crowd people lacks ability predict joint collision avoidance behaviors human pedestrians jiang lee department electrical computer engineering social robotics lab smart system institute ssi national university singapore singapore mingming rui jiang samge eleleeth end researches done understand principles humans joint collision avoidance strategies one pioneering works social force model sfm joint collision avoidance model reciprocal velocity obstacles rvo proposed underlying assumption involved agents adopt collision avoidance strategies ideas also applied visual tracking pedestrians recently several attempts made learn probabilistic models pedestrians trajectories joint collision avoidance based robot navigation decision generated able behave naturally correctly similar situations paper propose augment dimensions interaction social navigation endowing robot appropriate group behaviors travelling human companion capability highly desirable assistive mobile robots serve assistants companions expected travel along theirx human partners home environment also possibly crowded public areas words apart understanding collision avoidance behaviors pedestrians robot also needs consider motion companion maintain sense affinity travelling together towards certain goal call socially concomitant navigation scn challenging aforementioned social navigation problem robot assumed travel alone simpler pursuit reaching specific goal free collision address problem scn develop new learning scheme called role playing learning rpl particularly formulate problem framework partially observable markov decision process pomdp reinforcement learning neural network used parameterize navigation policy robot optimized gives proper steering commands next time instance based robot current previous observations surroundings facilitate process create simulative navigation environment mirroring collections real world pedestrians data sets develop optimization method called partially observable trust region policy optimization run optimization iteration robot attempt play companion randomly chosen pedestrian executing navigation policy policy optimized using based batch collected trajectories compared existing analytically derived approaches rpl scheme following advantages rpl scheme less restrictive rely assumption robot agents pedestrians share models navigation goals pedestrians known formulation rpl scheme generalizable flexible formulation contain manuallydefined feature domain knowledge statistics pedestrians behaviors easily modified incorporate kinematics different mobile robot platforms sensor specifications navigation objectives addition unlike learned navigation policy operates without assess global map environment therefore well generalizable unmet scenarios explicitly consider noise limitation robot sensor measurements approaches social navigation assume robot full accurate knowledge interested variables positions distance pedestrians obstacles contrary rpl schemes rooted situation robot perceive lie within sensor field view fov existence measurement noise approach rpl efficient although rpl aims solving tasks involve interaction among robot humans physical environment require participation human data collection learning known tedious timeconsuming instead learning process safely automated simulative yet realistic environment human intervention evaluate performance approach simulations experiments comparing baseline planner based rvo humans repectively also show tricks learned navigation policy still effective navigation scenario reduced aforementioned social navigation means robot travelling without human companion remainder paper organized follows related work first reviewed section section iii problem scn formulated pomdp associated definitions given rpl scheme algorithm described section sections provide extensive results simulation experiment followed concluding remarks section vii elated ork problems robot navigation populated dynamic environment addressed number angles largely classified two groups following subsections interactive behaviors models many researches proposed describe interactive navigation behaviors humans fitting computational model observed pedestrians trajectories way robot path planner able understand pedestrians intention joint collision avoidance actively calculate optimal route towards goal field robotics majority work direction done via inverse reinforcement learning irl learns cost function explains observed behaviors example maximum entropy irl adopted number works discrete human behavior prediction route planning however discrete representation less desirable modeling trajectories nature continuous higher order dynamics velocities acceleration instead adopts bayesian irl learn appropriate navigation behavior specific mobile robot set demonstration trajectories note demonstration data specific configurations robot sensor collected via human operation could hand learns probabilistic models composite trajectories pedestrians video data maximum entropy learning irl better capture characteristics observed trajectories propose develop models based set features according domain knowledge psychological studies addition features contain velocities accelerations pedestrians practice hard precisely measure besides interacting gaussian process igp derived model joint trajectories pedestrian explicitly considering effects observation noise nevertheless design igp also requires several kernels formulated based priori information specific application scenario researchers robotics community computer vision also possess great interest pedestrian modeling one important topics trajectory prediction video space linear trajectory avoidance lta developed dynamic model pedestrians video space shortterm trajectory prediction integrated visual tracking system gaussian process adopted learn motion pattern pedestrians recently social lstm proposed human trajectory prediction crowd space similarly feature social sensitivity developed analyze trajectories pedestrians bicyclists methods effectively predict navigation intention pedestrians videos still unclear apply model navigation robot real scenarios steering models contrast learning behavior models pedestrians direct perspective develop steering model outputs immediate navigation actions given robot current observation environment one pioneering work direction social force model sfm uses functions encode social status pedestrian navigation motivation pedestrian derived taking gradients energy functions following idea subsequent work propose infer optimal parameters energy function fitting video data however likely produce suboptimal results demonstration data humans imperfect authors integrate people tracker iterative planner robot actively follows pedestrian travelling similar direction navigate crowded environment follow idea formulate choice pedestrian follow decision making process hands develops hierarchical pomdp predictive navigation dynamic environment idea predict motion pedestrians generate cost map path planning obstacle avoidance navigating manner several reactive collision avoidance techniques also developed dwa velocity obstacles reciprocal velocity obstacles rvo common idea methods treat pedestrians moving obstacles reactively update planner every short periods achieve collision avoidance mentioned section methods less effective social navigation lack predictive abilities based restrictive assumptions accurate knowledge moving agents velocities agents adopt identical collision avoidance strategy proposed navigation policy belongs steering models takes observation vector input outputs navigation action stochastic neural networks rpl policy optimized algorithm derived based recent advances deep reinforcement learning drl drl exploits massive representation power deep neural networks dnn build complex yet sophisticated decision model agent directly learn raw signals instead carefully crafted feature tends act intelligently recently several attempts using dnn drl robot navigation example motion planner learned map raw sensor data laser range finder onto steering commands mobile robot decentralized collision avoidance policy learned via drl thought drl version original rvo approach finally visual navigation policy home environment learned via drl create set virtual home environments effective efficient training agent shares similar idea proposed rpl scheme iii roblem ormulation formulate problem socially concomitant navigation gives following rules scn robot reach goal fast possible robot collide pedestrians companion run obstacle robot run far away companion rules serve generic description robot desired performance navigation give concrete definitions consider navigation process discounted pomdp discrete time defined tuple finite set states reflecting navigation status robot finite set actions paper defined twosome translational rotational velocities mobile robot mapping characterized dynamics robot humans environment without loss generality assume deterministic state transition state action taken time set robot observation state denotes conditional observation probability distribution note practice robot observation incomplete access subject certain measurement noise implies initial state distribution scalar reward given robot reward discount factor robot motion dynamics paper mobile robots considered whose motion equation approximated assuming robot velocities constant within certain short time period length particularly let denote robot heading positions cartesian space time respectively represent robot translational rotational velocities define robot nonzero rotational velocity sin sin cos cos otherwise cos sin formulations goal optimizing stochastic navigation policy parameters order maximize expected discounted reward denotes whole trajectory specific definitions ingredients scn elaborated follows state given define distance direction point robot follows arctan robot distance goal located computed denotes offset angle robot current heading goal similarly define twosomes djped djcom djobs describe relative position pedestrians companion obstacle robot definitions state defined incorporate information related robot navigation status follows pped pcom pobs current action vector ped ped pped dped pcom dcom pobs obs dobs dobs dobs dobs vector pped includes distances directions nped closest pedestrians pcom includes robot companion vector pobs compact description robot perception surrounding environment particularly boundaries occupied space obstacles environment represented finite point set variables pobs defined based following assumption assumption obstacle effect robot navigation decision satisfies predefined finite constant assumption sufficient consider obstacles closed enough robot whose distances less practice limit may correspond robot perception range let components vector pobs described follows distance nearest obstacle located heading robot obs min small constant obs dobs represent distance direction closest farthest obstacles robot left obs right side respectively defined mathematically follows obs arg min obs arg min fig illustration state variables blue yellow green circles represent robot companion com pedestrians ped respectively red dashed circle radii represents boundary set black arrow shows current heading robot considering robot current position origin polar coordinates pedestrians companion closest farthest obstacles direction compactly represented vectors pped pcom pobs account order develop robust practical navigation system end define robot observation true state follows assume robot accurate information goal position current taken action velocity commands output robot motor observations pped pcom pobs may imperfect particularly consider field views fovs robot pedestrian obstacle detectors illustrated fig mathematically let finite point sets fped fobs denote current fovs pedestrian obstacle detectors terized threesomes ped dped dobs respectively robot observations pedestrians relative positions obtained nped nped obs arg max arg max variables pobs simply determined distance directions points according eqs figure provides comprehensive illustration state variables pped pcom pobs observation discussed previous sections sensors mounted robot always subject various kinds limitation measurement noise must taken obs djped fped else ped fped else nped measurement fig field views pedestrians green obstacles blue detectors arrow points towards current heading robot constants ped denote maximum minimum offset angles corresponding fovs finally ped dobs represent maximum detection ranges pedestrian obstacle detectors respectively values constants determined according specific configurations robot sensor corresponding detection algorithms outside fovs observable therefore omitted sensors range sensors tof omnidirectional cameras long interested positions sensor raw measurements remark mathematical definitions variables observations given better understanding required simulative rpl process practice clear values directly measured via robot onboard sensors without accessing actual cartesian coordinates considered point sets fped fobs example consider robot equipped laser range finder distances offset angles easily obtained returned ranges array reward function scalar reward given robot award reaching goal penalty colliding losing companions particularly time process scn terminated following three termination conditions true goal reaching condition collision conditions similarly define obs dobs dobs dobs dobs compared states obstacles within fobs observable thus obs formulated obs min measurement obstacle detection closest observed obstacles robot left right sides defined similar way obs arg min obs arg min obs arg max obs arg max distance directions robot calculated using eqs observation robot companions rely following assumptions com assumption companions always observable robot dcom remark eqs implied noises additive independent different observations typical example noise additive gaussian white noise agwn remark general formulations states observations applicable various types onboard min djped dcom min dobs dobs obs stray condition dcom based three given robot terminal conditions reward follows else clearly positive reward given robot reaches goal receive large negative reward collides anything stray companion otherwise robot receive intermediate reward penalizes robot rotational velocity encourage smoother trajectory less turning behaviors role laying earning section described rpl scheme learn effective navigation policy scn efficient manner core idea transform crowd trajectories data collected simulative dynamic navigation environment robot play virtual pedestrian iteratively improve performance via partially observable trusted region policy optimization consider set simulative navigation environment environment contains set pedestrian trajectories binary map annotates cartesian coordinates space environment abstract process rpl described following pseudo codes algorithm algorithm role playing learning initialize navigation policy iter maxiter number collected sample time steps batch size randomly choose environment trajectory initialize robot position initial velocities set choose robot heading choose scn mode probability scn mode assign trajectory robot companion arg mint else create synthesized companion moves along robot end assign trajectories pedestrians none termination conditions satisfied update states observations robot according eqs let robot execute policy update robot position according dynamics calculate current reward update positions companion pedestrians according trajectories end end update using end companion synthesization mode described algorithm rpl actually incorporates two different navigation scenarios scn proposed paper traditional social navigation scenario robot human companion helps develop navigation policy adaptable situations restrictive assumption existence companion particularly companion position vector pcom observation synthesized dcom every time step clear synthesized pcom equivalent situation companion travelling along robot constant distance guarantee termination conditions always false hand scn mode companion assigned truncated trajectory initial robotcompanion distance sufficiently large paper construct deep policy neural network parameterize navigation policy whose structure shown fig policy network trained trust region policy optimization trpo method fig structure deep policy network time observation vector input feature network feedforward perceptron mlp output feature network fed lstm network recurrent network aggregation information collected navigation process lstm network outputs assigned mean vector diagonal gaussian unit right covariance matrix however independent amd designed gradually decreasing training fixed tests experiments finally actions drawn according however original trpo method derived based fully observable mdp directly applied problem due imperfect observation formulation practice thus proposed extend original trpo algorithm described following subsections trusted region policy optimization trpo algorithm effective optimization method large nonlinear policies tends give monotonic improvement iterative optimization process specific fully observable mdp considered trpo therefore policy optimized formulated parameter vector policy note determines action directly true state differs policy let consider following standard definitions value function value function advantage function eai addition define discounted visitation frequencies generated according let denote old parameters last iteration trpo proposes optimize parameters iteratively regarding following objective function algorithm maximize subject dkl importance sampling distribution dkl divergence old current policies compute estimated advantages time steps using gae estimated value function update objective function constraints update objective function constraints imulation partially observable trpo mentioned navigation problem considered pomdp policy depends observation instead true state therefore write objective function constraint maximize subject dkl samples collected executing old policy generate set trajectories therefore next trajectory use generalized advantage estimation gae construct empirical estimation advantage function following approach deep neural network policy requires massive amount data learn socially concomitant navigation behavior section describe construct simulative environment according proposed rpl scheme particularly environments deep neural network policy algorithm algorithm developed framework rllab make use trajectories interacting pedestrians collected five different data sets includes eth hotel video clips eth walking pedestrians ewap motion capture data set well zara ucy video clips note zara ucy data sets multiple subsets thus totally different rpl environments details environments summarized tab table details rpl environments name trajectories name trajectories eth hotel trajectory environment provides sequences cartesian positions pedestrian sampling period second addition eight binary grid maps representing occupied obstacles given however maps kept unknown robot throughout training evaluation used simulate robot perception environment state pobs observation without loss generality use eth data set evaluation environment finally conditional observation probability distribuall data sets tab training environments tion independent parameters time obtain words learned policy performance assessed estimation objective function constraints rpl environment excluded training replacing expectations sample averages reflects whether properly generalize uncovered ptk aki situations maximize trajectories environments subject people wandering remained approximately stationary excluded candidates robot companion still considered pedestrians dkl robot navigating environment use neural network hidden layers feature network policy containing form one obtained except tanh units respectively output fed lstm policy conditioned observation instead network units variance gaussian output finally constrained optimization problem described unit chosen linearly decaying solved conjugate gradient algorithm training iterations effectively encourages exploration algorithm summarize pseudo code early stage learning ensures convergence update algorithm given navigation policy gae layer estimation value function parameters old parameters collecting set trajectories obtained solving following constrained regression problem ptk ptk minimize ski ptk ski ski subject network tanh units used update step size policy network adaptively chosen gae update fixed step size used update batch size batch size algorithm rpl consider pedestrians thus state pped observation describe closest pedestrians omit others situation less pedestrians perceived dummy static pedestrians created remote corner environment maintain dimensions pped considering kobuki turtlebot hokuyo laser range finder mounted top specify sensor limitation robot simulation follows ped obs ped obs measurement noises modeled gaussian variances specified follows finally maximum translational rotational velocities assigned example rpl environment constructed eth data set illustrated fig environment results trained deep policy network iterations data rpl environments except eth environment curve average discounted return obtained batch trajectories visualized fig fig average discounted return rpl progresses compare performance policy planner based rvo robot companion surrounding pedestrians treated agents every time steps positions velocities agents given planner note fair comparison agents positions subject noise described observations obstacles assume planner full perfect knowledge required original rvo algorithm protocol update robot position according planner output update positions agents according trajectories rpl environments termination conditions section iii applied robot directed planner determine whether robot conducted successful navigation policy planner conduct trials evaluation environment compute rates percentages different terminal conditions robot reaches goal successfully robot hits robot loses companion performance statistics policy planner scn scenarios listed tab simulative environment rpl fig illustrative example rpl simulative environment black curve represents trajectory robot navigating toward goal red dot yellow curve denotes trajectory robot companion besides number blue curves representing pedestrians perceived robot green lines denotes fences around entrance university bottom center note trajectories pedestrians synthesized captured video thus robot thought playing role extra person realistic environment table rates different terminal conditions policy planner scn scenarios terminal condition policy rvo bee seen tab policy performs much better planner scn planner much lower success rate rate suggesting frequently losses companion scn clearly due fact rvo nature collision avoidance algorithm thus simply takes robot companion another normal agent robot tends stay far behind companion avoid collision instead actively following contrary policy achieves much higher success rate indicates learns effectively balance objectives scn robot able reach prescribed goal maintaining distance companion avoiding collision agents environment addition scn scenarios without companion also tested analyzed previous sections reduces traditional social navigation scenarios comparative results shown tab iii table iii rates different terminal conditions policy planner traditional social navigation scenarios terminal condition policy rvo fig narrow corridor experiments performed situations without companion policy still outperforms planner higher success rate lower rate finally worth noting planner requires velocities accurate global map static obstacles conversely policy depends position measurements directly accessible robot onboard sensors therefore much simpler practical trajectories robot moving left right two pedestrians moving right left xperiments experiments assess performance developed navigation policy comparing humans scenarios particularly robot human repeat specific navigation scenario times respectively following two metrics calculated average minimum distance pedestrians average minimum distance human pedestrians throughout trajectory average maximum distance companion average maximum distance human companion throughout trajectory use mobile platform turtlebot kobuki base laser range finder hokuyo simulated last section pedestrian detection localization adopt leg tracker use ultra wideband uwb indoor positioning system localize companion navigation goal easily mapped observations based odometry robot finally laptop placed onboard processing unit policy operated period second experiments conducted narrow corridor width meters shown fig typical scenario requires pedestrians navigate cooperatively human control experiment similar navigation scenario black trajectory left right two right left fig comparison robot policy human control experiment social navigation scenario scenario traditional social navigation subsection examine method performance traditional social navigation scenario particularly robot required pass corridor two oncoming pedestrians arrive goal meters ahead addition control experiment humans one compared human two pedestrians conducted space metric computed example trajectories robot human control shown fig robotic experiments trajectories pedestrians obtained robot laser range finder robot trajectory based odometry sensor hand trajectories human control experiments captured using uwb localization system fig clear robot policy able understand human cooperative behavior collision avoidance navigate appropriate manner two pedestrians successfully pass corridor specifically observing two pedestrians blue purple meters ahead robot started approach wall left side create free space right pedestrians smoothly walk comparing figures fig see robot proactive human since black trajectories fig fig started make space oncoming pedestrians early stage cooperative avoidance process performance metrics average minimal distance pedestrians robot although smaller human control experiments value still indicates safe decent navigation behavior robot radius trajectories robot companion moving left right pedestrian moving right left scenario socially concomitant navigation subsection scenario scn studied human companion initially standing front robot start walk corridor another pedestrian passing end described previous sections robot policy closely navigate companion avoid oncoming pedestrian cooperatively additional metric used evaluate performance policy comparing statistics obtained another human control experiments example trajectories shown fig performance metrics summarized tab table performance metrics robot human controls scn scenarios robot compared human shown fig tab robot able achieve objectives scn one hand effectively engaged joint collision avoidance process resulted behavior similar observed last subsection robot even slightly larger hand average maximum distance within limit specified learning process nearly compared human showing robot actively navigate along companion instead deviating areas lagging behind shows robot driven policy able understand pace companion achieve similar sense companionship terms distance sum results demonstrate practical efficacy methods traditional social navigation complicated scn scenarios proves policy learned rpl simulative environment transferable uncovered situations vii onclusions paper problem socially concomitant navigation scn investigated formulated pomdp human control experiment similar scn black compared human orange companion trajectories left right blue pedestrian trajectory right left fig comparison robot policy human control experiment scn scenario framework explicit considerations limitation inaccuracy mobile robots onboard sensors partially observable trpo algorithm proposed optimization navigation policies role playing learning rpl scheme developed enable efficient safe reinforcement learning navigation policies mirroring large amount pedestrian trajectories simulative environments comparative simulation experiment studies demonstrated efficacy superiority policy scn traditional social navigation scenarios eferences thrun fox burgard dynamic window approach collision avoidance ieee transactions robotics automation vol hwang ahuja potential field approach path planning ieee transactions robotics automation vol cui new potential functions mobile robot path planning ieee transactions robotics automation vol trautman krause unfreezing robot navigation dense interacting crowds intelligent robots systems iros international conference trautman murray krause robot navigation dense human crowds statistical models experimental studies cooperation international journal robotics research vol helbing molnar social force model pedestrian dynamics physical review vol helbing farkas vicsek simulating dynamical features escape panic nature vol van den berg guy lin manocha reciprocal nbody collision avoidance robotics research van den berg abbeel goldberg optimized path planning robots motion uncertainty imperfect state information international journal robotics research vol van den berg lin manocha reciprocal velocity obstacles navigation robotics automation icra ieee international conference pellegrini ess schindler van gool never walk alone modeling social behavior tracking ieee international conference computer vision yamaguchi berg ortiz berg going computer vision pattern recognition cvpr ieee conference kuderer kretzschmar sprunk burgard prediction trajectories socially compliant robotics science systems kretzschmar spies sprunk burgard socially compliant mobile robot navigation via inverse reinforcement learning international journal robotics research kim pineau socially adaptive path planning human environments using inverse reinforcement learning international journal social robotics vol bicchi fagiolini pallottino towards society robots ieee robotics automation magazine vol gross schroeter mueller volkhardt einhorn bley martin langner merten progress developing socially assistive mobile home robot companion elderly mild cognitive impairment intelligent robots systems iros international conference wang liu adaptive shared control novel mobile assistive robot transactions mechatronics vol argall chernova veloso browning survey robot learning demonstration robotics autonomous systems vol abbeel apprenticeship learning via inverse reinforcement learning proceedings international conference machine learning ziebart maas bagnell dey maximum entropy inverse reinforcement aaai vol ratliff bagnell zinkevich maximum margin planning proceedings international conference machine learning ziebart ratliff gallagher mertz peterson bagnell hebert dey srinivasa prediction pedestrians intelligent robots systems iros international conference henry vollmer ferris fox learning navigate crowded environments robotics automation icra ieee international conference vernaza bagnell efficient high dimensional maximum entropy modeling via symmetric partition functions advances neural information processing systems kitani ziebart bagnell hebert activity forecasting european conference computer vision springer choi kim map inference bayesian inverse reinforcement learning advances neural information processing systems kim lee essa gaussian process regression flow analysis motion trajectories computer vision iccv ieee international conference alahi goel ramanathan robicquet savarese social lstm human trajectory prediction crowded spaces proceedings ieee conference computer vision pattern recognition robicquet sadeghian alahi savarese learning social etiquette human trajectory understanding crowded scenes european conference computer vision johansson helbing shukla specification social force pedestrian model evolutionary adjustment video tracking data advances complex systems vol lerner chrysanthou lischinski crowds example computer graphics forum vol wiley online library helbing johansson pedestrian crowd evacuation dynamics encyclopedia complexity systems science stachniss arras burgard socially inspired motion planning mobile robots populated environments proc international conference cognitive systems mehta ferrer olson autonomous navigation dynamic social environments using decision making intelligent robots systems iros international conference foka trahanias probabilistic autonomous robot navigation dynamic environments human motion prediction international journal social robotics vol seder petrovic dynamic window based approach mobile robot motion control presence moving obstacles robotics automation ieee international conference fiorini shiller motion planning dynamic environments using velocity obstacles international journal robotics research vol schulman moritz levine jordan abbeel highdimensional continuous control using generalized advantage estimation arxiv preprint schulman levine moritz jordan abbeel trust region policy optimization corr lecun bengio hinton deep learning nature vol pfeiffer schaeuble nieto siegwart cadena perception decision approach motion planning autonomous ground robots arxiv preprint chen liu everett decentralized noncommunicating multiagent collision avoidance deep reinforcement learning arxiv preprint zhu mottaghi kolve lim gupta farhadi visual navigation indoor scenes using deep reinforcement learning arxiv preprint choi bok kim kweon extrinsic calibration lidars using two orthogonal planes ieee transactions robotics vol miller murphey optimal planning target localization coverage using range sensing automation science engineering case ieee international conference endres hess sturm cremers burgard mapping camera ieee transactions robotics vol foix alenya torras object modeling using tof camera uncertainty reduction approach robotics automation icra ieee international conference liu siegwart topological mapping scene recognition lightweight color descriptors omnidirectional camera ieee transactions robotics vol kneip caprari siegwart characterization compact hokuyo laser range scanner robotics automation icra ieee international conference hochreiter schmidhuber long memory neural computation vol nocedal wright conjugate gradient methods numerical optimization duan chen houthooft schulman abbeel benchmarking deep reinforcement learning continuous control arxiv preprint leigh pineau olmedo zhang person tracking following laser scanners robotics automation icra ieee international conference
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jan bornes effectives des fonctions approximation des solutions formelles binomiales guillaume rond aim paper give effective version strong artin approximation theorem binomial equations first give effective version greenberg approximation theorem polynomial equations using weierstrass preparation theorem apply effective result binomial equations prove artin function system binomial equations bounded doubly exponential function general bounded affine function order approximated solutions bounded introduction dans greenberg suivant est corps quelconque soit alors existe une fonction telle que tels que tels que plus existe deux constantes telles que plus petite fonction cette est fonction approximation greenberg fonction dans artin cas est par nombre fini quelconque variables montrant existence une fonction approximation dans cas peut son sous forme suivante dans cas particulier est par deux variables mais seul cas nous par suite soit alors existe une fonction telle que tels que tels que plus petite fonction cette est fonction artin peut remarquer que est peut parler fonction fonction artin faisant fonction fonction artin malheureusement preuve par artin qui utilise essentiellement des fonctions implicites weierstrass apporte que peu information sur nature croissance cette fonction que est constructive pendant longtemps est question savoir toute fonction mathematics subject classification primary secondary guillaume rond artin par une fonction affine peut mentionner existe une celle artin pour montrer existence une fonction approximation qui utilise les ultraproduits qui est pas constructive donc qui apporte aucune information sur ces fonctions approximation dans est exemple dont fonction artin est pas par une fonction affine est seul exemple connu jusqu exemple dans concernant cas les variables sont par trois variables dans cet exemple est que fonction artin est par une fonction polynomiale sait rien plus sur cette fonction artin connait toujours aucune borne sur aucun exemple autre que quelques cas connus fonction artin est par une fonction affine pour mieux comprendre croissance des fonctions artin serait avoir des exemples pour lesquels ait des bornes effectives leur fonction artin peut remarquer que exemple principal par travail est justement donner des bornes pour fonction artin binomial principe est suivant est plus facile essayer donner des bornes sur les fonctions artin une famille que sur particulier effet preuve artin ses avatars utilisent toujours une sur hauteur est premier soit peut utiliser des fonctions implicites construit solution soit remplace par est mineur matrice jacobienne bien choisi augmente ainsi hauteur cependant nouvel obtenu est plus premier priori utilisation des fonctions implicites travailler avec premier faut remplacer par ses premiers borner fonction artin par celles ses premiers trouve que connait des bornes sur ces premiers fonction des est donc assez naturel essayer trouver une fonction qui majore toutes les fonctions artin des par des cette fonctionne comme artin peut choisir fonction dans pour tous les par des une valeur est pas possible avoir plus informations sur cette fonction artin uniforme que fait que soit constructible dans notre travail nous montrons que cette donne une borne effective des fonctions donne une borne effective sur les coefficients fonction des nombre variables est assez grande polynomiale doublement exponentielle mais uniforme ensuite lem nous utilisons weierstrass ramener majoration fonction artin binomial deux majorations section majoration fonction artin binomial dont les solutions lieu singulier donc dans cas cette fonction est par une fonction affine majoration des fonctions une famille par des nombre croissant variables mais dont est par des pour cela utilise donc deux choses sur corps clos nulle borne ordre des solutions alors fonction artin binomial est par une fonction affine fonction artin bornes effectives des fonctions approximation des solutions formelles binomiales binomial est par une fonction doublement exponentielle malheureusement aucune savoir cette borne est raisonnable pas effet famille est pas quelconque mais semble tout assez difficile ces sont pas est vite impossible calculer leur radical une primaire fait nombre rapidement important variables qui entrent jeu rappels sur certaines bornes effectives commutative nous allons commencer par rappeler quelques classiques commutative effective que nous allons utiliser librement dans suite soit avec deg pour soit une primaire minimale telle que soient tous distincts alors avec pour alors les suivants soit min min dmin existe une fonction polynomiale exponentiel tel que chaque est par des existe une fonction polynomiale exponentiel tel que chaque est par des proposition soient par des alors est par des deg pour tous entiers tout correspond pour par les par pour proposition existe tel tel que deg pour tous borne effective fonction dans cas polynomial dans cette partie nous allons greenberg suivant essentiellement preuve mais faisant attention chaque dans toute cette partie corps nulle pour tous existe tel que pour tout avec tel que deg pour tout pour tout pour tout tel que pour tout existe tel que pour tout plus peut choisie affine forme sont par une fonction polynomiale exponentiel guillaume rond nous noterons dans suite plus petit entier qui les conclusions nous allons noter plus petit entier qui les conclusions pour tout quelconque hauteur par des plus petit entier qui les conclusions pour tout premier hauteur par des pour tous nous avons les relations suivantes max implique directement nous allons donner une preuve montrant fur mesure les est premier alors est maximal supposons existe tel que pour tout notons par alors ker mais ker propre maximal nous avons ker donc particulier mais clairement ker qui contredit existence donc peut prendre ici est alors valable car est jamais supposons que que est vrai pour tout hauteur strictement plus grande que soit primaire radical notons alors entier satisfait les conditions pour effet soit tel que pour tout donc existe entier tel que pour tout effet dans cas contraire pour tout existerait tel que notons alors par des mais qui contredit qui donc existe entier tel que pour tout donc existe tel que pour tout tel que comme obtient conclusion voulue ainsi nous supposons donc maintenant que est premier hauteur alors est peut supposer que engendrent existe donc tel que par exemple mineur ordre matrice jacobienne proposition notons alors remarquons que deg notons alors soit une primaire bornes effectives des fonctions approximation des solutions formelles binomiales nous allons les telle sorte que pour pour est une primaire theorem chapter donc soit pose alors donc chaque est par des donc est par des lemma notons alors car soit soit tel que pour tout cas comme par existe tel que pour tout conclusion voulue cas supposons maintenant que relation par rapport obtient mod existence mineur ordre matrice jacobienne encore qui fait intervenir que des partielles par rapport aux quitte renommer les variables tel que particulier par utilisant des fonctions implicites tougeron lemma existe tel que pour alors donc donc comme pour voit que pour car donc pour pour comme qui conclut guillaume rond fonction artin binomial pour fonction artin binomial nous allons mettre les sous forme weierstrass nous allons utiliser lem suivant lem soit corps quelconque soient deux weierstrass avec tels que avec alors pour tout abord comme voit que car terme constant est terme constant ord ord autre part alors peut donc supposer que division weirstrass xdn par par rapport variable est suivante xdn division weierstrass xdn par par rapport variable xdn par dans division weierstrass pour cette division peut faire algorithmique effet construit les suites par qui suit pose xdn avec puis par induction pour xdn plus petit divisible par xdn pose alors suite ord est strictement croissante converge vers pourp topologie suite deg est strictement croisssante particulier comme est que donc pour nous allons maintenant fonction artin binomial soit corps clos soit par avec pour tels que peut supposer quitte faire changement que les sont variable donc est ordre avec pour supposons que pour tout utilisant lem alors est coefficient dans bornes effectives des fonctions approximation des solutions formelles binomiales notons maxk obtient alors deux autre chercher fonction approximation artin ces deux premier ayant une fonction artin par une fonction affine puisque pour tout que lieu singulier une torique est inclus dans union des axes second coefficients dans soit corps clos nulle alors les suivantes pour tout pour tout existe suivante soit binomial par des soit soient tels que ord pour tous alors existe tels que pour tout tels que pour tout pour tout existe une fonction doublement exponentielle telle que pour tout binomial par des fonction artin est par nous allons abord montrer dont ensuite supposons comme que est par les pour binomial est encore binomial est alors les sont des binomiaux une primaire minimale soit tel que supposons que qei pour tout alors pour tout donc existe entier tel que pour tout comme est premier binomial alors est car lieu singulier une torique est toujours inclus dans union des hyperplans donc est lisse donc est lisse ceci implique existe tels que pour tout soit fonction artin par les posons max les voit que peuvent par une fonction uniquement soit alors qui alors qei guillaume rond donc existe tels que pour tout existe tels que pose alors tdj tdj bien ceci prouve peut remarquer que apet sont par une fonction polynomiale maxk exponentiel donc existe une constante telle notons aed bed les plus petites constantes satisfaisant pour que par les est encore existe une constante telle que aed bed majore aed bed donc posant maxe pour tout soit tels que ord alors pose sinon pose alors selon entier peut donc remplacer les par les supposer que ord pour tout supposer que maxk choisissant assez grand comme applique alors existence tels que ceci prouve exemple dans est que fonction artin est pas par une fonction affine voit que fonction artin dans cas provient fait que les fonctions des sont par des fonctions affines dont les coefficients croissent vite fonction ordre des exemple peut remarquer que famille solutions dans sont des solutions dont ordre est exemple soit ord ord peut avec alors les sont solutions suivant modulo peut aide que par ces est pas ceci semble assez peut remarquer que les bornes effectives des fonctions approximation des solutions formelles binomiales par sont proches des espaces jets par qui sont pas exemple ici suivant soient deux entiers premiers entre eux est une formelle telle que est proche une puissance que est proche une puissance une fonction qui mesure rapport entre distance une puissance celle une puissance suivante proposition soient premiers entre eux existe une fonction telle que pour tout existe telle que alors existe telle que peut choisir pour une fonction doublement exponentielle pour tout restreint tous les dont ordre vaut alors peut choisir pour une fonction affine soit fonction artin supposons que alors existe tels que comme sont premiers entre eux que est factoriel existe tel que alors avec artin algebraic approximation structures complete local rings publ math ihes becker denef lipshitz van den dries ultraproducts approximation local rings inventiones eisenbud sturmfels binomial ideals duke math greenberg rational points henselian discrete valuation rings publ math ihes hermann die frage der endlich vielen schritte der theorie der polynomideale math lascar effectif des approximation artin acad sci paris grayson stillman macaulay software system research algebraic geometry available ttp rond propos fonction artin dimension math acad sci paris rond sur fonction artin annales scientifiques normale vol rond lem izumi fonction artin journal algebra vol seidenberg constructions algebra trans oct teissier commutative effective bourbaki vol exp tougeron fonctions ergebnisse der mathematik und ihrer grenzgebiete band york wavrik theorem solutions analytic equations applications deformations complex structures math zariski samuel commutative alegbra van nostrand company princeton new jersey guillaume rond iml campus luminy case marseille cedex france address rond
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estimation convolution structure density model part adaptation scale anisotropic classes apr lepski willer institut marseille rue marseille france abstract paper continues research started lepski willer framework convolution structure density model address problem adaptive minimax estimation scale anisotropic nikol skii classes fully characterize behavior minimax risk different relationships regularity parameters norm indexes definitions functional class risk particular show boundedness function estimated leads essential improvement asymptotic minimax risk prove selection rule proposed part leads construction optimally nearly optimally logarithmic factor adaptive estimator ams subject classifications keywords phrases deconvolution model density estimation oracle inequality adaptive estimation kernel estimators anisotropic nikol skii class introduction present paper interested adaptive estimation convolution structure density model considerations continue research started lepski willer thus observe vectors common probability density satisfying following structural assumption supposed function estimated recall two functions work carried framework labex midex project funded investissements avenir french government program managed french national research agency anr ver file date april furthermore denotes set probability densities ball radius lebesgue measure last let fourier transform convolution structure density model studied arbitrary except case function necessarily probability density want estimate using observations estimator mean map accuracy estimator measured kpp denotes expectation respect probability measure observations also objective construct estimator small adaptive estimation let given subset estimator define maximal risk supf minimax risk given inf infimum taken possible estimators estimator whose maximal risk bounded constant factor called minimax let collection subsets nuisance parameter may complicated structure problem adaptive estimation formulated follows possible construct single estimator would simultaneously minimax class lim sup refer question problem minimax adaptive estimation scale estimator exists call optimally adaptive using modern statistical language call estimator nearly optimally adaptive lim sup interested adaptive estimation scale anisotropic nikolskii class see definition explained part adaptive estimation scale viewed adaptation anisotropy inhomogeneity function estimated recall also simply means common density observations uniformly bounded easy see let briefly discuss another example let arbitrary priory chosen numbers assume considered collection anisotropic nikol skii classes obeys following suppose also kgks restrictions exists completely determined kgks additionally study adaptive estimation collection show boundedness underlying function allows improve considerably accuracy estimation historical notes minimax adaptive estimation active area mathematical statistics interested reader find detailed overview well several open problems adaptive estimation recent paper lepski discuss articles whose results relevant consideration density setting minimax perspective let start following remark one assumes additionally convolution structure density model interpreted follows observations written sum two independent random vectors random vectors common density estimated noise variables random vectors known common density last bernoulli random variables supposed known sequences supposed mutually independent observation scheme viewed generalization two classical statistical models indeed case corresponds standard deconvolution model another extreme case correspond direct observation scheme intermediate case considered first time hesse understood partially contaminated observations direct case vast literature dealing minimax minimax adaptive density estimation see example efroimovich hasminskii ibragimov golubev donoho devroye lugosi rigollet rigollet tsybakov samarov tsybakov nickl akakpo gach lepski among many others special attention paid estimation densities unbounded support see juditsky developed results found goldenshluger lepski goldenshluger lepski section compare detail results obtained papers intermediate case best knowledge adaptive estimation case partially contaminated observations studied yet able find two papers dealing minimax estimation first one hesse discussed model introduced dimension author evaluated proposed estimator functional class formally corresponding nikol skii class yuana chenb latter result developed multidimensional setting minimax timation intriguing fact accuracy estimation partially contaminated noise direct observation scheme however none articles studied optimality proposed estimators come back aforementioned papers section order compare assumptions imposed noise density deconvolution case first let remark behavior fourier transform function plays important role works dealing deconvolution indeed problems correspond fourier transforms decaying towards zero results established moderately ill posed problems detail results papers studying type operators assumption means exist fourier transform satisfies minimax minimax adaptive results dimension different classes smooth functions found particular stefanski carroll fan fan pensky vidakovic fan koo comte hall meister meister lounici nickl kerkyacharian results multidimensional setting seems masry first paper deconvolution problem studied multivariate densities worth noting masry considered general weakly dependent observations paper formally deal minimax setting however results obtained paper could formally compared estimation isotropic class regularity exactly setting yuana chenb case partially contaminated observations let also remark lower bound result masry developed results deconvolution model obtained comte lacour rebelles section compare detail results obtained papers lower bound minimax seen problem optimal adaptation collection formulated attainability family minimax risks single estimator although necessary following approach used majority problems related minimax adaptive estimation first step consists finding lower bound second one consists constructing estimator attaining least asymptotically bound adopt strategy investigations present several lower bound results recently obtained lepski willer assumptions function imposed lepski willer let denote set subsets set let denote cardinality denotes elements define operator let denote identity operator define note obviously assumption exists assumption exists exists assumption bounded function assumption exists supi sup moreover moreover worth noting bounds lepski willer obtained assumptions assumption used estimation unbounded functions considered come back assumption section assumption seems purely technical appear upper bound results also recall lower bounds lepski willer proved condition lower bounds lepski willer introduce set following quantities define general case remind set later assume implies particular recall also put last theorem lepski willer let fixed satisfying assumptions exists independent lim inf inf infimum taken possible estimators following terminology used lepski willer call set parameters satisfying tail zone satisfying dense zone satisfying sparse zone turn latter zone divided two sparse zone corresponding sparse zone corresponding bounded case introduce theorem lepski willer let fixed satisfying assumptions exists independent sup lim inf inf infimum taken possible estimators assumptions function selection rule family linear estimators oracle inequalities obtained part adaptive results presented paper established following condition imposed function assumption exists exists comparing condition assumption section assert coherent indeed case come following assumption literature referred moderately statistical problem particular assumption checked centered multivariate laplace law note first assumption sense weaker assumption since require regularity properties function moreover assumptions restrictive verified many distributions including centered multivariate laplace gaussian ones note also assumption always holds additionally holds real positive function latter true particular probability law obtained even number convolutions symmetric distribution next assumption weaker conditions imposed hesse yuana chenb papers adaptive estimation scale anisotropic nikol skii classes start section recalling definition pointwise selection rule proposed part pointwise selection rule let continuous function belonging set let recall hisotr set let later operations relations understood sense particular means let satisfy operator equation introduce let arbitrary subset introduce sup sup define arg inf final estimator fbh call pointwise selection rule remark note estimator depends later consider two choices parameter set namely hisotr present results write order underline aforementioned dependence choice used adaptation studied anisotropic nikol skii classes hisotr used considered scale consists isotropic classes anisotropic nikol skii classes let denote canonical basis function real number define first order difference operator step size direction variable uej induction order difference operator step size direction variable defined definition given vectors say function belongs anisotropic nikolskii class kgkrj every exists natural number corresponding nikolskii class denoted furthermore called isotropic construction kernel first recall results concerning risk pointwise selection rule established part proved following assumption imposed kernel assumption exist use following specific kernel definition estimator family next see kerkyacharian goldenshluger lepski let ber integer number let compactly supported continuous function satisfying put add following structural condition assumption assumption kernelrk constructed way bounded compactly supported belongs satisfies examples kernels satisfying simultaneously assumptions found instance comte lacour main results introduce following notations hisotr otherwise bounded case address adaptive estimation collection functional classes first problem conjectured lepski willer boundedness function belonging minimal condition allowing eliminate inconsistency zone results obtained theorem together theorem confirm conjecture theorem let satisfying assumption fixed let satisfy assumptions exists independent lim sup sup defined exists independent lim sup sup hisotr isotr remarks order estimation procedure completely assertions theorem completely new comparing pendent results obtained theorems assert estimator nearly optimally adaptive construction estimation procedure would open problem conjecture lower bounds asymptotics minimax risk found theorem sharp order conjecture case partially confirmed results obtained comte lacour rebelles since articles deal estimation unbounded functions discuss next section worth noting previous statements true convolution structure density model also view theorem observation scheme well note asymptotic minimax risk partially contaminated observations independent coincides asymptotic risk direct observation model first time phenomenon discovered hesse yuana chenb recent paper lepski particular case optimally adaptive estimator built easy check independently value corresponding set parameters belongs dense zone note however estimator zone applied much general collection functional classes worth noting estimator procedure used lepski nothing common pointwise selection rule direct observation scheme results coincide obtained recently goldenshluger lepski however tail zone bound slightly better since bound obtained latter paper contains additional factor interesting note although estimator constructions based upon local selections family kernel estimators selection rules different let finally discuss results corresponding tail zone first lower bound minimax risk given accuracy provided estimator mentioned passage seems unavoidable payment application local selection scheme interesting note additional factor disappears dimension first note onedimensional setting considered juditsky setting juditsky corresponds deal case settings rule sparse zone rates convergence found papers easily recovered results corresponding tail dense zones next remark aforementioned factor appears anisotropic functional classes considered indeed view second assertion theorem estimator nearly optimally adaptive tail zone isotropic case natural question arising context whether unavoidable payment anisotropy underlying function last note isotropic case results remain true corresponding nikol skii class defined worth noting analysis proof theorem allows assert first statement remains true logarithmic factor however asymptotic maximal risk estimator remains unknown finish discussion following remark assumption implies many cases uniformly bounded therefore theorem particular always case model considered indeed case implies another case recall assumption used proofs theorems assumption obviously generally since definition nikol skii class implies latter condition verified particular kgkq saying explains study estimation unbounded functions case unbounded case problem address adaptive estimation collection functional classes already mentioned additionally therefore view theorem discussed section consistent estimator either analyzing proof latter theorem come following assertion conjecture let assume assumption fulfilled suppose additionally assumption holds assertion theorem remains true one replaces latter result formulated conjecture prove present paper proof postponed part iii adaptive estimation collection introduced part studied reason later consider parameters belonging set defined max given latter set consists class parameters uniform consistent estimation possible theorem let satisfying assumption fixed let satisfy assumptions exists independent lim sup sup defined exists independent lim sup sup hisotr isotr remarks order note implies therefore parseval identity together assumption allows assert hence condition automatically checked also worth noting considering adaptation collection isotropic classes require coordinates would latter true second assertion theorem well last analyzing proof theorem assert second assertion remains true slightly weaker assumption assertion theorem analogue existing literature except results obtained comte lacour rebelles comte lacour deals particular case rebelles studied case easy check papers whatever value corresponding set parameters belongs dense zone note also estimation procedures used comte lacour well rebelles based global version method attain asymptotic minimax risks corresponding dense zone found theorem method nearly optimally adaptive however global selection family standard kernel estimators leads correct results considered see instance goldenshluger lepski hand estimation procedures based local selection scheme applied estimation functions belonging much general functional classes often lead optimally adaptive method fortunately loss accuracy inherent local procedures logarithmic number observations together theorems theorems provide full classification asymptotics minimax risks nikolskii classes class parameters belonging sparse zone logarithmic factor belonging tail dense zones well boundaries mean results theorems valid indeed given one choose fixed number used kernel construction integer strictly larger open problems let briefly discuss unresolved adaptive estimation problems convolution structure density model construction estimator already mentioned proposed pointwise selection rule leads optimal adaptive estimator class parameters belonging sparse zone bounded unbounded case conjecture construction estimator values nuisance parameters via pointwise selection impossible methods invented worth noting estimator known neither density model density deconvolution even dimension dimension larger one intriguing questions related eventual price pay anisotropy discussed remark theorem adaptive estimation unbounded functions able study unbounded case estimation unbounded densities direct well partially contaminated observations remain open problems conjecture results obtained case true anymore neither upper bounds lower bound correct nearly correct upper bounds asymptotics minimax risk still deduced oracle inequalities proved part case least two interesting problems first results valid condition absence assumption may effects accuracy estimation absolutely unclear next let mention lower bound result proved theorem holds consideration convolution structure density model could bounds established deconvolution model adjustment lower upper bound assumptions comparing assertions theorems theorem remark obtention corresponding lower bounds minimax risk requires additional rather restrictive assumptions function weakened even removed proof theorems proofs based application theorem part auxiliary assertions presented subsequent proof stand constants depend independent constants different different appearances important concepts part proof outline section recall definition important quantities appeared theorem part discuss facts established make theorem applicable theorem part deals minimax result class arbitrary subset defined section part theorem consider therefore makes theorem part applicable case show one theorem consider therefore theorem part find applicable latter inclusions mostly based embedding anisotropic nikol skii spaces used proof proposition lemma application theorem part case requires compute inf inf remind universal constant completely determined kernel dimension next section propose quite sophisticated constructions vectors show propositions defined defined defined given proposition prove additionally last definition defined independent together allows assert see thus putting obtain view get used large enough follows assertions established proof independent proposition deduce following bound moreover get view special set bandwidths bandwidth construction presented well auxiliary statements next section exploited proving theorems also consideration forming part iii work reason formulate bit general form needed current purposes set let max recall appeared number satisfying recall recall introduce put complimentary constant chosen differently accordance special relationships parameters determine relations max max set auxiliary statements results formulated proved section let remark note also introduce following notations recall define set let put finally proposition let given assume large enough exists independent large enough exists independent either remark note condition simply means since hand condition holds whatever values since also note indeed since follows get last inequality used strictly decreasing particular deduce condition always fulfilled case recall defined introduce following quantities define also note proved proposition however shown proof proposition formulae large enough also last since moreover introduce finally proposition let given let exists independent large enough current paper use statements proposition context remark proposition let satisfying assumption fixed one find independent holds additionally fulfilled well last remain true one replaces quantity quantities introduced part reader find proof proposition let also present following auxiliary results useful sequel proofs postponed appendix lemma let exists finish section following observations useful sequel one one concluding remarks let collect bounds several terms appearing theorem part used proofs theorems simultaneously follows first remark deduce definition yields together definitions choosing elementary computations taking account obtain bpn bpn bounds surprising last get thanks definition presentation proved last choosing obtain bpn bpn yields proof theorem already mentioned apply theorem part consider cases choose remark statements propositions hold indeed suffices note since case apply bounds obtained particular get bpn since considered cases view second equality applying third assertion theorem part obtain assertion theorem follows considered cases consider case choose remark statements propositions hold hold indeed implies therefore deduce applying first assertion theorem part also used completes proof theorem proof theorem following assume since implies definition anisotropic nikol skii class hence results case follow theorem since moreover remark imposed condition implies view proved remark first makes second assertion proposition applicable next allows recall rewrite appeared proposition consider case remark view nikol skii theorem taking account independent thus theorem part section applicable choose remark statements propositions hold since assertion theorem obtained first assertion theorem part computations led consider case recall case necessary existence uniformly consistent estimator since definition anisotropic assert second assertion theorem nikol skii class implies part applicable choose note considered case thus deduce sup bpn assertion theorem follows case remains study case let arbitrary number satisfying lemma since assert view nikol skii theorem independent thus theorem part section applicable choosing deduce second assertion theorem part sup since either get simple algebra shows using obtain spy since satisfies lemma thus large enough bpn assertion theorem case follows first equality theorem proved proofs propositions proof lemma postponed appendix lemma following true proof proposition start proof several remarks useful sequel first obviously exists independent lim sup sup sup next let proceed proof first assertion first remark indeed since therefore one view definition note construction proof completed since set remark view large enough taken account since denoting assert first assertion established proving second assertion let make several remarks following true since remind first equality follows directly definition thus let prove second equality used using definition get using definition obtain obtain applying lemma second formula established next let prove equivalent definition implies large enough since view deduce first equality proved remains note since therefore large enough view lemma second equality since completes proof one denoted inf llj llj indeed view definition put note established simple algebra shows deduce recall let also prove large enough latter inclusion follows indeed view note last let proceed proof second assertion let choose view similarly thus prove assertion need show let distinguish three cases let remark definition case yields large enough obtain view similarly let assumption thus get follows let previously used put goal show large enough view definition order establish suffices show since assumed necessarily since strictly decreasing hence required results follows thus proved choosing obtain large enough second assertion proved proof proposition start proof several remarks useful sequel let show large enough view definition therefore one taking account remains note large enough therefore also view definition together proves cases noting equivalent deduce thus together yields case whatever value let view definition view hence holds case let view definition therefore holds case let first note imply since either thus therefore note therefore yields remains note therefore implies case already treated completes proof independent remark obviously exists lim sup sup sup hence view one large enough setting taking account obtain choose yields view since holds finish proof proposition need show let distinguish three cases let first note cases next view second inequality obtain get last inequality used let view second inequality since simple algebra shows implies result follows sup let view second inequality denoted goal show large enough sup easily compute denoting right hand side obtained inequality obviously sup max remarking easily compute moreover obviously consider case deduce sup thanks definition implies last results together prove case consider case moreover since hence view lemma view definition note get last inequality used thus conclude together implies considered case moreover view definition routine computations come following equality hence large enough together allows assert considered case consider case required result follows strictly increasing therefore view completes proof finally conclude case choosing deduce large enough aav proof proposition view lemma lepski note also independent without mentioning couple used let either condition obviously uhej sup sup last equality follows definition order difference operator hence view definition nikol skii class remind kbh sup yields sup first second assertions proposition proved let choosing relation recall sup lim sup view monotone convergence theorem triangle inequality lim sup minkowski inequality integrals see folland section obtain taking account proves first second assertions proposition recall set uhej sup sup hence third assertion follows appendix proof lemma note follows hand checked since note first necessarily since strictly decreasing hence established let prove first note obvious case thus assume next holds indeed case implies hence number interval satisfies last note since view obtained contradiction completes proof proof lemma indeed moreover view latter inequality remains note lemma follows references akakpo adaptation anisotropy inhomogeneity via dyadic piecewise polynomial selection math methods statist model selection density estimation http comte rozenholc taupin penalized contrast estimator adaptive density deconvolution canad comte lacour anisotropic adaptive kernel deconvolution ann inst probab statist fan optimal rates convergence nonparametric deconvolution problems ann fan adaptively local subproblems application deconvolution problem ann fan koo wavelet deconvolution ieee trans inform theory devroye lugosi nonasymptotic universal smoothing factors kernel complexity yatracos classes ann statist donoho johnstone kerkyacharian picard density estimation wavelet thresholding ann statist efroimovich estimation density unknown smoothness theory probab appl folland real analysis second edition wiley new york gach nickl spokoiny spatially adaptive density estimation localised haar projections ann inst nickl exponential inequality distribution function kernel density estimator application adaptive estimation probab theory related fields goldenshluger lepski bandwidth selection kernel density estimation oracle inequalities adaptive minimax optimality ann statist goldenshluger lepski adaptive minimax density estimation probab theory related fields golubev estimation smooth probability densities probl inform transm guzman differentiation integrals appendices antonio crdoba robert fefferman two roberto moriyn lecture notes mathematics vol york hall meister approach deconvolution ann hasminskii ibragimov density estimation view kolmogorov ideas approximation theory ann statist hesse deconvolving density partially contaminated observations journal multivariate analysis juditsky minimax density estimation bernoulli kerkyacharian lepski picard nonlinear estimation anisotropic denoising probab theory related fields kerkyacharian thanh picard localized spherical deconvilution ann statist lepski multivariate density estimation loss oracle approach adaptation independence structure ann statist lepski adaptive estimation anisotropic functional classes via oracle approach ann statist lepski willer lower bounds convolution structure density model bernoulli lepski willer estimation convolution structure density model part oracle inequalities annals submitted lepski new approach estimator selection bernoulli appear http lounici nickl global uniform risk bounds wavelet deconvolution estimators ann statist masry strong consistency rates deconvolution multivariate densities stationary processes stochastic processes applications meister deconvolution problems nonparametric statistics lecture notes statistics berlin nikol skii priblizhenie funktsii mnogikh peremennykh teoremy vlozheniya russian approximation functions several variables imbedding theorems second edition revised supplemented nauka moscow pensky vidakovic adaptive wavelet estimator nonparametric density deconvolution ann rebelles structural adaptive deconvolution math methods statist rivoirard adaptive density estimation curse support statist plann inference rigollet adaptive density estimation using blockwise stein method bernoulli rigollet tsybakov linear convex aggregation density estimators math methods statist samarov tsybakov aggregation density estimators dimension reduction advances statistical modeling inference ser world sci hackensack stefanski rates convergence estimators class deconvolution problems statist probab stefanski carroll deconvoluting kernel density estimators statistics yuana chenb deconvolving multidimensional density partially contaminated observations journal statistical planning inference
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designs ordered categorical data nov jie liping abhyuday university illinois chicago advocate health care university georgia abstract cumulative link models widely used ordered categorical responses uniform allocation experimental units commonly used practice often suffers lack efficiency consider designs ordered categorical responses cumulative link models predetermined set design points derive necessary sufficient conditions allocation locally develop efficient algorithms obtaining approximate exact designs prove number support points minimally supported design depends number predictors much less number parameters model show minimally supported allocation case usually uniform support points addition provide designs highly efficient surrogate bayesian designs much robust uniform designs key words phrases approximate design exact design multinomial response cumulative link model minimally supported design ordinal data introduction paper determine optimal efficient designs factorial experiments qualitative factors ordered categorical responses simply ordinal data design experiment multinomial response ordered categories particular becoming increasingly popular rich variety scientific disciplines especially human evaluations involved christensen examples include wine bitterness study randall potato pathogen experiments omer radish seedling study krause polysilicon deposition study beef cattle research osterstock toxicity study agresti research motivated odor removal study conducted textile engineers university georgia scientists studied manufacture containing odorous volatiles need removed commercialization purpose factorial experiment conducted using algae synthetic plastic resin blends factors types algae raffinated solvent extracted algae catfish pond algae synthetic resins polyethylene jie yang liping tong abhyuday mandal polypropylene response three ordered categories serious odor medium odor almost odor following traditional factorial design theory pilot study equal numbers case replicates experimental setting conducted uniform design results summarized table yij represents number responses falling jth category ith experimental setting demonstrated later section best design identified research could improve efficiency three experimental settings involved table pilot study odor removal study experimental setting factor algae level resin summarized responses odor serious medium odor kind ordinal response categories predictors popular model practice first proportional odds model also known cumulative logit model see liu agresti detailed review mccullagh extended cumulative link model also known ordinal regression model general link function proportional odds model special case logit link examples include complementary link polysilicon deposition study see example cauchit link toxicity study see example adopt cumulative link model two categories cumulative link model essentially generalized linear model binary data mccullagh nelder dobson barnett optimal designs generalized linear models growing body literature see khuri atkinson stufken yang references therein case known minimum number experimental settings required nondegenerate fisher information matrix equals number parameters fedorov designs ordered categorical data yang mandal design least number experimental settings known minimally supported design practical significance specified regression model due cost changing settings also known experimental units uniformly assigned minimally supported design adopted binary response univariate generalized linear model yang mandal cumulative link model special case multivariate generalized linear model mccullagh relevant results optimal design literature meagre restricted logit link function zocchi atkinson perevozskaya obtain theoretical results efficient algorithms general link functions reveal optimal designs quite different cases prove minimum number experimental settings still strictly less number parameters theorems result due responses single experimental setup summarized responses degrees freedom requiring fewer distinct experimental settings minimally supported design reason allocation replicates minimally supported design usually uniform section differs traditional factorial design theory generalized linear models information matrix cumulative link models depends unknown parameters different approaches proposed solve dependence optimal designs unknown parameters including local optimality chernoff bayesian approach chaloner verdinelli maximin approach pronzato walter imhof sequential procedure ford pointed ford torsney locally optimal designs important good initial parameters available previous experiments also benchmark designs chosen satisfy experimental constraints mainly focus locally optimal designs situations local values parameters difficult obtain experimenter idea range parameters without prior distribution recommend optimal designs fisher information matrix replaced expected values atkinson yang mandal majumdar compare bayesian doptimal designs chaloner verdinelli designs ordinal data surrogate bayesian designs design much easier find retains high efficiency respect bayesian criterion section jie yang liping tong abhyuday mandal among various optimal design criteria maximizes determinant fisher information matrix frequently used zocchi atkinson often performs well according criteria atkinson study designs design literature one type experiment deals quantitative continuous factors design problem includes identification set design points corresponding weights see example atkinson stufken yang numerical algorithms typically used cases two factors see example woods another type experiment employs qualitative discrete factors set design points predetermined weights optimized see example yang mandal one pick grid points continuous factors turn first kind problem second tong volkmer yang section also bridged gap two types problems way results involving discrete factors applied cases continuous factors concentrate second kind design problems assume given fixed paper organized follows section describe preliminary setup obtain fisher information matrix cumulative link model general link generalizing perevozskaya also identify necessary sufficient condition fisher information matrix positive definite determines minimum number experimental settings required sections provide theoretical results numerical algorithms searching locally approximate exact designs section identify analytic designs special cases illustrate minimally supported design usually uniform support points section illustrate examples design highly efficient respect bayesian make concluding remarks section relegate additional proofs results supplementary materials fisher information matrix determinant suppose predetermined experimental settings ith experimental setting corresponding predictors xid experimental units assigned among experimental units kth one generates response vik belongs one ordered categories shown example dimension predictors significantly larger number factors considered experiment allows flexible models designs ordered categorical data general setup many applications vini regarded discrete random variables let vik let yij vik number vik falling jth category yij multinomial assumption let vik based assumption consider independent multinomial observations yij corresponding predictors cumulative link model ordinal regression model mccullagh agresti christensen exists link function parameters interest xti leads equations parameters assumption link differentiable derivative assumption satisfied commonly used link functions including logit log probit log log complementary log log cauchit tan mccullagh nelder christensen relevant formulas link functions provided supplementary materials section according assumption strictly increasing example consider logit link log two predictors three ordered categories model consists equations parameters assumptions example suppose model consists three covariates predictors number predictors cumulative link model function constant yij log jie yang liping tong abhyuday mandal xti perevozskaya obtained detailed form fisher information matrix logit link one predictor result general link predictors proof relegated supplementary materials section theorem assumptions fisher information matrix written matrix xis xit cit xit symmetric matrix diagonal entries entries gij gij gij cit git git git uit git bit git contains one entry fisher information matrix always positive fedorov special case fisher information experimental setting also known design point support point thus positive determinant fisher information matrix among different criteria optimal designs looks allocation maximizing determinant design predetermined design points could either integerpm valued allocation maximizing fixed known exact design maximizing known approximate design theorem determinant fisher information matrix designs ordered categorical data homogeneous polynomial proof theorem relegated supplementary materials section given map matrix whose kth row kth row take order obtain analytic properties need lemmas first covers lemma perevozskaya special case lemma rank rank furthermore positive definite gis gis xti example suppose link function according theorem homogeneous polynomial based lemma lemma supplementary materials section remove terms form therefore cijk cijkl coefficients cijk cijkl based lemmas order keep largest possible fewest possible number positive theorem determine whether experimental settings support points enough keep fisher information matrix positive definite study leading term example lemma must exist different lemma provides explicit formula coefficient jie yang liping tong abhyuday mandal lemma suppose eis matrix consisting rows proof lemma supplementary materials section find allocations write order homogeneous polynomial function exact design problem find allocation maximizing subject given positive integer denote according theorem due theorems directly applied approximate design problems find allocation maximizing subject according lemma thus lemma positive long full rank theorem implies minimally supported design contains least support points following theorem states necessary sufficient condition minimum number support points exactly theorem extended design matrix full rank minimal number experimental settings required thus strictly less number parameters odor removal study example cumulative link model involves four independent parameters two covariates two intercepts minimally supported design could involve three experimental settings multinomial responses categories get two degrees freedom experimental setting optimal allocation experimental units often uniform see section contrary case binary responses yang mandal majumdar yang mandal designs ordered categorical data approximate design locally approximate design allocation maximizing values parameters solution always exists since continuous set feasible allocations convex compact nontrivial approximate design problem requires assumption assumption rank assumption adopted throughout set valid allocations nonempty since linear log concave positive matrices silvey thus also convex theorem feasible allocation satisfies rank consisting rows direct conclusion theorem contains whose coordinates strictly positive special case uniform allocation necessary sufficient condition approximate design doptimal type kiefer pukelsheim atkinson stufken yang fedorov leonov yang mandal majumdar convenient searching numerical solutions following yang mandal majumdar given set well defined long theorem suppose following algorithm proposed yang mandal majumdar parallel results algorithm case simplicity also call algorithm jie yang liping tong abhyuday mandal theorem given allocation attains maximum algorithm start allocation satisfying set random order going determine according theorem determinants calculated according use method gradient defined find maximizing let take replace repeat theorem algorithm converges resulting maximizes example odor removal study response ordinal nature serious odor medium odor odor fit cumulative link model data presented table estimated values model parameters experiment planned estimated parameter values regarded true values approximate allocation found algorithm efficiency uniform far satisfactory example wine bitterness study christensen table aggregated wine data randall contains output factorial experiment two treatment factors two levels temperature cold warm contact yes affecting wine bitterness response ordinal five levels least bitter bitter original design employed uniform allocation estimated parameter values logit link experiment planned regarding estimated values parameters true values approximate allocation found algorithm efficiency original designs ordered categorical data design nevertheless corresponding efficiency may drop larger see figure case allocations minimally supported see figure discussed section figure wine bitterness study assumed true parameter values contour plot efficiency original design regions design minimally supported examples studied algorithm often converges within iterations yang mandal majumdar algorithm guaranteed converge applied algorithm converge number iterations exact design design literature different discretization methods proposed round approximate design exact design given including quota method kiefer pukelsheim efficient rounding procedure pukelsheim pukelsheim rieder usually work well large enough guarantee small sample size imhof wong section provide direct search exact designs theorem result follows corollary rank assume throughout section maximize adopt exchange algorithm idea fedorov jie yang liping tong abhyuday mandal used adjust simultaneously randomly chosen keeping constant start satisfying following yang mandal majumdar let fij fij theorem lemmas result follows theorem suppose satisfies given fij fij obtained using matrix fij fij matrix theorem shares form theorem according theorem maximize fij one obtain exact polynomial form fij calculating fij fij fij practical need find exact form fij since one simply calculate fij following yang mandal majumdar exchange algorithm see supplementary materials section based theorem could used search exact allocation example odor removal study continued conduct experiment experimental units using exchange algorithm obtain exact designs across different table expected exact allocation consistent approximate allocation last row table large time costs seconds last column table recorded cpu memory rerun experiment exact design efficiency uniform design example polysilicon deposition study considered experiment studying polysilicon deposition process six factors described details phadke due inconvenience counting number surface defects major evaluating characteristic treated ordinal variable defects designs ordered categorical data table exact designs approximate design odor removal study iterations time sec original design denoted includes experimental settings based orthogonal array apply cumulative link model represent factor say levels linear component taking values quadratic component taking values hamada fitted model complementary link chosen aic bic criteria see example agresti involves four twelve coefficients true parameter values assumed estimated ones used exchange algorithm find design denoted see supplementary materials section list experimental settings compared efficiency original design order check efficiency rounded design used algorithm find approximate design contains positive distinct experimental settings case quota method efficient rounding procedure end rounded design see section efficiency minimally supported design practical significance experiment run minimal number different settings example experimental settings polysilicon deposition study example run sequential way two settings arranged day phadke less experimental settings often indicate less time less cost another practical application minimally supported design optimal allocation restricted support points obtained easily even analytically according theorem minimally supported design contains least support points hand according theorem corollary minimally supported design could contain exactly support points extended design matrix full rank jie yang liping tong abhyuday mandal example let binomial response parameters general link function satisfying assumptions xti theorem contains entry thus simply lemma still holds assume design matrix satisfies assumption according theorem lemmas given pid eid essentially lemma yang mandal minimally supported design contain support points doptimal one keeps equal weight support points yang mandal theorem univariate responses including binomial ones generalized linear model minimally supported design must keep equal weights support points order keep yang mandal however multinomial responses usually case section use cases illustration order check minimally supported design need condition since conditions karush kuhn tucker also sufficient theorem allocation satisfying exists minimally supported designs one predictor start corresponding parameters consider designs supported two points minimally supported invoke theorem lemmas theorem objective function matrix designs ordered categorical data actually according lemma predictor levels theorem provides way find exact form calculating different allocations problem maximize polynomial special case allocation solved explicitly follows corollary objective function two levels predictor design furthermore setup corollary general following result provides conditions minimally supported design proof relegated supplementary materials section corollary suppose let distinct levels predictor minimally supported design defined corollary example consider factor levels logit link parameters satisfy investigate design minimally supported according theorem deign satisfies figure shows cases general parameter values figure four regions occupied minimally supported jie yang liping tong abhyuday mandal figure regions design logit link note required designs required example regions labeled indicates minimally supported design satisfying given triple figure design supported far example toxicity study agresti table reported data developmental toxicity study one factor concentration diegdime five levels per day ordinal response status mouse fetus nonlive malformation normal case fit cumulative link model cauchit link chosen aic bic criteria estimated parameter values regarded true parameter value approximate allocation found algorithm minimally supported alternatively pair indices obtain best design supported according corollary check whether using corollary minimally supported design also respect efficiency original design roughly uniform one minimally supported designs two predictors section consider experiments two predictors response parameters cases similar conclusions could obtained messier notation designs ordered categorical data according theorem minimally supported design needs three support points example assumption matrix full rank following theorem lemmas objective function since need consider satisfying according theorem maximizes following tong volkmer yang obtain analytic solution theorem without loss generality allocation maximizing exists unique satisfies obtained analytically follows special case iii proof theorem relegated supplementary materials section corollary suppose defined example consider factorial design problem response four design points denoted jie yang liping tong abhyuday mandal take five special cases iii distinct distinct theorem provides analytic forms minimally supported designs corollary suppose let distinct level combinations two predictors matrix minimally supported design obtained according theorem sum example consider experiments design points figure provides boundary lines regions parameters best design particular figure shows region given clearly indicates best design tends absolute values large region tends larger absolute values increase hand figure displays region given symmetry boundary lines due logit link symmetric interesting conclusion based corollary case design never remark extra degrees freedom play important role uniformity allocation minimally supported design multinomialtype responses categories total degrees freedom distinct experimental settings cumulative link model contains parameters minimally supported design see example objective function pid allocation designs ordered categorical data figure boundary lines design logit link region given outside boundary lines panel region given boundary lines panel pid however degrees freedom strictly larger number parameters extra degrees freedom case distinct experimental settings may play different roles estimating parameters values example objective function according corollary according equation uniform allocation depends true general design previous sections mainly focus locally designs require assumed parameter values many applications experimenter may little information values parameters bayes chaloner verdinelli maximizes log given prior distribution parameters provides reasonable solution alternative yang mandal majumdar atkinson essentially maximizes log according yang mandal majumdar simulation study across different models choices priors designs much easier calculate still highly efficient compared bayes designs based theorem design maximizes viewed locally design cit uit bit replaced expectations replacement lemma still holds therefore almost results previous sections applied directly designs jie yang liping tong abhyuday mandal exception lemma provides formula terms gij order find designs needs calculated terms uit bit example formulas lemma corollary corollary corollary need written terms uit bit according lemma need calculate uit bit cit uit bit cit use algorithm section exchange algorithm section find designs example odor removal study continued instead assuming parameter values consider true values parameters satisfy assume four parameters independently uniformly distributed within intervals use function constroptim maximize log find bayes allocation procedure costs seconds computational time order get design need seconds total calculate uit bit find using algorithm even terms bayes optimality chaloner larntz song wong abebe relative efficiency respect exp relative efficiency uniform allocation order check robustness towards misspecified parameter values let run points use algorithm find allocation corresponding determinant calculate efficiency respectively table shows summary statistics efficiencies implies comparable much better terms robustness table summary efficiency odor removal study design bayes uniform min quartile median discussion mean quartile max designs ordered categorical data paper use real experiments illustrate much improvements experimenter could make compared designs efficiencies original designs often far satisfactory example example example interestingly designs recommended example example minimally supported two surprising findings different cases univariate generalized linear models yang mandal minimum number experimental settings strictly less number parameters allocation experimental units support points minimally supported design usually uniform cumulative link models widely used modeling ordinal data nevertheless models used responses including logit model nominal response logit model ordinal data logit model hierarchical response see liu agresti agresti review methods developed paper could extended models well extensions approaches could used planning experiments one categorical response example paper feeder experiment pcb experiment analyzed joseph involved multiple binomial responses supplementary materials proofs theorems lemma corollaries available supplementary materials also tabularized formulas commonly used link functions additional lemmas section section maximization section exchange algorithm exact allocation section results example acknowledgements thank suraj sharma providing details odor removal study john stufken valuable suggestions early version paper also thank associate editor reviewers comments suggestions substantially improved quality manuscript research part supported las award faculty science uic references abebe tan van breukelen serroyen berger choice prior bayesian designs logistic regression model single predictor communications statistics simulation computation jie yang liping tong abhyuday mandal agresti categorical data analysis third edition wiley new jersey atkinson donev tobias optimum experimental designs sas oxford university press new york chaloner larntz optimal bayesian design applied logistic regression experiments journal statistical planning inference chaloner verdinelli bayesian experimental design review statistical science chernoff locally optimal designs estimating parameters annals mathematical statistics christensen link models estimation analysis ordinal data ordinal cumulative available via http dobson barnett introduction generalized linear models third edition chapman london fedorov theory optimal experiments academic press new york fedorov leonov optimal design nonlinear response models chapman new york ford titterington kitsos recent advances nonlinear experimental design technometrics ford torsney use canonical form construction locally optimal designs problems journal royal statistical society series imhof maximin designs exponential growth models heteroscedastic polynomial models annals statistics imhof wong efficiencies rounded optimal approximate designs small samples statistica neerlandica joseph failure amplification method information maximization approach categorical response optimization discussions technometrics karush minima functions several variables inequalities side constraints dissertation department mathematics university chicago kiefer role symmetry approximation exact design optimality statistical decision theory related topics gupta yackel eds academic press new york kiefer general equivalence theory optimum designs approximate theory annals statistics khuri mukherjee sinha ghosh design issues generalized linear models review statistical science designs ordered categorical data krause madden hoitink effect potting mix microbial carrying capacity biological control rhizoctonia radish rhizoctonia crown root rot poinsettia phytopathology kuhn tucker nonlinear programming proceedings berkeley symposium berkeley university california press liu agresti analysis ordered categorical data overview survey recent developments test mccullagh regression models ordinal data journal royal statistical society series mccullagh nelder generalized linear models second edition chapman boca raton omer johnson rowe recovery verticillium dahliae north american certified seed potatoes characterization strains vegetative compatibility aggressiveness american journal potato research osterstock macdonald boggess brown analysis ordinal outcomes carcass data beef cattle research journal animal science perevozskaya rosenberger haines optimal design proportional odds model canadian journal statistics phadke quality engineering using robust design englewood cliffs pronzato walter robust experiment design via maximin optimization mathematical biosciences pukelsheim optimal design experiments john wiley sons new york pukelsheim rieder efficient rounding approximate designs biometrika randall analysis sensory data generalised linear model biometrical journal silvey optimal design chapman hall london song wong optimal designs model heteroscedastic errors communications statistics theory methods stufken yang optimal designs generalized linear models design analysis experiments volume special designs applications hinkelmann wiley new york tong volkmer yang analytic solutions factorial designs generalized linear models electronic journal statistics woods lewis eccleston russell designs generalized linear models several variables model uncertainty technometrics jie yang liping tong abhyuday mandal simultaneous optimization robust design quantitative ordinal data international journal industrial engineering theory applications practice hamada experiments planning analysis optimization second edition wiley new york yang mandal factorial designs generalized linear models communications statistics simulation computation yang mandal majumdar optimal designs factorial experiments binary response statistica sinica zocchi atkinson optimum experimental designs multinomial logistic models biometrics university illinois chicago phone fax advocate health care lipingtong university georgia amandal supplementary materials designs ordered categorical data jie liping abhyuday university illinois chicago advocate health care university georgia supplementary materials commonly used link functions cumulative link models link function logit probit cauchit log log log log log tan exp exp arctan exp exp cumulative distribution function probability density function stands complementary example continued logit link thus gij xti additional lemmas section since yij independent random vectors function constant cumulative link model yij log jie yang liping tong abhyuday mandal xti xti xti xti yij yit since yij come multinomial distributions know yij yis yit following lemma lemma let fst fisher information matrix gij fst xis xit gij gij git git git iii gis gis gis gis max supplementary materials perevozskaya obtained detailed form fisher information matrix logit link one predictor expressions good fairly general link predictors simplify notations denote gij cit uit git git git bit git note gij defined lemma obtain following lemma plays key role calculating lemma cit uit bit uit bij cit lemma rank true based lemmas obtain two lemmas significantly simplify structure polynomial lemma thus proof lemma without loss generality assume implies case due lemma otherwise thus due lemma thus according provided theorem lemma thus proof lemma without loss generality assume indicates let satisfy matrix written jie yang liping tong abhyuday mandal matrix either single entry symmetric diagonal entries upper entries lower entries note asymmetric general exists case according lemma may assume otherwise according lemma suppose order show first replace replace changes new matrix note according lemma sum columns elementwise sum columns secondly add tth column denote resulting matrix note consider consists first columns sth row simply jth row proportional xid therefore rank leads thus according theorem lemma always positive positive definite furthermore log concave section procedure seeking analytic solutions follows tong volkmer yang direct conclusion conditions see also theorem necessary condition maximize equivalent terms supplementary materials denote since implies terms equivalent lemma suppose maximizes constrains proof lemma straightforward otherwise one could exchange strictly improve ready get solutions equations case case case implies plugging positive solution case implies plugging positive solution iii ratio leads plugging positive solution distinct without loss generality assume otherwise previous elimination procedure order could easily changed accordingly based lemma maximizes thus ratio leads implies note plugging get jie yang liping tong abhyuday mandal lemma suppose equation one one solution furthermore proof lemma order locate roots equation let hand first derivative therefore verified two cases case strictly decreases since one one solution case increases strictly decreases due one one solution either case conclusion justified additional proofs proof theorem direct conclusion lemmas examples theorem include respectively proof theorem study structure polynomial function denote entry akl given row map define whose kth row given kth row matrix akl supplementary materials power index denote terms construction says rows matrix according leibniz formula determinant sgn permutation sgn sign signature therefore sgn sgn proof lemma simplify notations let two types may type exist following similar procedure proof lemma obtain sgn clq type verified sgn jie yang liping tong abhyuday mandal according theorem type type sgn clq sgn sgn set permutations general case obtained similarly proof theorem suppose rank exist according lemma regarded polynomial let based lemma written therefore close enough supplementary materials order justify condition rank also necessary need show rank actually construct proof lemma similar proof lemma add tth column denote resulting matrix note consider consists first columns sth row simply sth row row claim rank otherwise rank exist consisting rows nonsingular consisting rows nonsingular implies rank contradiction implies rank thus based theorem thus proof theorem combining theorem theorem straightforward rank need show rank due theorem need verify case otherwise may simply remove support points suppose rank exist according proof theorem exists long hand denote jth row real numbers since least one without loss generality assume verified following proof theorem also exists long let min mini mini denote verified jie yang liping tong abhyuday mandal choice according lemma proof corollary order check minimally supported design supported add one support point according theorem lemmas objective function approximate design based theorem design similar conclusions could justified proof theorem according solutions provided software mathematica largest root equation simplification note calculation thus regarded operations among complex numbers since expression square root could negative nevertheless end would real number thus able provide analytic solution maximizing proof corollary order check minimally supported design first add four design points consider supplementary materials four design points check design could constructed first three design points let defined lemma case matrix following theorem lemmas objective function minimally supported design given according theorem minimally supported design case equivalent corollary leads corollary since forms change four design points added consideration note corollary equal could replaced since three equal maximization section according theorem polynomial determine coefficients need calculate determinants defined note matrix determined example jie yang liping tong abhyuday mandal respectively determined maximization numerically straightforward since polynomial derivative given jaj exchange algorithm exact allocation section exchange algorithm allocation given start initial design set random order going pairs let let otherwise two cases case one calculate fij defined directly find maximizes fij case two first calculate fij secondly determine according theorem thirdly calculate fij based fourthly find maximizing fij cases define note fij replace repeat convergence step supplementary materials example polysilicon deposition study table shows list experimental settings polysilicon deposition study factors decomposition temperature decomposition pressure nitrogen flow silane flow setting time cleaning method column provides original indices experimental settings distinct ones experimental setting labelled design responses collected phadke assumed independent table polysilicon deposition study experimental settings original rounded approximate exact designs index original rounded table shows model matrix design found polysilicon deposition study table factor represented linear component quadratic component thus level combinations predictors jie yang liping tong abhyuday mandal table polysilicon deposition study model matrix design index
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contextual bandits stochastic experts rajat karthikeyan sanjay feb university texas austin ibm research thomas watson center february abstract consider problem contextual bandits stochastic experts variation traditional stochastic contextual bandit experts problem problem setting assume access class stochastic experts expert conditional distribution arms given context propose bound ucb algorithms problem employ two different importance sampling based estimators mean reward expert estimators leverage information leakage among experts thus using samples collected experts estimate mean reward given expert leads instance dependent regret bounds log term depends mean rewards experts smallest gap mean reward optimal expert rest quantifies typically log mation leakage among experts show assumptions implement algorithm stochastic experts generated classification oracles show superior empirical performance datasets compared state art contextual bandit algorithms introduction modern machine learning applications like recommendation engines computational advertising testing medicine inherently online settings task take sequential decisions profitable also enable system learn better future instance computational advertising system task sequentially place advertisements users webpages dual objective learning preferences users increasing rate fly key attribute systems exploration searching space possible decisions better learning exploitation taking decisions profitable principled method capture study bandit problems stochastic bandit problems studied several decades formulated sequential process time step one selected upon selection arm arm returns stochastic reward expected reward starting work major focus regret difference total reward accumulated genie optimal policy one always selects arm maximum expected reward chosen online policy current algorithms achieve regret log optimal corresponds gap expected reward best arm next best one additional side information incorporated setting framework contextual bandits stochastic setting assumed nature draws fixed unknown distribution represents context vector rewards context revealed decides choose arm reward revealed computational advertising example context thought browsing history age gender etc user arriving system generated according probability user clicking advertisements task learn good mapping space contexts space arms decisions taken according mapping mean reward observed high popular model stochastic contextual bandits literature experts setting task compete best expert class experts expert function mapping mean reward expert defined random variable denoting context expectation taken unknown distribution best expert naturally defined expert highest mean reward expected difference rewards genie policy always chooses best expert online algorithm employed defined regret problem literature popular approach reduce contextual bandit problem supervised learning techniques polylog time leads powerful algorithms regret bounds practice class experts generated online training classification oracles trained resulting provide reliable confidence scores given new context especially confidence scores effectively probability vector entry probability choosing arm best given context motivated observation propose variation traditional experts setting term contextual bandits stochastic experts assume access class stochastic experts deterministic instead expert conditional probability distribution arms given context expert conditional distribution denoted random variable denoting arm chosen context additional benefit setting allows derive regret bounds terms closeness soft experts quantified divergence measures rather terms total number arms task compete expert class highest mean reward expected reward stochastic expert defined mean reward observed arm drawn conditional distribution propose ucb style algorithms contextual bandits stochastic experts problem employ two importance sampling based estimators mean rewards various experts prove regret guarantees algorithms main contributions paper listed next section main contributions contributions paper importance sampling based estimators key components approach two importance sampling based estimators mean rewards experts estimators based observation samples collected one expert reweighted ratios averaged provide estimate mean reward another expert sharing information termed information leakage utilized various settings first estimator use adaptive variant clipping technique proposed estimator presented however carefully adapt clipping threshold online manner order achieve regret guarantees also propose importance sampling variant classical median means estimator see estimator also designed utilize samples collected experts together estimate mean reward given expert define estimator best knowledge importance sampling used conjunction median means technique literature provide novel confidence guarantees estimator depends divergences conditional distributions various experts may independent interest instance dependent regret bounds propose contextual bandits stochastic experts problem design two ucb based algorithms problem based two importance sampling based estimators mentioned show utilizing information leakage experts leads regret guarantees scale number experts information leakage two experts first estimator governed pairwise measure def second estimator divergences def characterize leakage show regret ucb algorithm based two estimators scales log related largest pairwise divergence values two divergence measures used parameter gap mean rewards optimal expert second best depends gaps mean rewards optimum experts various ones normalized sum difference squares gaps adjacent experts ordered gaps assumption suboptimal gaps except second best arm uniformly log expectation distributed bounded interval show parameter define parameter explicitly section clipped estimator show largest pairwise associated clipped estimator median means estimator largest pairwise divergence naively treating expert arm would lead regret scaling log however ignores information leakage existing bounds contextual bandits scale poly log problem dependent bounds near optimal dependence depend numbers arms however depends divergence measure associated information leakage problem parameters besides analysis empirically show divergence based approach rivals performs better efficient heuristics contextual bandits like bagging etc data sets iii empirical validation empirically validate algorithm three real world state art contextual bandit algorithms implemented vowpal wabbit implementation use online training classification oracles generate class stochastic experts show algorithms better regret performance compared algorithms related work contextual bandits studied literature several decades starting simple setting discrete contexts linear contextual bandits finally general experts setting work focus experts setting contextual bandits first studied adversarial setting algorithms optimal regret scaling log paper interested stochastic version problem context rewards arms generated unknown fixed distribution first strategies explored setting style strategies achieve regret scaling log case following several efforts design adaptive algorithms achieve polylog regret scaling notable among algorithms map contextual bandit problem supervised learning assume access classification oracles algorithms heavily optimized vowpal wabbit study contextual bandits stochastic experts problem experts deterministic functions mapping contexts arms conditional distributions arms given context tighter regret bounds derived theorems mention corollaries approach easy state show achieve regret guarantees problem scale log log assumptions gap mean reward best expert second best divergence term experts algorithms based importance sampling based estimators leverage information leakage among stochastic experts use adaptive clipped importance sampling estimator mean rewards experts introduced estimator studied explore setting study cumulative regret problem need adjust parameters estimator online manner addition introduce importance sampling based median means style estimator paper leverage information leakage among experts problem setting definitions general stochastic contextual bandit problem arms defined sequential process discrete interest time nature draws vector unknown fixed probability distribution reward arm context vector revealed whose task choose arm possibilities reward chosen arm revealed use place notational convenience figure bayesian network denoting joint distribution random variables given contextual bandit setting denotes context denotes chosen arm denotes reward chosen arm also depends context observed distribution reward given chosen arm context marginal context remain fixed time slots however conditional distribution chosen arm given context dependent stochastic expert stochastic experts consider class stochastic experts conditional probability distribution random variable denoting arm chosen context use shorthand denote conditional distribution corresponding expert notational convenience observation model time step follows context observed chooses stochastic expert arm drawn probability distribution stochastic reward revealed joint distribution random variables denoting context arm chosen reward observed respectively time modeled bayesian network shown fig joint distribution factorizes follows reward distribution given arm context marginal distribution context determined nature distribution fixed hand distribution arm chosen given context depends expert selected round time conditional distribution encoded stochastic expert chosen time position define objective problem regret objective contextual bandit problem perform well best expert class experts define distribution corresponding random variables expert chosen expected reward expert denoted epk denotes expectation respect distribution best expert given arg objective minimize regret till time defined note analogous regret definition deterministic expert setting let define optimality gap terms expected reward expert let define divergence metrics important describing algorithms theoretical guarantees divergence metrics section define metrics important analyzing estimators similar divergence metrics defined analyze clipped estimator context best arm identification problem addition divergence metric also define divergence metric useful analyzing median means based estimator first define conditional definition let convex function two joint distributions associated conditionals conditional given eqx recall conditional distribution given thus conditional conditional distributions note definition marginal distribution marginal given nature inherent distribution contexts work concerned two specific metrics defined follows definition mij measure consider function exp define following measure mij log mij crucial analyzing one estimators clipped estimator defined section definition measure known divergence respective ditional distributions let important analyzing second estimator median means defined section section propose general bound ucb style strategy utilizes structure problem converge best expert much faster naive ucb strategy treats expert arm bandit problem one key observations framework rewards collected one expert give valuable information mean another expert owing bayesian network factorization joint distribution propose two estimators mean rewards different experts leverage information leakage among experts importance sampling estimators defined section propose algorithm designed use estimators corresponding confidence intervals control regret algorithm divergence based ucb contextual bandits stochastic experts time step observe context choose random expert play arm drawn conditional distribution observe context let arg maxk select arm distribution observe reward end denotes estimate mean reward expert time denotes upper confidence bound corresponding estimator time propose two estimators utilize samples observed various experts provide estimate mean reward expert first estimator denoted section clipped importance sampling estimator inspired estimator used set equation second estimator denoted section median means based importance sampling estimator estimator used set equation estimators confidence bounds section define two estimators estimating mean rewards given expert estimators effectively leverage information leakage samples collected various experts importance sampling one key observation enables following equation epj termed information leakage leveraged literature identification settings recall subscript denotes expectation taken joint distribution distribution imposed expert however even distribution technically estimate mean reward expert equation motivation behind estimators introduce first estimator clipped estimator estimator introduced context pure exploration problem analyze estimator cumulative regret setting parameters estimator need adjusted differently let denote number times expert invoked algorithm till time define fraction also define subset among expert selected let estimate mean reward expert samples collected till time estimator given log mkj mkj value random variable time drawn using expert set adjustable term controls estimator intuition clipped estimator weighted average samples collected different experts sample scaled importance ratio suggested also clip importance ratios larger clipper level clipping introduces bias decreases variance clipper level carefully chosen bias variance clipper level values weights dependent divergence terms mkj divergence mkj large means samples expert valuable estimating mean expert therefore weight applied similarly clipper level set log mkj restrict aditive bias upper confidence term algorithm estimator chosen sck time log log set analysis upper confidence bound derived using lemma lemma consider estimator following confidence bound time exp log lemma implied theorem include proof section appendix lemma shows clipped estimator pool samples experts order estimate mean expert variance estimator depends depends number times expert invoked median means estimator introduce second estimator based wellknown median means technique estimation median means estimators popular statistical estimation underlying distributions estimator mean expert time obtained following steps divide total samples log groups fraction samples expert preserved choose analysis let index groups means least bni samples expert group calculate empirical mean expert samples group importance sampling iii median means estimator setup notation let indices samples expert lie group let number samples expert group let mean expert estimated group given median means estimator expert given median intuition mean every group weighted average samples expert rescaled importance ratios similar clipped estimator however importance ratios clipped particular level estimator controlled taking median means groups number groups needs carefully set control upper confidence bound used conjunction estimator time given log set algorithm choice inspired following lemma lemma let estimator following confidence bound log provide proof lemma section appendix constant theoretical results section provide instance dependent regret guarantees algorithm two estimators proposed clipped estimator median means estimator let gap expected reward optimum expert second best define parameter later section depends gaps expected rewards various experts optimal one log simifor algorithm uses clipped estimator regret scales log maxlarly case median means estimator regret scales imum maximum divergence two experts respectively gaps optimum expert ones distributed uniformly random parameter log expectation contrast show experts used separate arms naive application bounds would yield regret scaling log prohibitively large number experts large ease exposition results let experts using indices regret guarantees clipped estimator provided following assumption assumption assume terms bounded let maxi mij position present one main theorems provides regret guarantees algorithm using estimator theorem suppose assumption holds regret algorithm time using estimator bounded follows log log universal constant defer proof theorem appendix present theorem provides regret guarantees algorithm using estimator theorem holds following assumption assumption assume terms bounded let maxi theorem suppose assumption holds regret algorithm time using estimator bounded follows log log universal constant average loss average loss average loss figure plots progressive validation loss till time plotted function time performance algorithms drug consumption dataset performance algorithms stream analytics dataset performance algorithms letters dataset legend proof theorem deferred appendix delve deeper instance dependent terms theorems theorem imply following corollary proofs corollary let following regret bounds algorithm estimator log similarly algorithm estimator log min min corollary leads next result corollary show gaps uniformly log expectation distributed corollary consider generative model order statistics random variables drawn uniform interval let denote measure following log log algorithm estimator log algorithm estimator remark note guarantees term containing number arms dependence implicitly captured divergence terms among experts fact number arms large expect divergence based algorithms perform comparatively better algorithms whose guarantees explicitly depend phenomenon observed practice empirical validation real world section also show empirically term grows slowly number experts empirical result included appendix empirical results section empirically test algorithms three classification datasets state art algorithms contextual bandits experts classification dataset converted contextual bandit scenario features contexts feature context sample point revealed following contextual bandit algorithm chooses one classes reward observed correct class otherwise bandit feedback correct class never revealed chosen method widely used benchmark contextual bandit algorithms fact implemented vowpal wabbit algorithm run batches starting batch add experts trained prior data classification oracles also update divergence terms experts estimated data observed far batch algorithm deployed current set experts procedure provided algorithm use xgboost algorithm batched classification experts let expert chooses arms randomly time steps choose arm sampled expert add experts trained observed data update divergences deploy algorithm experts end let add experts trained observed data update divergences end logistic regression calibration base classifiers oracles bootstrapping used generate different experts starting batch new experts added constants set practice settings held fixed three without parameter tuning provide details appendix appendix also show gap dependent term theoretical bounds grows much slower compared bounds fig number experts increase stream analytics dataset implementation algorithm found compare vopal wabbit implementations following algorithms parameter set first greedily selects best expert parameter set iii online cover parameter set bagging simulates thompson sampling bagged classifiers parameter set drug consumption data dataset part uci repository data respondents dimensional continuous features contexts history drug use drugs study arms entry bandit algorithm selects drug recently used reward observed reward observed performance algorithms shown fig see algorithm median moments clearly performs best terms average loss followed clipped estimator median moments converges average loss within samples stream analytics data dataset collected using stream analytics client samples dimensional mixed features contexts classes arms entry bandit algorithm selects correct class reward observed reward observed performance algorithms shown fig bagging performs best closely followed two versions algorithm bagging strong competitor empirically however algorithm lacks theoretical guarantees bagging converges average loss median moments converges average loss letters data dataset part uci repository samples english letters visual features contexts classes arms corresponding letters entry bandit algorithm selects correct letter reward observed reward observed performance algorithms shown fig versions significantly outperform others median moments based version converges average loss clipped version converges average loss https stochasticexperts conclusion study problem contextual bandits stochastic experts propose two ucb style algorithms use two different importance sampling estimators leverage information leakage stochastic experts provide regret guarantees ucb based algorithms algorithms show strong empirical performance datasets believe paper introduces interesting problem setting studying contextual bandits opens opportunities future research may include better regret bounds problem acknowledgment work partially supported nsf satc aro dot supported tier university transportation center references stream analytics dataset http accessed vowpal wabbit https accessed alekh agarwal daniel hsu satyen kale john langford lihong robert schapire taming monster fast simple algorithm contextual bandits international conference machine learning pages shipra agrawal navin goyal analysis thompson sampling bandit problem conference learning theory pages peter auer using confidence bounds journal machine learning research nov peter auer nicolo yoav freund robert schapire nonstochastic multiarmed bandit problem siam journal computing peter auer ronald ortner ucb revisited improved regret bounds stochastic bandit problem periodica mathematica hungarica alina beygelzimer john langford offset tree learning partial labels proceedings acm sigkdd international conference knowledge discovery data mining pages acm alina beygelzimer john langford lihong lev reyzin robert schapire contextual bandit algorithms supervised learning guarantees proceedings fourteenth international conference artificial intelligence statistics pages bottou jonas peters joaquin denis charles max chickering elon portugaly dipankar ray patrice simard snelson counterfactual reasoning learning systems example computational advertising journal machine learning research djallel bouneffouf amel bouzeghoub alda lopes algorithm mobile recommender system international conference neural information processing pages springer bubeck nicolo regret analysis stochastic nonstochastic multiarmed bandit problems foundations trends machine learning bubeck nicolo lugosi bandits heavy tail ieee transactions information theory lars buitinck gilles louppe mathieu blondel fabian pedregosa andreas mueller olivier grisel vlad niculae peter prettenhofer alexandre gramfort jaques grobler api design machine learning software experiences project arxiv preprint tianqi chen carlos guestrin xgboost scalable tree boosting system proceedings acm sigkdd international conference knowledge discovery data mining pages acm wei chu lihong lev reyzin robert schapire contextual bandits linear payoff functions international conference artificial intelligence statistics pages ira cohen moises goldszmidt properties benefits calibrated classifiers pkdd volume pages springer miroslav dudik daniel hsu satyen kale nikos karampatziakis john langford lev reyzin tong zhang efficient optimal learning contextual bandits arxiv preprint elaine fehrman awaz muhammad evgeny mirkes vincent egan alexander gorban five factor model personality evaluation drug consumption risk data science pages springer peter frey david slate letter recognition using adaptive classifiers machine learning tze leung lai herbert robbins asymptotically efficient adaptive allocation rules advances applied mathematics john langford tong zhang algorithm bandits side information advances neural information processing systems pages finnian lattimore tor lattimore mark reid causal bandits learning good interventions via causal inference advances neural information processing systems pages lei dingding wang tao daniel knox balaji padmanabhan scene scalable personalized news recommendation system proceedings international acm sigir conference research development information retrieval pages acm lihong wei chu john langford robert schapire approach personalized news article recommendation proceedings international conference world wide web pages acm lugosi shahar mendelson estimators mean random vector arxiv preprint rajat sen karthikeyan shanmugam alexandros dimakis sanjay shakkottai identifying best interventions online importance sampling proceedings international conference machine learning volume proceedings machine learning research pages international convention centre sydney australia pmlr liang tang romer rosales ajit singh deepak agarwal automatic format selection via contextual bandits proceedings acm international conference conference information knowledge management pages acm cem tekin onur atan mihaela van der schaar discover expert expert selection medical diagnosis ieee transactions emerging topics computing cem tekin jinsung yoon mihaela van der schaar adaptive ensemble learning confidence bounds personalized diagnosis aaai workshop expanding boundaries health informatics using clipped estimator mentioned section motivating equation guiding design estimators equation tells even statistics samples observed governed distribution expert infer mean expert observations made context best arm identification problems suppose observe samples expert guided one might come following naive importance sampled estimator mean expert however possible derive good confidence interval estimator even though reward variable bounded reweighing term unbounded case key idea come robust estimators good variance properties one approach controlling variance estimators clip samples large leads following clipped estimator clipping makes estimator biased however helps controlling variance clipper level depends relationship needs set carefully control shown measure mkj defined bounded good choice log mkj want additive bias theorem idea generalized estimating mean expert observing samples experts leads clipped estimator lemma provides confidence guarantees estimator proof lemma follows theorem include completeness follows abbreviate epj section let proof lemma note lemma follows sake analysis let consider rescaled version written log mkj mkj since every random variable sum bounded log let therefore chernoff bound following chain exp log exp log exp log exp log combine equations obtain exp log prove theorem note experts note throughout proof algorithm defined equations respectively proceed let prove key lemmas first prove high enough probability upper confidence bound estimate optimal expert greater true mean lemma following confidence bound time proof following chain last inequality obtained setting lemma next prove large enough time ucb estimate expert less log lemma following confidence bound time log proof following chain iii log follows fact zkt log definition definition finally concentration bound iii follows lemma note lemmas together imply log proof theorem let log regret algorithm bounded log log log tnx median means estimator median means estimator popular estimating statistics distribution shall see median means based estimator good variance properties divergence assumption bounded proving lemma establishing intermediate results lemma consider quantity variance quantity upper bounded follows var proof following chain var var var apply chebyshev conclude prove lemma proof lemma light probability median within distance bounded bin concludes proof note experts note throughout proof algorithm defined equations respectively proceed let prove key lemmas prove lemmas analogous lemmas lemma following confidence bound time proof follows directly lemma lemma following confidence bound time log proof following chain iii log follows fact concentration bound iii follows lemma log definition finally note lemma together imply log proof theorem let tnx log regret algorithm bounded log log log instance dependent terms section devote attention instance dependent terms theorems first prove corollary proof corollary prove two statements two estimators separately going back lemma proof theorem get log simply follows fact smallest gap therefore chain leading follows new definition log hence regret algorithm estimator bounded follows log log analyze terms alternate manner proof theorem follows regret algorithm clipped estimator bounded using definition obtain log combining equation get desired result theorem immediately implies log median means estimator alternately analyze regret follows proof theorem follows regret algorithm median means estimator bounded using definition obtain log combining equations get desired result work assumption gaps means experts generated according generative model corollary log proof corollary light corollary need prove assume order statistics uniform interval note jensen following let joint pdf given therefore following chain dxdy dadb dadb combining yields log empirical results instance dependent term section provide details empirical results following term term number experts figure plot terms bounds term theorem involving gaps bounds number experts grows stream analytics dataset observed instance dependent term grows much slower pace number experts fact stops increasing certain point training stochastic experts algorithm new experts added starting new batch stochastic experts classifying functions trained using classification oracles data observed far uses ideas key idea reduce classification problem importance weighted classification solved using binary classifiers providing weights samples suppose context observed algorithm chooses expert draws arm conditional distribution suppose reward observed training sample sample weight added dataset training next batch experts shown importance weighing yields good classification experts classifiers provide confidence scores arms given context hence serve stochastic experts different experts added beginning batch three trained xgboost baseclassifier one trained logistic regression diversity maintained among experts added training bootstrapped versions data observed far also selecting different note parameter selection scheme tuned per dataset held fixed three datasets estimating divergence parameters divergence metrics mij estimated data observed far run algorithm divergences depend arm chosen context distribution conditional distributions encoded expert therefore easily estimated data observed suppose contexts observed far interested estimating ispthe divergence estimator would empirical mean note distribution arms nothing confidence scores observed evaluation classifying oracle order robust use median means estimator instead simple empirical mean estimating divergences empirical analysis instance dependent terms section empirically validate instance dependent terms theorem indeed much smaller compared corresponding terms regret bounds even real problem generative assumptions hold order showcase plot term theorem given along corresponding term bounds given number stochastic experts grow stream dataset experiments section true means experts estimated hindsight whole dataset plot shown fig observed term bounds grows much slower pace fact stops increasing number experts certain point
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transactions american mathematical society volume number pages jan generalized algebras discrete complex reflection groups apoorva khare abstract define study generalized algebras associated coxeter groups motivated question coxeter construct first examples algebras usual algebras novel type family call explore several combinatorial properties including coxeter word basis length function series show corresponding generalized coxeter group flat deformation algebras yield symmetric semigroup module categories necessarily monoidal write krein duality motivated bmr freeness conjecture reine angew math define generalized algebras discrete real complex reflection groups finite infinite provide complete classification algebras remarkably turn either usual nilcoxeter algebras algebras proves special case strengthens lack equidimensional analogues finite complex reflection groups particular generic hecke algebras flat deformations complex contents introduction main results background motivation proof theorem tannakian formalism semigroup categories proof theorem distinguished basis words proof theorem primitive elements categorification proof theorem generalized acknowledgments references introduction main results throughout paper denote fixed unital commutative ground ring paper define study generalized algebras associated coxeter groups generally discrete complex reflection groups mathematics subject classification primary secondary key words phrases complex reflection group generalized coxeter group generalized nilcoxeter algebra length function xxxx american mathematical society apoorva khare finite infinite algebras map onto associated graded algebras generic hecke algebras complex reflection groups algebras coxeter groups discuss working algebras allows broader class corresponding reflection groups begin real groups coxeter groups associated hecke algebras play important role representation theory combinatorics mathematical physics group defined coxeter matrix symmetric integer matrix mij finite mii mij artin monoid associated generated modulo braid relations mij precisely mij factors either side artin group generalized braid group group generated relations typically use denote generators three algebras associated matrix group algebra coxeter group algebra algebra also called nilcoxeter algebra nil coxeter algebra nil hecke ring literature free coxeter word basis length function satisfy quadratic relation sense usual algebras generic hecke algebras scalars see hum chapter words unique lengths form monoid together said differently algebras generic hecke algebras graded homogeneous positive degree indeed deg polynomial exponents introduce main objects interest present work generalized coxeter matrices associated algebras always graded definition define generalized coxeter matrix symmetric integer matrix mij finite mij mii fix matrix given integer vector define matrix replacing diagonal generalized coxeter group quotient braid group order relations coxeter matrix remark used familiar artin group associated coxeter group could well written since diagonals matrices play role braid diagram coxeter graph vertices indexed pair vertices mij edges define corresponding generalized algebra follows kbm khti timii timii mij times mij times generalized algebras complex reflection omit braid relation mij given define interested family generalized algebras multiple reasons category theory real reflection groups complex reflection groups deformation theory elaborate motivations section next duality semigroup categories kho representation categories rep used categorify weyl algebra highlight two properties generalized algebras also categorical content choice coproduct bialgebra shown every algebra equipped cocommutative coproduct viewed prism representation categories coproduct equips rep structure symmetric semigroup category note simple module serve unit object whence rep necessarily monoidal natural apply tannakian formalism categories tensor structure record answer surprising formulated unable find literature definition category semigroup category also additive theorem let associative unital algebra field rep veck forgetful functor structure together tensor structure equips coproduct algebra map structure braided respectively symmetric respectively triangular algebra coproduct simply means exists invertible element satisfying hexagon relations triangularity means notice generalized algebras indeed examples triangular algebras cocommutative coproduct counit algebras interesting theory pbw deformations smash product algebras see section also show obtain honest symmetric tensor category algebra via central extension noted theorem sense expected serves act motivation algebras provide concrete examples symmetric nonmonoidal categories novel main results focus algebras real reflection groups novel family nilcoxeter algebras next result constructs novel family generalized nilcoxeter algebras type classifying finitedimensional objects settings algebra combinatorics apoorva khare subject tremendous classical modern interest including weyl coxeter complex reflection groups associated hecke algebras also finite type quivers kleinian singularities correspondence simple lie algebras etc recent setting involves classification nichols algebras prominent ingredients study algebras common present work see ghv details another famous recent classification pointed hopf algebras ansc turn arise generalized small quantum groups motivations goal similarly classify generalized algebras next result presents first novel family examples remark equation generalizing order relations timii also motivated another setting classical work coxeter investigated generalized coxeter matrices group finite specifically coxeter considered type artin braid group instead quotienting relations obtain symmetric group worked spi coxeter interested computing quotient group finite group order showed see also finite case size quotient group coxeter result extended koster thesis completely classify generalized coxeter groups finite turn precisely finite coxeter groups shephard groups parallel classical works wish understand matrices algebra finitely generated coxeter group dim answers known instance marin mar shown algebra finitely generated however apart usual algebras knowledge finitely generated algebras known date following result following coxeter construction type exhibit first family algebras theorem given integers define words generated relations tnd free generators tnk particular subalgebra generated isomorphic usual algebra cal generalized algebras complex reflection remark adopt following notation sequel without reference let denote respective longest elements symmetric group corresponds algebra algebras studied previously begin explore properties specializes usual nilcoxeter algebra type vein present three properties akin usual algebras theorem fix integers algebra length function restricts usual length function theorem tnk unique longest word length field local unique maximal ideal generated ideal nilpotent also study algebra connection khovanov categorification weyl algebra see proposition complex reflection groups bmr freeness conjecture determining algebras strongly motivated study complex reflection groups hecke algebras recall groups enumerated see also coh subsequently popov classified infinite discrete groups generated affine unitary reflections sequel term infinite complex reflection groups groups see mal ors res references therein complex reflection groups important program study generic hecke algebras well associated bmr freeness conjecture malle rouquier see also recent publications mar thesis conjecture connects dimension generic hecke algebra order underlying reflection group study connection corresponding algebras define follows given definition suppose discrete finite infinite complex reflection group together finite generating set complex reflections order relations set braid relations involving words least two distinct reflections infinite complex reflection groups listed mal tables one order relation define infinite complex reflection apoorva khare groups otherwise given integer vector define corresponding generalized algebra khti tidi braid relations replaced corresponding relations alphabet similarly also notion corresponding braid diagram tables mal tables longer always coxeter graph note definition one work specific presentation complex reflection groups canonical minimal set generating reflections see related work known generalized algebra associated finite complex reflection group indeed marin mentions mar key difference real complex reflection groups lack algebras dimension precisely latter verified cases complex reflection groups loc cit final result shows assertion fact stronger statement discrete finite infinite real complex reflection groups even stronger priori provide complete classification generalized algebras groups notice theorem suffices consider groups whose braid diagram connected theorem suppose irreducible discrete real complex reflection group words real reflection group connected braid diagram complex reflection group connected braid diagram presentation given tables mal tables table also fix integer vector including possibly additional order relation mal following equivalent generalized algebra finitely generated kmodule either finite coxeter group type ideal generated nilpotent assertions hold exists length function unique longest element say length words examples field usual algebras algebras note also results key tool proving theorems diagrammatic calculus akin crystal theory combinatorics quantum groups questions organization paper knowledge algebras novel construction light theorem generalized algebras usual ones particular exploration properties warranted conclude section discussing directions generalized algebras complex reflection algebras related flag varieties bgg categorification kho symmetric function theory bss also recall divided difference operator representation usual type algebra used define schubert polynomials polynomials simultaneously annihilate precisely symmetric polynomials interesting determine similar natural representation operators polynomial ring consider polynomials one obtains analogously see mar related calculation observe algebra come finite reflection group larger dimension corresponding generalized coxeter group equation given connection coxeter groups well crystal methods used interesting explore algebras connected crystals lie super algebra proof theorem involves argument running discrete complex reflection groups proof result would desirable paper organized follows section elaborate motivations make additional remarks following four sections prove turn four main theorems background motivation section elaborate aforementioned motivations studying generalized algebras first algebras interesting categorical perspective module categories symmetric categories see definition monoidal discuss next section duality categories well central extension symmetric tensor category second motivation comes real reflection groups provide novel family algebras type akin work coxeter koster context remarkable theorem algebras usual algebras examples theorem shows algebras similar usual analogues note however algebras also differ key aspects see theorem proposition show particular multiple maximal words words killed every generator fundamental difference arises considerations flat deformations make precise remarks around equation third motivation comes complex reflection groups much recent interest bmr freeness conjecture discusses equality dimensions generic hecke algebras group algebra underlying finite complex reflection group paper study associated graded algebra deformation parameters set zero shown marin mar cases reflection groups come equipped analogues make precise strong way apoorva khare theorem complex reflection groups particular theorem shows generic hecke algebras flat deformations underlying associated graded analogues complex property shared algebras algebras coxeter groups flat deformations indeed denotes generalized coxeter matrix corresponding claim even dim odd see mij odd generalized coxeter matrix conjugate whence sgi sgj gcd hand surjects onto algebra coprime say generates trivial subgroup vanish generic hecke algebras discussed fit broader framework deformation theory provides fourth motivation behind paper addition question flatness discussed theory deformations associative algebras area sustained activity subsumes drinfeld algebras graded affine hecke algebras symplectic reflection algebras rational cherednik algebras infinitesimal hecke algebras programs literature also highlight program shepler witherspoon see references therein settings bialgebra usually hopf algebra acts vector space hence quotient tensor algebra one characterizes deformations algebra flat also termed pbw deformations regard significance generalized algebras manifold first bialgebra settings extended recent work kha framework cocommutative algebras also include algebras moreover characterized pbw deformations sym thereby extending loc cit pbw theorems previously mentioned works significance framework incorporating along previously studied algebras full structure specifically antipode even counit required order characterize flat deformations sym coming shown program shepler witherspoon see kha algebra coproduct field possible characterize graded sym whose fiber pbw property consideration directly motivates classification result theorem conclude third connection aforementioned active program pbw deformations studied kha case local setting nilpotent ideal one obtains lot information deformations sym including understanding pbw deformations well center abelianization generalized algebras complex reflection modules especially simple modules generated explains interest understanding nilpotent theorem shows condition fact equivalent generalized nilcoxeter algebra proof theorem tannakian formalism semigroup categories remainder paper devoted proving four main theorems opening section begin studying representation category generalized coxeter matrix first assertion category never monoidal category characteristic zero follows following result proposition suppose field characteristic zero generalized coxeter matrix bialgebra result fails hold positive characteristic indeed prime algebra bialgebra coproduct counit proof note unique possible counit suppose setting ker ideal generated follows note constitutes terms higher total degree multiplicative raising mii power yields mii mii tik timii higher degree terms impossible long image nonzero assuming follows multiplicative hence coproduct finally surjects onto usual algebra coxeter word basis indexed follows indeed nonzero consequence proposition tannakian formalism theorem generalized coxeter matrix module category rep necessarily tensor category said map coproduct coassociative algebra map cocommutativity implies rep symmetric semigroup category outline show first theorem seeks understand duality categories possibly without unit objects proof theorem proof part follows theorem one ignores last statement proof additional data required two braided versions part deduced proof proposition apoorva khare conclude section passing rep honest tensor category say field alternately via tannakian formalism theorem produce bialgebra surjects onto namely generated additional generator subject braid relations former set well timii note longer central extension asking grouplike yields unique bialgebra structure hence monoidal category structure rep claimed proof theorem distinguished basis words prove main theorems algebras specifically family beginning theorem note algebra usual algebra algebra theorems easily verified cases using hum chapter thus assume throughout proofs begin showing notice spanned words claim word either zero equal braid relations word occurrences successive monomial tnk show claim consider word tna tnb tik word rewrite using braid relations required minimal length say may assume else would done using braid relations assume otherwise factors may moved past using braid relations similarly thus assume minimality next claim following relation holds artin braid group hence shown descending induction hence tna tnb tna tnb max claim follows last expression contains substring similarly shows claim generalized algebras complex reflection prove upper bound notice generate subalgebra relations satisfied hence map algebra map notice equation every nonzero word form tnk hence similar reasoning assuming minimal length may rewrite carrying operation yields tnk reduced word nonzero thus tnk generators follows generators shows desired upper bound hard part proof involves showing words tnk form require following technical lemma symmetric group algebra proof included completeness lemma suppose symmetric group simple reflections labelled usual every element written reduced form unique given element usual algebra otherwise note equation thought statement lengths symmetric group proof first claim reduced expression occurs exactly proof induction clearly claim true given claim consider reduced expression contains induction hypothesis hence braid relations equals reduced expression one less occurrence similar analysis works repeatedly carrying procedure proves claim prove uniqueness lemma previous paragraph write smallest possible length say sik using braid relations clearly hence minimality choose smallest produce contradiction assuming integer exists may move sil past preceding terms contradicting minimality clearly else apoorva khare reduced thus whence form sil sil sil verify sil sil sil sil sil sil sil sil sil sil sil sil contradicts minimality thus integer exist proves next claim integer unique wan first make reduced wsn see first recall hum lemma corollary together imply finite coxeter group simple root follows applying result successively wsn next define follows hence shows wan proves uniqueness claim write reduced form obtain also unique remains show equation using analysis write since commutes may assume first suppose suffices prove without loss generality may work subalgebra generated hence suppose prove induction clear next suppose may suppose prove result induction base case easy thus compute remark notice equation holds algebra containing elements satisfy braid relations particular holds returning proof theorem introduce diagrammatic calculus akin crystal theory first write case order provide intuition case general let free basis given nodes graph figure figure node thought applied unit generating basis vector corresponding similarly nodes arrows denote action remaining generalized algebras complex reflection figure regular representation generator actions nodes yield zero one verifies inspection defining relations satisfied action therefore since generated basis vector corresponding node surjection sends corresponding basis vectors free result follows upper bound proved strategy similar general uses following detailed notation let denote corresponding welldefined word alphabet let denote subalgebra generated letters define free basis elements set words observe basis vectors thought corresponding respectively words tnk definition expression word form said standard form define structure via defining directed graph structure precisely describe following figure figure may help visualizing structure figure thought analogous central hexagon either two arms figure begin explaining figure node wkm corresponds basis vector notice vectors bijection coxeter word basis usual algebra let denote span given define span basis elements note special case figure central hexagon spans nodes span span define apoorva khare figure regular representation let denote distinguished basis bijection holds equip spaces corresponding module structure usual algebra type structure uniquely determined given sil set next define action via lemma write unique using previous paragraph follows correspondingly define hand suppose define otherwise define lemma see equation otherwise remains show graph structure indeed defines module structure similar argument case completes proof following argument occasionally use lemma well remark without reference first notice algebra relations involving clearly satisfied module construction verify relations involving hold notice via figure partitioned three subsets recall opening remarks section first show relation tnd holds equality linear operators vector hence separately consider cases generalized algebras complex reflection lies top rows figure easily verified killed desired reasoning shows tnd kills let relation holds since correspond vectors lie top rows figure finally let thus write lemma follows remark tnd next show relation holds consider three cases fix verify using aforementioned action equal otherwise similarly let compute since follows done since else note whence finally let write lemma first suppose hard show vanish otherwise terms equal similar analysis shows otherwise next show braid relation holds involved computation carry consider three cases fix easily verified sides braid relation kill instead compute side first notice hence braid relation holds last equality follows definition thus suffices case verify braid relation holds done considering following four commutes easily seen equal apoorva khare suppose using remark structure compute whence done since commutes last two previous two first suppose similar computations one verifies vanish finally suppose one verifies next suppose form definition show kills consider two done otherwise suppose compute using remark relations verified penultimate equality uses finally suppose analysis first case need consider hard show vanish hand repeated use remark equation shows generalized algebras complex reflection notice calculation shows action strings type similarly one shows verifies last braid relation holds last case thus algebra relations hold making generated claimed particular analysis first part proof completes proof last assertion theorem finally algebra cal surjects onto free hence cal desired proof theorem primitive elements categorification section continue study algebras starting theorem proof theorem retain notation theorem via isomorphism identify basis element tnk let equation claim til nonzero word precisely length expressed uniquely standard form proof induction already standard form nonzero given word length length satisfies claim write via induction hypothesis word standard form length proof theorem shows applying standard form either yields zero length precisely proves claim analysis shows suppose field algebra maximal ideal fact proof theorem moreover local element invertible particular one understands representations algebra kha aforementioned claim also proves nonzero word expressed standard form without changing length immediate consequence corollary field graded degree series polynomial proof also uses standard result series usual algebra see hum next discuss property explored kho usual algebras algebras always frobenius study apoorva khare algebras also frobenius following result shows happens degenerate case theorem suppose field given algebra frobenius one checks via equation conditions equivalent group algebra generalized coxeter group flat deformation proof theorem crucially uses knowledge maximal primitive words algebra formally given generalized coxeter matrix say element left respectively right primitive respectively theorem primitive denote sets elements respectively priml primr prim proposition every generalized algebra equipped fixes generator isomorphism priml primr moreover following hold finite coxeter group unique longest word priml primr prim priml primr prim priml spanned words tnk prim spanned words tnk cases map fixes prim well lengths nonzero words proof first two statements obvious since preserves defining relations assertion standard see hum chapter easily verified next classify elements suppose tnk clearly since discussed proof theorem similarly also computed proof theorem hence complete proof suffices show nonzero linear combination remaining words form tnk suppose first word coefficient nonzero case choose element smallest length linear combination discussed proof theorem kills terms tnk moreover hum chapter generalized algebras complex reflection also kills terms length thus left linear combination case words linear combination form tnk choose smallest length corresponding word nonzero coefficient yields nonzero linear combination analysis theorem proves assertion next identify primitive elements first claim tnk fixed indeed compute using braid relations type fixes hence tnk tnk using claim indeed tnk compute tnk claim linear combination remaining elements listed indeed let denote minimum mvalues words nonzero coefficients analysis kills elements since commutes hence taken past multiply killed terms equal killed analysis theorem follows linear combination rightprimitive completes classification primitive elements next prim fixed shown equation moreover equals finite equipped suitable length function preserves length algebra relations preserved remark light proposition natural ask write rightprimitive words standard form generally given unique via lemma tnk twe tnk proposition hand turn frobenius property following proof reveals frobenius prim proof theorem finite coxeter groups corresponding nilcoxeter algebras indeed frobenius see kho also easy verify frobenius using symmetric bilinear form uniquely specified thus remains show algebra frobenius indeed frobenius nondegenerate invariant bilinear form nonzero primitive exists vector pap follows may take linear functional prim nonsingular whence dimk prim thus proposition apoorva khare conclude section discussing connection categorification khovanov kho weyl algebra zhx namely usual type algebra bimodule structure studied loc leading construction tensor functors categorifying operators explain algebra fits framework proposition isomorphism bimodules result shown kho proposition general using notation kho result implies category algebra corresponds including previously known case particular proposition strengthens theorems explained left structure namely free rank proof proposition proof theorem algebra regular representation sending also recall subspaces defined discussion following equation theorem free left rank one also free right rank one using proposition remark fact uniqueness standard form shown proof theorem implies map atnk isomorphism result follows proof theorem remark notice proof proposition also categorifies corollary proof theorem generalized algebras prove theorem classifies generalized algebras finite bulk proof involves showing employ diagrammatic calculus used show theorem applied five diagrams figure begin assuming generalized coxeter group classify algebras finite following classification address remaining finite complex reflection groups followed infinite discrete complex reflection groups presentations case suppose mii case coxeter group hum chapter bijection must therefore finite generalized algebras complex reflection fig fig fig fig fig figure modules generalized nilcoxeter algebras case suppose case appeal figure work proof mar proposition thus fix free basis given countable set define via figure namely kills basis vectors following exceptions apoorva khare head arrow refers precisely index increasing easy verify defining relations hold endk hold therefore module generated finitely generated also finitely generated approach used remainder proof obtain diagrams figure thus mention figure corresponding cases case figure actually special case figure included demonstrate simpler case suppose generally two nodes since coxeter graph connected exist nodes figure write path successive pair nodes connected least single edge appeal figure define structure free spank bmr bmr kills basis vectors except actions obtained figure generated proceed show finitely generated case previous cases reduce situation unique vertex coxeter graph next two steps show adjacent unique node first suppose adjacent appeal figure setting define module structure spank proceed case next suppose adjacent coxeter graph two nodes previous case appeal figure define structure spank proceed previous cases observe finitely generated corresponds coxeter group hence coxeter graph finite type graphs classified coxeter rule cases type case analysis shows case first notice dihedral types types already ruled cases cases also rule one possibility types may set remaining cases types assume coxeter graph labelled case construct appealing figure proceed case next case type notice extremal pendant vertex analysis first assume extremal node generalized algebras complex reflection long arm coxeter graph appeal figure construct one two extremal nodes define quotient algebra whose coxeter graph type kill generators long arm furthest away repeat construction previous paragraph using figure easy verify space module hence algebra allows proceed previous show finitely generated whence neither case coxeter graph type may reduce analysis previous case hence follows using figure finitely generated case coxeter graph type may reduce analysis case follows case finitely generated completes classification generalized coxeter groups appeal classification presentation finite complex reflection groups whose coxeter graph connected groups presentations listed tables follows adopt following notation corresponding generalized algebras denoted similarly work follows often claim finitely generated omitting phrase unless usual algebra finite coxeter group case exceptional types finite coxeter graph coxeter graph type case addressed thus possibility finite rank equals desired next coxeter graph type also addressed never yields algebra finite suppose form type whence quotient algebra generated finitely generated arguments case follows also finitely generated next case case coxeter graph dihedral type also addressed case exceptional types remaining exceptional values finite coxeter group appeal figure three cases first suppose case set figure define basis proceed next set figure define proceed finally fix case use figure define proceed apoorva khare case infinite families remains consider six infinite families enumerated make family three families consist finite coxeter groups types considered consider three families suppose table consider quotient algebra generated killing generators generators satisfy relations tsds ttdt tudu times times thus use figure define structure proceed show finitely generated suppose see table apply similar argument previous using generators figure suppose finite coxeter group hence addressed next killing reduces quotient addressed finally suppose setting generators satisfy tsds ttdt tudu times times use figure define structure proceed completes proof finite complex reflection groups next hum chapter infinite coxeter group finitely generated whence result holds use classification remaining infinite complex reflection groups associated connected braid diagram groups described subsequently mal thus exists complex affine space group translations choosing basepoint identify semidirect product group affine transformations moreover define lin image factor group tran subset tran lin tran remains consider three cases irreducible infinite complex reflection groups case group noncrystallographic compact theorem exists real form whose complexification moreover theorem restricting elements yields affine weyl group hence finitely generated cwr impossible generalized algebras complex reflection case group genuine crystallographic group compact lin complexification real reflection group groups studied malle mal presentations groups provided tables loc cit specifically malle showed groups quotients free monoid set braid relations order relations together one additional order relation show none groups algebra defined definition finitely generated proceed specifying figure corresponds groups three suppose group mal table mal table groups appeal figure proceed case notice suffices show claim given coxeter graph nodes labelled clockwise fashion corresponding algebra finitely generated construct module using figure proceeding shows claim hence result figure module cafn remaining cases mal tables appeal figure case three suitably chosen generators case case finally consider remaining nongenuine crystallographic cases table thus compact lin complexification real reflection group cases verify inspection table cocycle always trivial thus lin ntran lin finite weyl group tran lattice rank indexes simple reflections weyl group claim corresponding family generalized algebras finitely generated show claim requires presentation terms generating reflections following recipe presentation communicated popov notice table tran direct sum two lin lattices rank thus lin finite real reflection group moreover semidirect product real crystallographic reflection group whose fundamental domain simplex yields presentation via generating reflections faces simplex one combines presentations obtain system generators apoorva khare see context remarks following mal theorem setting follows theorem isomorphic real reflection group since coxeter type determined affine weyl group coxeter types thus sense double affine weyl group also easy verify inspection table isomorphic root lattice whence equipped presentation analyze follows fix isomorphism choose affine reflections corresponding respectively together upon simple reflections generate quotienting relation using presentation via corresponding generators last term affine weyl algebra hence finitely generated therefore neither desired shows converse follows hum chapter theorem show equivalent note analysis finitely generated either surjects onto affine weyl algebra one define module exists word twr expressed using generators sends vector follows cases nilpotent next finite coxeter group see hum chapter nilpotent finally nilpotent theorem shows final statement length function longest element also follows hum chapter theorem remark generalized coxeter matrix mij similarly work kill follows infinite acknowledgments author grateful ivan marin vladimir popov victor reiner informative stimulating correspondences james humphreys carefully going earlier draft providing valuable feedback author also thanks daniel bump gunter malle eric rowell travis scrimshaw bruce westbury useful references discussions author partially supported young investigator award infosys foundation references ansc andruskiewitsch schneider classification finitedimensional pointed hopf algebras annals mathematics joachim assion proof theorem coxeter math acad sci canada generalized algebras complex reflection tathagata basak coxeter diagrams complex reflection groups transactions american mathematical society bss chris berg franco saliola luis serrano pieri operators affine nilcoxeter algebra transactions american mathematical society bgg joseph bernstein israel gelfand sergei gelfand schubert cells cohomology spaces russian mathematical surveys joseph bernstein ossip schwarzman complex crystallographic coxeter groups affine root systems journal nonlinear mathematical physics david bessis zariski theorems diagrams braid groups inventiones mathematicae michel gunter malle rouquier complex reflection groups associated braid groups representations groups banff cms conference proceedings american mathematical society providence michel gunter malle rouquier complex reflection groups braid groups hecke algebras journal die reine und angewandte mathematik roger carter representation theory algebra journal algebra eirini chavli conjecture exceptional groups rank thesis coh arjeh cohen finite complex reflection groups annales scientifiques normale harold coxeter discrete groups generated reflections annals mathematics harold coxeter complete enumeration finite groups form kij journal london mathematical society harold coxeter factor groups braid group proceedings canadian mathematical congress banff university toronto press vladimir drinfeld degenerate affine hecke algebras yangians functional analysis applications pavel etingof proof conjecture characteristic zero losev pfeiffer arnold mathematical journal pavel etingof victor ginzburg symplectic reflection algebras space deformed homomorphism inventiones mathematicae pavel etingof olivier schiffmann lectures quantum groups lectures mathematical physics international press somerville sergey fomin richard stanley schubert polynomials algebra advances mathematics ghv heckenberger leandro vendramin nichols algebras group type many quadratic relations advances mathematics heckenberger leandro vendramin classification nichols algebras modules groups journal european mathematical society heckenberger leandro vendramin classification nichols algebras groups finite root system rank two journal european mathematical society mervyn hughes complex reflection groups communications algebra mervyn hughes extended root graphs complex reflection groups communications algebra hum james humphreys reflection groups coxeter groups cambridge studies advanced mathematics cambridge university press yorkmelbourne kha apoorva khare generalized algebras cocommutative algebras pbw property ams contemporary mathematics kho mikhail khovanov nilcoxeter algebras categorify weyl algebra communications algebra mal mar ors res apoorva khare mikhail khovanov aaron lauda diagrammatic approach categorification quantum groups representation theory bertram kostant shrawan kumar nil hecke ring cohomology group advances mathematics david koster complex reflection groups thesis university wisconsin alain lascoux des schubert invariant theory fossum haboush hochster lakshmibai eds contemporary mathematics volume pages american mathematical society providence gustav lehrer donld taylor unitary reflection groups australian mathematical society lecture series volume cambridge university press cambridge ivan losev quotients hecke algebras algebra number theory george lusztig affine hecke algebras graded version journal american mathematical society gunter malle presentations crystallographic complex reflection groups transformation groups ivan marin freeness conjecture hecke algebras complex reflection groups case hessian group journal pure applied algebra ivan marin pfeiffer bmr freeness conjecture groups mathematics computation published online norton algebras journal australian mathematical society peter orlik victor reiner anne shepler sign representation shephard groups mathematische annalen vladimir popov discrete complex reflection groups communications mathematical institute vol rijksuniversiteit utrecht vladimir popov personal communication victor reiner anne shepler invariant derivations differential forms reflection groups preprint geoffrey shephard john todd finite unitary reflection groups canadian journal mathematics anne shepler sarah witherspoon theorem quadratic algebras group actions transactions american mathematical society anne shepler sarah witherspoon theorems commutative algebra noncommutative algebraic geometry volume expository articles eisenbud iyengar singh stafford van den bergh eds mathematical sciences research institute proceedings volume pages cambridge university press cambridge department mathematics indian institute science bangalore india address khare
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feudal reinforcement learning dialogue management large domains stefan lina milica department engineering university cambridge polyai limited london mar abstract reinforcement learning promising approach solve dialogue policy optimisation traditional algorithms however fail scale large domains due curse dimensionality propose novel dialogue management architecture based feudal decomposes decision two steps first step master policy selects subset primitive actions second step primitive action chosen selected subset structural information included domain ontology used abstract dialogue state space taking decisions step using different parts abstracted state combined information sharing mechanism slots increases scalability large domains show implementation approach based networks significantly outperforms previous state art several dialogue domains environments without need additional reward signal introduction spoken dialogue systems sds form personal assistants recently gained much attention academia industry one important modules sds dialogue manager policy module charge deciding next action dialogue turn reinforcement learning sutton barto studied several years promising approach model dialogue management levin henderson pietquin young casanueva however dialogue state space increases number possible trajectories needed explored grows exponentially making traditional methods scalable large domains hierarchical hrl form temporal abstraction proposed order mitigate problem budzianowski peng however proposed hrl methods require task defined hierarchical structure usually handcrafted addition usually require additional rewards subtask space abstraction instead successfully applied dialogue tasks dialogue state tracking dst henderson policy transfer domains wang dst set binary classifiers defined slot shared parameters learning general way track slots policy transfer method presented wang named domain independent parametrisation dip transforms belief state fixed size representation using handcrafted feature function idea could also applied large domains since used learn general way act slot dialogues hrl method relies space abstraction feudal frl dayan hinton allow scale domains large number slots frl divides task spatially rather temporally decomposing decisions several steps using different abstraction levels framework especially useful tasks large discrete action spaces making attractive large domain dialogue management paper introduce feudal dialogue policy decomposes decision turn two steps first step policy decides takes slot independent slot dependent action state slot abstracted account features related slot primitive action chosen previously selected subset model require modification reward function hierarchical architecture fully specified structured database representation system ontology requiring additional design main belief state discount factor objective find optimal policy policy maximizes expected return belief state algorithms optimal policy found greedily taking action maximises sdss belief state space defined ontology structured representation database entities user retrieve talking system entity set properties refereed slots slots take value set belief state defined concatenation probability distribution slot plus set general features communication function used user database search method henderson set defined set summary actions actions either slot dependent request food confirm area slot hello inform belief space defined ontology therefore belief states different domains different shapes order transfer knowledge domains domain independent parametrization dip wang proposes abstract belief state fixed include summary actions dependent slots inform group argmax master actions argmax slot independent primitives background dialogue management cast continuous mdp young composed continuous multivariate belief state space finite set actions reward function given time agent observes belief state executes action receives reward drawn action taken decided policy defined function policy function defined expected discounted return starting state taking action following policy end dialogue time step argmax slots slot dependent primitives figure feudal dialogue architecture used work surrounded dashed line shared parameters simple lines show data flow double lines decisions size representation action either slot independent dependent slot feature function defined stands slot independent actions therefore order compute policy approximated slot associated action wang presents handcrafted feature function includes slot independent features belief state summarised representation joint belief state summarised representation belief state slot section gives detailed description function used work feudal dialogue management frl decomposes policy decision turn several using different abstracted parts belief state subdecision objective task oriented sds fulfill users goal goal observable sds sds needs gather enough information correctly fulfill therefore turn decompose decision two steps first decide taking action order gather information user goal information gathering actions taking action fulfill user goal part information providing actions second select primitive action execute previously selected subset dialogue set information gathering actions defined set slot dependent actions set information providing actions defined remaining actions architecture feudal policy proposed work represented schematically figure primitive actions divided two subsets slot independent actions hello inform slot dependent actions request confirm addition set master actions defined corresponds taking action taking action feature function defined slot well slot independent feature function master feature function feature functions handcrafted dip feature function introduced section function approximator used neural networks trained jointly policy finally master policy slot independent policy set slot specific policies one defined contrary feudal policies slot specific shared parameters order generalise slots following idea used henderson dst differences slots size value distribution accounted feature function therefore defined argmax run argmax else selected policy runs slot specific policy choosing pair maximises function slot argmax summary action constructed joining summary action request food feudal dialogue policy algorithm given appendix domain cambridge restaurants san francisco restaurants laptops ser masks user env std code sfr lap env std constraint slots env std env std requests values env unf env std table sumarised description domains environments used experiments refer casanueva detailed description experimental setup models used experiments implemented using pydial toolkit ultes evaluated pydial benchmarking environment casanueva environment presents set tasks span different size domains different semantic error rates ser different configurations action masks user model parameters standard std unfriendly table shows summarised description tasks models developed paper compared algorithms handcrafted policy presented benchmarks baseline implementation dip based networks dqn mnih implemented additional baseline papangelis stylianou policy named uses hyperparameters dqn implementation released pydial benchmarks dip feature function based description wang used accounts general features belief state database search method accounts features joint belief state entropy joint belief accounts features marginal distribution slot entropy appendix shows detailed description dip features used work feudal dqn policy feudal policy based architecture described sec implemented named fdqn constructed dqn policy note actions set composed communication function slot dependent actions thus reducing number actions compared implementation models obtained env env env env env env task sfr lap sfr lap sfr lap sfr lap sfr lap sfr lap suc rew suc rew bnch rew hdc rew table success rate reward benchmarking tasks compared reward best performing algorithm task bnch handcrafted policy hdc presented casanueva policies hyperparameters baseline dqn implementation except two hidden layer sizes reduced respectively feature functions subsets dip features used original set summary actions benchmarking environment size number slots set divided two size size including trained sparse reward signal used baselines getting reward dialogue successful otherwise minus dialogue length results results tasks benchmarking environment training dialogues presented table evaluation procedure benchmarks used presenting mean different random seeds testing every seed dialogues fdqn policy substantially outperforms every policy environments except env figure learning curves dipdqn env compared two best performing algorithms casanueva dqn gpsarsa shaded area depicts mean standard deviation ten random seeds performance increase considerable two largest domains sfr lap gains points accumulated reward challenging environments env lap compared best benchmarked policies addition fdqn consistently outperforms handcrafted policy hdc environments traditional methods could achieve env however results fdqn rather low specially surprisingly results env differs env absence action masks thus principle complex environment outperform every algorithm analysing dialogues individually could observe environment policies prone overfit action performance fdqn env also better env difference environments also lies masks suggests specific action mask design helpful algorithms harm performance others especially severe case shows good performance challenging environments unstable prone overfit fdqn however main purpose action masks reduce number dialogues needed train policy observing learning curves shown figure fdqn model learn nearoptimal policy large domains dialogues even additional reward used making action masks unnecessary additional pass action added subset taken whenever executed simplifies training algorithm model overestimates value incorrect action continuously repeating user runs patience conclusions future work presented novel dialogue management architecture based feudal substantially outperforms previous state art several dialogue environments defining set slot dependent policies shared parameters model able learn general way act slots increasing scalability large domains unlike hrl methods applied dialogue additional reward signals needed hierarchical structure derived flat ontology substantially reducing design effort promising approach would substitute handcrafted feature functions used work neural feature extractors trained jointly policy would avoid need design feature functions could potentially extended modules sds making learning tractable addition single model potentially used different domains papangelis stylianou different feudal architectures could make larger action spaces tractable adding third subpolicy deal actions dependent slots acknowledgments research funded epsrc grant open domain statistical spoken dialogue systems references budzianowski stefan ultes nikola wen inigo casanueva lina rojas barahona milica subdomain modelling dialogue management hierarchical reinforcement learning proc sigdial casanueva budzianowski nikola wen stefan ultes lina steve young milica benchmarking environment reinforcement learning based task oriented dialogue management arxiv preprint inigo casanueva thomas hain heidi christensen ricard marxer phil green knowledge transfer speakers personalised dialogue management proceedings annual meeting special interest group discourse dialogue pages heriberto steve renals oliver lemon hiroshi shimodaira evaluation chical reinforcement learning spoken dialogue system computer speech language heriberto seunghak ashley williamson jacob carse deep reinforcement learning dialogue systems nips workkshop peter dayan geoffrey hinton feudal reinforcement learning advances neural information processing systems pages milica catherine breslin matthew henderson dongho kim martin szummer blaise thomson pirros tsiakoulis steve young pomdpbased dialogue manager adaptation extended domains proceedings sigdial conference milica nikola david vandyke wen steve young policy committee adaptation multidomain spoken dialogue systems automatic speech recognition understanding asru ieee workshop ieee pages james henderson oliver lemon kallirroi georgila hybrid learning dialogue policies fixed data sets computational linguistics henderson thomson williams second dialog state tracking challenge proc sigdial henderson thomson young dialog state tracking recurrent neural networks proc sigdial esther levin roberto pieraccini wieland eckert using markov decision process learning dialogue strategies acoustics speech signal processing proceedings ieee international conference ieee volume pages volodymyr mnih koray kavukcuoglu david silver alex graves ioannis antonoglou daan wierstra martin riedmiller playing atari deep reinforcement learning arxiv preprint alexandros papangelis yannis stylianou dialogue management deep learning international workshop spoken dialogue systems peng gao celikyilmaz lee wong composite dialogue system via hierarchical deep reinforcement learning arxiv olivier pietquin matthieu geist senthilkumar chandramohan sampleefficient batch reinforcement learning dialogue management optimization acm transactions speech language processing tslp pawel budzianowski stefan ultes milica gasic steve young reinforcement learning supervised data dialogue management proceedings sigdial milica gasic nikola mrksic lina rojasbarahona stefan ultes david vandyke tsunghsien wen steve young continuously learning neural dialogue management arxiv preprint richard sutton andrew barto reinforcement learning introduction mit press stefan ultes lina david vandyke dongho kim casanueva budzianowski nikola wen milica steve young pydial statistical dialogue system toolkit acl demo association computational linguistics dip features section gives detailed description dip feature functions used work differences features used wang papangelis stylianou following priority importance features used potential contribution search features used joint belief features extended account aspects feature function feature description feature size last user dialogue act bin search method bin requested slots bin offer happened last action inform venue normalised slots slots normalised avg slot length values prob top values prob none value entropy diff top value probs bin slots top value none bin prob top values prob none value diff top value probs bin entropy values prob bin normalised slot length values slot length bin entropy distr values zhuoran wang wen yannis stylianou learning domainindependent dialogue policies via ontology parameterisation sigdial conference pages steve young milica blaise thomson jason williams statistical spoken dialog systems review proceedings ieee feudal dialogue policy algorithm algorithm feudal dialogue policy dialogue turn observe argmax total argmax drop else drop slot act argmax join slot act end execute end table list features composing dip features tag bin denotes binary encoding used feature joint features extracted joint belief computed cartesian product beliefs individual slots denotes features exist original belief state
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identification dynamical systems consistency fisher information jul jeremie sumeetpal ajay abstract learning model parameters dynamical system partial perturbed observations challenging task despite recent numerical advancements learning parameters theoretical guarantees extremely scarce article study identifiability parameters consistency corresponding maximum likelihood estimate mle assumptions different components underlying system order understand impact various sources observation noise ability learn model parameters study asymptotic variance mle associated fisher information matrix example show specific aspects tracking mtt problem detection failures unknown data association lead loss information quantified special cases interest key words identifiability consistency fisher information ams subject classifications introduction dynamical system comprised unknown randomly varying number objects partially observed markov process tracking refers problem estimating state objects noisy observations also corrupted detection failures false detections false alarms type problem arises many different fields systems biology robotics computer vision surveillance different formulations tracking exist including extensions approach multiple targets well formulations based simple point processes one main challenges tracking uncertainty data association refers problem finding right pairing targets recorded observations time task confounded corruption observations false positives detection failures inferentially multitarget tracking notoriously difficult solve involves exponentially growing numbers possible configurations data association past decade significant advancements towards practical solutions inference problem include solutions based sequential monte carlo smc hierarchical smc gaussian mixtures article mtt observation model motion model constituent individual targets assumed unknown instead parameterised inferred data although tracking active research field decades questions concerning identifiability consistency corresponding model parameter estimates received appropriate attention paper aim address gap shed light issue building results literature markov processes see prove identifiability consistency mle mtt model parameters theorem specifically constituent target mtt model partially observed markov process theorem show identifiability transfers dsap national university singapore email stahje engineering university cambridge alan turing institute email dsap national university singapore email staja department houssineau singh jasra single multiple targets appropriate assumptions practical implications results regarding identifiability include understanding behaviour markov chain monte carlo mcmc techniques tracking conditioned likelihood ratio correct parameter value possible values consistency maximum likelihood estimator raises question asymptotic normality corresponding variance turns motivates study fisher information matrix class problems demonstrated theorem strict loss information presence data association uncertainty detection failures characterise fisher information precisely specific illustrated cases show increasing number targets gain fisher information model parameters common targets large uncertainties origin corresponding observations persist see subsection fisher information matrix useful applications sensor management aims optimising position sensor finding best ratio probability false alarm probability detection proof identifiability mtt model well approach studying asymptotic variance mle mtt model parameters original best knowledge first kind consistency data association problem mtt studied context estimation multiple splitting merging targets observed without noise fixed time interval observations multiple targets made discrete times result limited case number observation tends infinity effectively amounts saying targets observed infinitely many times fixed interval scenario typically encountered practice case theoretical results proof techniques entirely different pertain mtt model parameters data association theoretical studies mtt also conducted stability specific inference methods structure article follows introducing required notations background concepts section section consistency maximum likelihood estimator established along asymptotic normality large class systems section finally order better understand effect various parameters asymptotic variance fisher information matrix computed important special cases systems section article concludes section notations random variables defined probability space expectation random variable probability measure denoted probability densities denoted letters probability measures denoted capital letter similarly random variables denoted capital letters whereas realisations time indexed set positive integers every time finite sequence observation points observation space made available space assumed subset euclidean space sequences observations form denoted standard formulation tracking one observation associated given object given time step conversely observations originated one object identification dynamical systems objects states modelled elements set assumed subset euclidean space usually satisfying propagated independently according markov kernel density state space depends parameter compact set densities defined reference measure true value parameter denoted random variable describing state time depends state time follows transition depend time associated markov chain said homogeneous observation process time given state modelled likelihood function also parametrised observation time independent states observations times process usually referred hidden markov model hmm law parameter denoted initialised stationary distribution assuming exists initialised background definition specific properties markov chains used following sections given completeness let markov chain transition density let probability measure cylinder characterising chain initialised point also let return time set defined inf consider following concepts set said accessible positive probability markov chain said exists density subset implies accessible set said harris recurrent event happens almost surely markov chain said harris recurrent accessible set harris recurrent density called invariant holds markov chain called positive admits invariant probability density details notions expressed formulation found concepts useful considering behaviour markov chains involved tracking problems consistency maximum likelihood estimator tracking model throughout section true number objects considered system assumed fixed denoted consider markov chain components independently evolving via markov transition observations time gathered vector space notation set containing empty sequence observation superposition houssineau singh jasra independent observation components via likelihood followed bernoulli thinning parameter corresponding detection failure false alarms clutter generated independently observations assumed come process whose cardinality time poisson parameter common distribution depends parameter compact set true value denoted number objects assumed known also considered parameter model parameter model defined vector transposition compact subsets respectively standing target clutter respectively true parameter assumed interior point special parameter sets subsets also introduced fixing one several parameters special values instance correspond respectively cases parameters known values outside domain definition alternatively value parameter known inside domain definition known corresponding hyperplane expressed although poisson distribution defined parameter parameter value simply assumed represent case false alarm markov transition associated process simply expressed likelihood however takes sophisticated form additional notations required let sym symmetric group letters uniform distribution sym also let distribution characterised variable target detected likelihood observations time given state characterised umt denotes poisson distribution parameter number detected targets ith detected target integer verifying choice likelihood corresponds marginalisation data association note number identification dynamical systems observations denoted strictly greater law joint markov chain parameter denoted initialised stationary distribution assumed start state corresponding densities written accordingly letters objective study ratio assumptions considered purpose detailed next section assumptions transferability order bring better understanding systems combination systems corrupted clutter assumptions primarily made individuals systems properties systems deduced whenever possible constants inf inf sup sup satisfy condition assumption ensures point state space reached point single time step otherwise would hold least one pair condition ensures transition sufficiently regular compared reference measure transition diffuse sense concentration probability mass single point state space assumption also holds straightforwardly satisfies type conditions since compact hence finite let transition kernel joint markov chain defined property sufficient ensure joint kernel defined positive aperiodic next assumption expectations taken respect respectively also bikp denotes binomial distribution success probability trials constant sup satisfies functions inf sup inf sup satisfy houssineau singh jasra well log log holds log inf bik assumption ensures points observation space reached least states via although might hold equation ensure boundedness calculations related identifiability upper bound likelihood function also assumed finite concentration probability mass allowed point demonstrated following lemma upper lower bounds considered assumption single target clutter common distribution sufficient guarantee type result multiple targets proof appendix lemma transfer boundedness assumption holds constant sup sup finite functions inf sup verify well log important result follows assumptions introduced far uniform forgetting conditional markov chain proved assumptions parameter holds dxk dxk dxt probability densities sequences observations forgetting rate generally smaller rate although mixing still guaranteed since finite hence also possible conclude pointwise convergence function function defined follows defined realisation observation process lim log limit depend indeed assumptions holds log lim identification dynamical systems result shows realisation observation process empirical average log converge irrespectively assumed initial state continuity assumption required order turn pointwise convergence result uniform convergence result mappings continuous follows directly assumption mappings continuous hyperplane made parameters number targets equal since mappings sums products continuous functions although continuity markov kernel likelihood function limited hyperplanes result lemma extended lim sup since hyperplane small enough addition continuity assumption enables derivation following result regarding uniform convergence function assumptions holds lim sup sup log since conditional function log continuous uniformly bounded follows also continuous hyperplanes constant target number following identifiability assumption considered order show consistency maximum likelihood estimator assumption fundamental since would chance discriminate true value among possible parameters yield law observations instance colour target considered parameter likelihood observations depend characteristics target observations come radar obtained changing colour would induce law equal assumption would verified shown next theorem identifiability problem deduced identifiability one important special cases proof appendix theorem transfer identifiability assumption holds true parameter holds true parameter subset made parameters form holds challenging prove identifiability transfers whole parameter set property assumed hold rather demonstrated assumption stringent condition since theorem shows identifiability sufficient ensure identifiability houssineau singh jasra important special cases moreover exists satisfy specific equations including bik assumption would hold since identifiability would clearly lost case absence observations targets remark made identifiability since obviously way learn dynamics observation targets none present different assumptions considered combined next section order prove consistency maximum likelihood estimator consistency asymptotic normality consequence dominated convergence theorem holds infinite observation sequence initial states lim log lim log inequality holds since conditional expectations divergences yet could happen would verify surely would compromise identifiability however assumption equivalent log objective show turn equivalent since term appears following line arguments proposition find assumptions holds conclude considered approach allows studying identifiability applying strict jensen inequality conditional expectation indeed follows lim log implies likelihood observation sequence parameter decreases exponentially fast compared likelihood irrespectively assumed initial states denoting argument maximum log consistency maximum likelihood estimator expressed theorem theorem also states asymptotic normality estimator makes use fisher information latter involves differentiation respect parameter however since number target natural number differentiations performed identification dynamical systems fixed understood default writing assumptions fisher information matrix expressed log log lim matrix transposition theorem assumptions holds lim considering additionally assumptions see appendix assuming positive definite holds denotes convergence distribution tends infinity normal distribution mean variance proof theorem follows lemma combined theorems demonstrated result theorem also holds special parameter sets sets used understand behaviour fisher information matrix simple cases next section analysis fisher information theorem guarantees convergence maximum likelihood estimator certain conditions proves asymptotic normality estimator variance latter inverse fisher information matrix therefore interest understand fisher information behaves different configurations section structured follows equivalent observation model fisher information matrix easier study introduced subsection yields characterisation configurations information loss induced data association uncertainty detection failures strictly positive qualitative estimates information loss obtained isolating different sources loss subsection subsection qualitative estimates confirmed numerical results simulated data obtained direct monte carlo integration original expression fisher information confirm validity derived alternative expressions henceforth two square matrices dimensions understood positive stand positive definite example assuming data association known joint probability observations becomes houssineau singh jasra score found log log log independence targets clutter fisher information distribution one target one clutter point respectively gradient taken lim log log log log spite simplicity example yields important remarks unsurprisingly missing information data association uncertainty information increases number targets similarly fisher information clutter distribution increases overall information increases interpretation poisson parameter less straightforward main objective however study fisher information targets rather false alarms interest compute score without differentiating respect although fisher information becomes difficult compute conclusions drawn focusing cardinality since parameter affect cardinality term log log remains computing fisher information matrix random variable induced term minimal increases goes toward sufficient conclude since fact information lost detection failures happen taken account cardinality information equal indeed equally easy estimate observation always never received reason useful consider information objective therefore characterise fisher information lim log log dynamical system behaves compared information unperturbed system excludes false alarms detection failures data association known refer difference latter information loss since fisher information unperturbed system quantity depends number objects system aim express information loss function fisher information matrix fisher information matrix unperturbed system clearly equal independence targets observation absence data association uncertainty order compute take identification dynamical systems logarithm probability density function however presence sum term prevents analysing fisher information general setting avoid directly dealing sums equivalent observation model depends explicitly assignment introduced next section observation model important contribution since allows understand behaviour fisher information multitarget tracking alternative observation model let hamming metric symmetric group sym characterised letting number points moved given sym instance given cauchy notation since set let vector concatenation operator let matrix size many lines detected targets seen mask matrix removes observations ones let permutation matrix corresponding sym matrix defined row vector ith position elsewhere observation model known data association written observation function observation noise respectively across components false alarms defined random variable independent independent observation model interest defined given integers random element distribution restriction random permutation drawn uniform law set defined sym houssineau singh jasra denoting identity function henceforth letter used random permutation realisation case considered avoid redundancy holds since permutations different identity move least two points case example recovered considering almost surely whereas full problem corresponds choice cardinality found subfactorial equal number derangements letters derangement refers permutation moves elements domain subfactorial defined via recurrence relation factorial initialisation expression justified follows number permutations moving number points less equal also number permutations moving exactly points number derangements points multiplied number ways picking points among holds nkk since alternative observation model brings insight fisher information matrix corresponding observation model compared unperturbed case corresponding information loss defined iloss cases relative information loss iloss used instead next theorem central result section proof found appendix theorem assumptions information loss iloss verifies iloss inequality strict either notice condition would sufficient make inequality theorem strict equal since data association might influence specific configurations individual likelihood depend objects state theorem provide quantitative characterisation information loss challenging general case yet behaviour information loss analysed special cases objective remainder section one advantages modified observation model fisher identity utilised alternative way computing score function based unobserved random variables model log log identification dynamical systems simplification expression log notational random variables conditioned event respective distributions conditional distributions given observations complex priors yet fisher identity enabled move sums integrals outside logarithm hence making easier analysis fisher information matrix single static target false alarm consider case one almostsurely detected static target state observation corrupted false alarms unknown data association corresponding hyperplane special parameter set composed parameters sufficient study one time step since observations different times become independent case holds lim log log log log making use fisher identity fisher information matrix expressed log log identifying parameter challenging distribution false alarm equal one observation since observations look alike close case holds log log expectation taken random variable follows relative information loss equal strictly increasing houssineau singh jasra relative loss gaussian log fig information loss function poisson parameter calculated samples gaussian scenario uniform distribution tends tends infinity result supported experiments displayed figure observation one static target corrupted false alarms observation model assumed linear gaussian variance cases false alarm uniform subset also considered scenario false alarm distributed way observation also confirmed scenario next two sections focus understanding role played specifically unknown data association detection failures unknown data association order set focus data association assumed belongs special parameter set conditions joint probability observations states becomes sum previous expression makes difficult directly compute fisher information matrix insight however obtained considering static objects following example example let known position static objects joint distribution observations found simplified setting assume finite support objects state chosen far enough hold whenever case expected loss information identification dynamical systems compared case known data association less intuitive result found objects state equal given situation holds permutations equally probable loss information two cases correspond extreme configurations uncertainty data association either resolvable irrelevant fisher identity used provide expression fisher information static objects follows fixed fisher information fully unknown association deduced log log log fisher information matrix without false alarm found log log scoi scoj scoi scoj scoi log term conditional probability object state generated observation given observations order obtain quantitative characterisation information loss special likelihood introduced consider observation model form one displayed figure compact exists collection disjoint subsets uniformly distributes probability mass outside example distribution given figure two objects objects houssineau singh jasra probability density fig example likelihood two objects states scoi scoj objective understand behaviour large order term order summand determined different values instance since increases needs augmented least linearly ensure family disjoint shows inverse proportionality order terms form sum sum terms order since case holds scoi scoj follows facts scoi depend therefore order following principles values find order since information idealised observation model data association known equal follows relative loss constant words large number targets adding targets increases information rate idealised model validation via simulations special likelihood taken form otherwise characterised via case displacement considered parameter true value relative information loss associated likelihood displayed figure two different configurations first one constant observation space identification dynamical systems relative loss constant observation space adaptive observation space number targets varying number objects varying association uncertainty special likelihood runs spatial separation runs experimental relative loss relative loss number targets varying number objects varying probability detection aration runs compared fig information loss association uncertainty detection failures figure corresponds case observation space large enough meet requirements associated relative loss seen increase linearly number targets second case adaptive observation space figure corresponds case observation space augmented fit new targets shows constant relative information loss last result consistent conclusion information loss order number targets observation space augmented simulations five static objects positions observed via linear gaussian model variance equal objective understand fisher information matrix evolves position objects assumed parametrises variance gaussian observation model scalar relative information loss displayed figure confirms intuition information loss increases except case loss houssineau singh jasra definition since data association known case also loss increased individual likelihoods overlap increasingly different decreases overlap becomes negligible maximum reached distance two consecutive objects fact loss follows irrelevance data association uncertainty objects position explained example better understand behaviour number targets figure displays relative information loss targets case full data association uncertainty results two sets simulations consistent show trend relative information loss increases number targets tend stabilise sum loss target construction loss linear number targets sufficiently many increases fastest transition two modes detection failures section case detection failures analysed assuming false alarms specialparameter set establish main result section theorem use concept missing information see instance context approximate bayesian computation theorem assuming information loss iloss known data association unconstrained detection failures found iloss proof found appendix follows theorem considered configuration fisher information matrix made arbitrary close making tend also loss expected order verify result theorem practice scenario detection failures without false alarms considered object starts time position evolves according random walk standard deviation time observation linear gaussian variance equal integral state space expression score computed monte carlo simulation samples expectation fisher information utilises samples relative information loss displayed figure confirms coefficient found analytically theorem next example shows fisher information evolves general adding new objects without involving data association uncertainty example fisher information problem related information new objects perturbed uncertainties random variable observation model verifies almost surely follows theorem example gives upper bound increase fisher information number objects increased since depicts case data association uncertainty objects would correspond practice case added objects area identification dynamical systems false alarm objects far existing objects well far far depends likelihood conclusion first important result article proof consistency maximum likelihood estimator tracking weak conditions weak means conditions often possible applying dynamics observation asymptotic normality holds additional assumptions second part article brings understanding asymptotic variance maximum likelihood estimate analysing fisher information matrix corresponding tracking qualitative results obtained general case fisher information decreases data association uncertainty detection failures presence false alarms quantitative results also derived important special cases one static target false alarm unknown data association multiple static targets unknown data association particular observation model multiple targets detection failures future works include study identifiability specific associations instead marginalising possibilities considered article approach involves additional challenges since parameters learned increase dimensionality time special case results presented acknowledgement authors supported singapore ministry education tier grant number appendix assumptions theorem following assumptions required proof asymptotic normality maximum likelihood estimator tracking norm defined matrix mappings twice continuously differentiable hyperplane made parameters number target equal holds sup sup log sup sup log sup sup log sup sup log exists integrable function exist integrable functions appendix proof lemma follows assumption supremum clutter density characterised houssineau singh jasra sup verifies since sup sup sup since terms sum positive holds sup sup concludes first part proof bipd bik inf bik also holds inf bik since support guarantees convolution also support infimum strictly greater zero follows considering log similarly sup bik finite finite infinite case noticing leading term convolution find lim sup sup lim kpk sup sup sup lim finite constant concludes proof lemma appendix proof theorem two cases theorem proved separately follows identification dynamical systems joint probability observations system initialised stationary distribution characterised measurable subset assuming measurable subset form sum collapses single term targets detected terms sum equal second case considered represents configuration observations without considering locations case holds bik show hold time alternatively follows easily identifiability two cases considered together show distributions associated differ subset observation space likelihood becomes marginalising location observations time step considering case observations gives assuming considering follows hold time concludes proof appendix proof theorem houssineau singh jasra lemma given integers let family probability measures ymk indexed let denote corresponding probability density common reference measure ymk assume integers let random vectors ditionally independent given law defined via process thinning augmentation clutter density random permutation assume consider constant almost surely let probability measure corresponding probability density assume densities entiable log log log log inequality strict strictly greater proof part lemma corresponds random permutation observation association uncertainty random thinning result follows lemma remark corresponds random thinning random permutation result follows corollary random thinning random permutation present result lemma based extended random variable follows noticing generated coarser one generated indeed finitely many equal holds inclusion follows countable union family subsets form generating family result follows result extends straightforwardly collection thinned observation vectors independently perturbed false alarm permutation since corresponding generated set rectangles form consequence fact implies since independent holds assumption lemma implies constant almost surely result holds collection observation vectors identification dynamical systems proof part lemma let measurements targets clutter random permutation dimensional vector deletions let missing target generated observations let denote joint depends implicitly using change variable formula noting mapping permutation hence jacobian transformation determinant follows log log deduce since holds log log projection coordinates describing follows almost surely log log let denote density log log applying joint random variables defined lemma follows log log let jensen inequality applied function random variable log yields log log almost surely log log log log log proves since jensen inequality applied strictly convex function case equality log log log log holds log log part lemma shows log measurable constant given log follows function equal since constant hence proving lemma houssineau singh jasra observation model imply equality gradients log log since depends parameters included interest however information loss result lemma satisfying proof theorem considered perturbed observation model properties one studied log log holds almost surely result lemma remark therefore used directly context interest give iloss integer log log log log objective prove implies lemma applied involved almost surely ditional laws implies log almost almost following principle lemma follows holds log almost surely turn implies supporting results proof part lemma supporting results proof lemma permutation uncertainty deletion lemma randomly permuting random let let denote randomised permutation independent let assume permits least exchange two indices furthermore assume law satisfies probability measure product probability measure constant almost surely proof lemma proof completed case easily generalised likewise present arguments employ clearest way consider case case sheds little light make generalisation apparent identification dynamical systems let since independent thus almost surely consider case preceding statements imply almost everywhere indeed also hold almost every due assumption mutual absolutely continuity statements holds almost everywhere hold almost every show implication holds almost everywhere may derived done complete proof manipulate assumption random variables independently identically distributed respect measure show constant almost everywhere first equality function collapses function variable denoted using second equality thus must constant independent verify implies change variable gives procedure applied shows second equality corollary extends lemma situation therein follows law clutter process defined section see corollary let infinite sequence independent random variables let vector random variables independent let random variable independent let random permutation matrix defined section assume constant almost surely proof let since independent random permutation matrix independent given law houssineau singh jasra conditioned satisfies assumptions lemma thus lemma implies almost surely constant clear independent next results extends corollary setting sequence vectors observed indirectly sequence vectors generated corollary augmenting clutter randomly permuting corollary let sequence random vectors let conditionally independent given pzm pzi defined corollary assume constant almost surely particular holds positively valued probability densities dominating measures measure proof due conditional independence assumption since almost invoking corollary thus function proceeding way similarly show invoke assumption show must constant almost surely manner similar proof lemma details omitted final statement corollary straightforward supporting results proof lemma deletion permutation uncertainty following result intermediate result intended convey main idea analysis actual full blown version contained lemma lemma let independent assume measurable set whenever implies almost surely constant remark random variables always relabelled assumption met assumption interpreted mean concentrated event imply assures independence random variables remark assumption satisfied without need densities example assumption satisfied independent density assumption satisfied indeed case identification dynamical systems proof lemma conditional expectation always expressed follows exists function version conditional expectation thus follows independence since expectation terms must zero hence almost surely measurable set let since almost surely show constant almost surely assuming converse true arrive contradiction hold almost surely exist set let thus implies violates assumption lemma multiple deletions let random vector denote thinned version components removed assume subsets furthermore assume following let means implies almost surely constant remark assumption lemma easily understood follows let unfortunately stating assumption via measurable rectangle sets sufficient product sigma algebra much richer measurable rectangles fact set assumption lemma measurable rectangle assumption reduces remark remark main assumption lemma satisfied probability measures mutually absolutely continuous proof fact uses lemma fact trace set measurable omitted proof remark condition translates lemma implies measurable sets range taking implies houssineau singh jasra clearly probability assumption lemma rectangle proof lemma random variable belongs disk joint union thus write constant measurable functions independence invoked trivial since implies assume assumed subsets focus terms also implies almost surely measurable set example get henceforth refer simply need show constant almost surely case exists subsets variables recall interpretation potentially function reduced set variables asserted first equality must genuinely function least variable clarity simplicity assume consider terms sum due lemma exists measurable set implies violates main assumption lemma following two lemmas concerns random variables declared statement lemma lemma exits set almost surely identification dynamical systems proof let denote let clearly show almost surely set measure zero comprised disjoint sets thus result follows since differ precisely first two sets reverse implication obvious lemma exists measurable function proof proof complete show implies exists set turn implies write rather smallest respect random variable measurable note every exists measurable set assume almost surely lemma set almost surely violate hence also simple function replace indicator function measurable set simple function almost sure counterpart becomes measurable general case result may established since may approximated sequence simple functions tending simple function measurable version say thus letting see lim appendix proof theorem case first considered number observations time equal zero one joint probability observations states becomes empty sequence size time made explicit expression sake clarity follows let noisy version original observation hmm equal law hmm sequence random variables common law uniform distribution ball radius centre switching process also introduced follows target detected otherwise order study fisher information easily introduce alternative observation model detection failure time replaced observation target law observation model houssineau singh jasra quantity interest compare case justify equivalence two observation model considered purpose verify score log equal score log required modifications theorem follows loss information iloss replacing original observations ones expressed iloss log log log log log log considering limit follows iloss lim iloss log log case simply holds information loss equal since targets detection independent data association known terminates proof proposition references blackman tracking radar applications artech house chenouard objective comparison particle tracking methods nature methods dean singh jasra peters parameter estimation hidden markov models intractable likelihoods scandinavian journal statistics del moral mean field simulation monte carlo integration del moral houssineau particle association measures multiple target tracking theoretical aspects modeling springer douc moulines ryden asymptotic properties maximum likelihood estimator autoregressive models markov regime annals statistics doucet andrieu davy particle filtering tracking sensor management proceedings fifth international conference information fusion vol ieee jiang singh bayesian tracking parameter learning nonlinear multiple target tracking models ieee transactions signal processing leroux estimation hidden markov models stochastic processes applications mahler multitarget bayes filtering via multitarget moments ieee transactions aerospace electronic systems meyn tweedie markov chains stochastic stability springer science business media identification dynamical systems mullane adams approach bayesian slam ieee transactions robotics russell sastry markov chain monte carlo data association tracking ieee transactions automatic control okuma taleghani freitas little lowe boosted particle filter multitarget detection tracking european conference computer vision springer pace del moral phd filters based generalized flow journal selected topics signal processing special issue tracking pailhas houssineau petillot clark tracking mimo sonar systems applications harbour surveillance iet radar sonar navigation storlie hannig lee statistical consistency data association problem multiple target tracking electronic journal statistics gaussian mixture probability hypothesis density filter ieee transactions signal processing singh doucet sequential monte carlo methods multitarget filtering random finite sets ieee transactions aerospace electronic systems
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advanced quantizer designs systems using uniform planar arrays nov jiho song member ieee junil choi member ieee taeyoung kim member ieee david love fellow ieee upa structure known mimo ability exploit vertical horizontal domain beamforming fully exploit accurate channel state information csi domains critical channel estimation techniques relying upon time division duplexing tdd leverage channel reciprocity transmit receive arrays calibrated current cellular systems however exploit frequency division duplexing fdd receiver estimate quantize feed back downlink csi transmitter high dimensionality massive mimo channel could cause large overheads downlink channel training quantization processes focus csi quantization paper refer references therein massive mimo downlink training problem majority csi quantization codebooks designed assumption spatially uncorrelated rayleigh fading channels uniformly distributed unit hypersphere normalized quantize channels codewords codebook cover unit sphere index mimo mimo uniform uniformly possible spatially correlated channels codebooks carefully shaped based planar arrays kronecker product codebooks prior knowledge channel statistics although previous codebooks designed based ntroduction analytical channel models difficult represent assive mimo properties true channels fdtems strong candidate fulfill throughput mimo spatial channel model scm requirements fifth generation cellular networks extension scm extensively maximize number antennas limited area used mimic measured channel variations dimensional antenna arrays uniform planar arrays upas standardization although scm stochastic channel cylindrical arrays host antennas vertical model provides limited insights practical csi quantizer horizontal domains prominently considered practice designs therefore necessary develop simple channel among various array solutions upas great model accurately represents properties interest simplify signal processing channel upa antennas paper define simple channel model using channels massive mimo employing sum finite number scaled array response vectors based song love school electrical computer simplified channel model develop csi quantizers engineering purdue university west lafayette jihosong upa scenarios first carry performance analysis djlove choi department electrical engineering postech pohang kronecker product csi quantizers analytical gyeongbuk korea junil studies csi quantizers provide design guidelines kim samsung electronics suwon korea develop quantizer narrowband single frequency email parts paper presented globecom washington usa tone csi using limited feedback resources proposed december quantizer concentrate dominant work supported part communications research team crt radio paths true channels maximize quantization quality samsung electronics ict program development mimo transceivers also develop codebook combiners cophases beyond mobile communication systems scales quantized beams vertical horizontal mimo systems utilize large number antennas base station expected enhance network throughput enabling improved multiuser mimo techniques deploy many antennas reasonable form factors base stations expected employ antenna arrays horizontal vertical dimensions known mimo popular array uniform planar array upa antennas placed grid pattern exploit full benefit massive mimo frequency division duplexing fdd downlink channel state information csi estimated quantized fed back receiver transmitter however difficult accurately quantize channel computationally efficient manner due high dimensionality massive mimo channel paper develop narrowband wideband csi quantizers fdmimo taking properties realistic channels upa consideration improve quantization quality focus quantizing dominant radio paths channel also combining quantized beams also develop hierarchical beam search approach scans vertical horizontal domains jointly moderate computational complexity numerical simulations verify performance proposed quantizers better previous csi quantization techniques mains searched beam quantization involves heavy computational complexity thus develop hierarchical beam search approach reduce complexity also develop wideband quantizer broadband communication evolving dual codebook structure lteadvanced dual codebook structure first layer quantizer used search correlated csi multiple frequency tones unless dominant paths gathered single cluster codebook effective adjacent radio paths selected quantized using resolution codebook thus concentrate detecting adjacent sperate paths within wideband resource block based proposed hierarchical beam search approach addition second layer quantizer designed refine beam direction quantized wideband csi according channel vectors narrowband comparing approach codebook refined beams cophased scaled approach codebook cophase adjacent beams without considering beam refinement section describe systems employing upas discuss simple channel model mimics true scm channels section iii review previously reported codebooks section develop narrowband csi quantizer takes multiple radio paths account conduct performance analysis develop design guideline csi quantizers section also propose wideband csi quantizer assuming framework section present simulation results conclusion follows section vii throughout paper denotes field complex numbers denotes semiring natural numbers denotes complex normal distribution mean variance closed interval denotes uniform distribution closed interval ceiling function denotes complete gamma function expectation independent random variable hadamard product kronecker product zeros vector identity matrix ones matrix eign denote entry dominant eigenvector principal eigenvalue matrix also denote conjugate transpose complex conjugate entry subvector including entries column vector respectively antenna structure assuming framework expression defined received baseband symbol ratio snr block fading miso channel transmit beamformer data symbol power constraint additive white gaussian noise note denotes frequency tone framework facilitate quantizer designs mimic define simplified channel model radio paths according true used present numerical results section number dominant paths subcarrier spacing excess tap delay radio path channel gain radio path radio path given angles upa scenario radio path frequency tone represented dmv dmh array response vector dma defined dma sin sin cos note antennas spacing angle array vector wavelength frequency tone csi center frequency satisfying speed light without loss generality narrowband representation channels defined ystem odel consider miso employing transmit antennas base station single receive antenna user number rows number columns upa although mainly discuss miso channel quantization simplify presentation proposed channel quantizer easily extended multipleinput mimo systems extension miso channel quantizer discussed section set radio paths set complex channel gains narrowband assumption dropped simplicity assume beam directions uniformly distributed vertical horizontal domains independent channel gains consider subcarrier ignore beam squinting effects limited feedback beamforming approach user chooses transmit beamformer among codewords codebook vectors channel vector decomposed vertical horizontal domains based singular value decomposition yielding arg max rank denotes total feedback overhead based assumption transmitter receiver know reshaped channel given matrix form fined codebook index selected beamformer cmv fed back transmitter feedback link majority channel quantization codebooks denotes dominant singular value cmv designed spatially correlated uncorrelated rayleigh denotes dominant left singular vector cmh fading channels analytical channel models rely denotes dominant right singular vector final upon rich scattering environments radio path codeword obtained limited effect channel characterization thus previous arg max arg max beamformer codebooks focus covering unit hypersphere uniformly possible without considering radio path individually however analytical channel models much despite advantage codebook issues different realistic channel models assume even los channel dominant radio path dominant scatterers therefore codebook may accurately quantized searching domain design approach may effective number separately also always effective quantize antennas large accurately quantize single radio path even may consist massive mimo channels important tailor codebook multiple paths although quantizer considers adding realistic channels consisting limited number radio two beams performance improvement limited paths beams combined properly iii ronecker roduct odebook eview ingle eam ase roposed narrowband uantizer ultiple eams ase critical fdd massive mimo systems quantize feedback information channels transmitter thus csi quantization codebooks developed tailor feedback link limited overhead massive mimo channels among various csi quantization techniques codebooks great interest quantize channels computationally efficient manner considering antenna structure based channel model codebooks designed quantize radio path upa structure codebooks based assumption covariance matrix channels approximated covariance matrices vertical horizontal domains hhh prior work verified scm channel realizations well modeled resolvable radio paths thus assume channel vector single frequency tone represented combination set multiple radio paths corresponding channel gain vector array response vectors channel gain vector contain different types channel information thus focus quantizing using different codebooks paper narrowband quantizer aim find thus codebook form avbt ahbt quantizes single dominant path discrete fourier transform dft codebook aab ama ama consisting codewords ama previous codebooks quantize first dominant vector domain separately find singular constructing set beams weight vector radio paths constituting channels represented kronecker product array response vectors thus beam defined combination quantized array response vectors vertical horizontal domains cvn chn assume reserved quantize array response vector domain reserved quantize weight vector construct condition limited feedback overhead following questions properly addressed beam quantization radio paths channel chosen quantized algorithm beam quantization initialization create initial empty matrix beam quantization given dft codebooks avbn ahbn construct beam set defined effectively quantize combine limited quantize radio path cvn chn number radio paths cvn chn arg following subsections address channel quantin hhh zation procedure across two separate quantization phases update beam set cvn chn evaluate quantization loss phase end remark quantized channel vector viewed final update representation channel using analog beamsteering quantized radio paths matrix realized set radio frequency phase unquantized vector shifters baseband beamformer therefore proposed hweight hhh approach follows hybrid beamforming architecture beam combining quantized beams cophased scaled feedback resource allocation given total feedback overhead allocation scenario phase beam quantization beam quantization phase aim construct selected set quantized beams well known dft codebook effective solution quantize array response vectors quantize array response vector dft codebook aabn beam quantization approach select quantize dft beams sequentially update quantized dft vectors unquantized weight obtained solving maximization problem cvn chn arg max includes previously selected dft beams quantize weight vector update practical construct codebook weight vectors constantly change dimension problem choosing dft beam simplified arg max max arg max derived unquantized weight vector computed based generalized rayleigh quotient arg max hhh bar top weight vectors denotes unquantized weight vectors beam quantization approach gives set quantized beams unquantized weight vector hhh separate beam quantization performed compute codeword candidate based algorithm following section practical beam search technique proposed quantize single dominant beam moderate computational complexity well compute multiple codeword candidates hierarchical fashion also evaluate quantization loss due beam quantization function number beams feedback overhead dft codebooks assuming unquantized weight vector beamforming gain channel vector set dft beams gbq max averaged channel realizations lemma presenting lemma make following assumption assumption assuming channel vector already decomposed set radio paths channel gains column vectors domain separately selected ama aabn arg max ama considering antenna spacing beamforming gain array vector selected dft vector derived appendix sin hhh cvn chn hhh lemma lower bound beamforming gain gbq approximated gbq please check appendix proof phase beam combining beam combining phase aim compute weight vector used combine beams quantize weight vector arg max design codebook including combiners zbc study codebook design framework model effective channel vector based kronecker correlation model covariance matrix hhh analytically computed appendix random variables denotes weight vector subject equal gain subset codebook design approach pick set codewords maximize inequality based kczu kzu kzu approximated plugging kronecker correlation model maximizer based correlated grassmannian beamforming algorithm codewords obtained setting note denotes phase quantization level also evaluate quantization loss due beam combining function number codewords codebook zbc quantizes baseband combiner analyze quantization performance zbc normalized beamforming gain normalized effective channel selected combiner gbc ehh max reu averaged effective channel easy compute closed form derive normalized beamforming gain special case based following assumption assumption simple analysis assume combiner selected arg max note arccos lemma special case normalized beamforming gain gbc approximated gbc sin please check appendix proof mimo channel quantizer mimo channel quantization approach dft beams chosen maximize sum channel gains receive antenna dft vectors quantized solving rewritten problem cvn chn arg max hmimo hmimo picking set equal gain vectors maximizing min min equal gain vectors restricted investigating feedback allocation solutions extend proposed quantizer mimo channel scenarios assuming mimo system employing receive antennas user channel matrix defined hmimo czu kczu czu min min min hhh cvn chn constructing selected set dft beams next compute orthogonal weight vectors compute precoding matrix spatial multiplexing given set beams compute beamformer layer transmission unquantized weight vector hmimo mimo beamforming gain beamforming gain fig cross correlation different allocation scenarios computed based generalized rayleigh quotient number antennas number beams precoding matrix constructed number dominant beams quantized allocation scenario proposed quantizer feedback scenario chosen denotes maximum transmission arg max feedback resource allocation codebook procedure quantizing beams high resolution codebook increases beamforming gain cost increased feedback overhead effectively allocate limited feedback overhead resources must derive beamforming gain randomly generated channel vectors selected codeword using function allocation however across quantization phases section section make hard compute beamforming gain closed form simplify analysis make following assumption assumption assuming quantization phases section section work independently channel quantization quality proposed codebook procedure evaluated combination quantization losses phases based assumption quantization phases independent beamforming gain proposed quantizer defined mixture gbq gbc gbq gbc gbc note designing beam combining codebook feedback resource allocation algorithm support multi layer mimo transmission interesting topics future research denotes size bits dft codebooks domain denotes size bits codebook combiners zbc evaluating possible scenarios considers beams note possible feedback scenarios subject total feedback overhead expectation taken number dominant paths since varies depending channel environments assuming equally probable plot arithmetic mean fig different numbers antennas feedback bits shown figure quantizing one two beams give best performance practical upa scenarios feedback overheads therefore construct codebook quantizing single dft beam codebook combining two quantized dft beams based predefined allocation respectively final codebook defined remark channel realizations interference due remained paths negligible channel gains contained first second dominant beams based codebook subset restriction algorithm severe interference could mitigated reporting remained paths considerable amount channel gains total feedback overhead feedback scenarios bits normalized cross correlation wideband resource blocks narrowband resource blocks fig overview wideband model multiple tones beam search approach necessary search vertical horizontal domains jointly scan dominant beams channel vector however joint approach increases computational complexity example required carry vector computations scan single dft beam allocation scenario reduce heavy computational complexity comes detecting single dominant beam propose beam search technique follows round channel vector first dominant beam chosen using dft codebooks dft vectors dft beam given arg max later selected dft beam baseline guides generation two codeword candidates round round assigned constructing two codeword candidates support channel realizations single dominant beam codeword computed based allocation scenario scanning beam directions near round first codeword given support channel realizations multiple dominant beams codeword computed based allocation scenario choosing additional dft beam combine second codeword given arg max using size dft codebooks size codebook zbc developed combine two dft beams explained section considering csi quantization technique second codebook defined zbc round using two codeword candidates final codeword selected additional bit arg max arg max fig normalized beamforming gains subcarrier channel vectors defined shift beam size codebook designed refining beam directions dft beams csi quantization approach first codebook defined roposed ideband uantizer develop wideband quantizer takes multiple frequency tones account developing practical quantizers overview broadband system model adopted shown fig total frequency tones divided wideband rbs wideband includes channels wideband written matrix form depicted fig wideband divided narrowband rbs hadamard product formulation satisfies following formulation denotes narrowband written matrix form numerical numerical numerical analytical analytical analytical beamforming gain beamforming gain numerical numerical numerical analytical analytical analytical fig beamforming gain comparison numerical results analytical results next correlation channel vectors studied numerically based cross correlations codeword quantizes channel vectors wideband narrowband refining beam direction according arg max denotes channel vectors codebooks denotes dominant dft beam chosen round second codeword computed support dft codebooks subcarrier shown channel scenario two dominant beams proposed fig verified dominant dft beams quantizer refines direction well combines different subcarriers channel vectors highly correlated second dft beam second codeword based empirical studies wideband quantizer designed way correlated information dominant dft beam shared neighboring subcarriers level wideband resource block choose two dft beams close channel vectors arg max wideband supporting wideband first dft beam chosen zbc allocated bits assigned refining first dft beam bits assigned arg max combining two dft beams round among two codeword candidates final dft codebooks next second dft beam codeword selected additional bit according chosen using dft codebooks arg max arg max max imulation esults level narrowband resource block within wideband set dft beams baseline guide quantization channel vectors narrowband construct set two codeword candidates allocated narrowband round first codeword computed support channel scenario single dominant beam first verify performance csi quantizers evaluating proposed quantizers pause validate accuracy approximated beamforming gain beamforming gain simplified channel quantized channel computed numerical simulations channels generated assuming fixed number beams fig shown approximated formula gives tight lower bound numerical results prop prop prop enh enh prop prop enh enh prop normalized beamforming gain normalized beamforming gain fig normalized beamforming gain narrowband quantizers table simulation parameters antennas antennas scenario carrier frequency subcarrier spacing vertical antenna spacing horizontal antenna spacing upa upa umi nlos ghz khz evaluate narrowband quantizers using simulations numerical results obtained monte carlo simulations channel realizations generating channels use parameters table evaluate normalized beamforming gain narrowband quantizer final chosen compare beamforming gains proposed quantizer codebooks enhanced codebook note quantizer listed table computational feedback summarized table iii figs normalized beamforming gains three quantizers plotted different antenna spacing perfect csi beamformer gives normalized beamforming gain one allocation scenarios proposed narrowband quantizers predefined count number vector computations evaluate complexity feedback overhead per frequency tone csi assessed combination overheads first second rounds section table eedback configurations narrowband quantizer prop prop prop enh enh codebook round round table iii eedback overheads complexity comparisons prop quantizer enhanced codebook feedback overhead vector computations scenarios proposed quantizer searches vertical horizontal domains jointly codebooks search beams lying domain independently integrate results later dft beams quantized proposed quantizer aligned cophasing scaling beam contrary quantized beams enhanced codebook simply added together without considering phase alignment reasons proposed quantizer generates higher beamforming gains codebooks next evaluate normalized beamforming gain wideband quantizer according chosen codeword figs normalized beamforming gains wideband quantizer compared narrowband quantizers legend first alphabet denotes type quantizer normalized beamforming gain normalized beamforming gain fig normalized beamforming gain comparison narrowband wideband quantizers table eedback overheads wideband narrowband level wideband level narrowband table ideband configurations narrowband wideband quantizers narrowband quantizer codeword tones codeword tones wideband quantizer second alphabet denotes allocation scenario table final digit represents wideband table total feedback overhead proposed wideband quantizer defined numerical results verify wideband quantizers outperforms narrowband quantizers exploits correlation frequency tone csis wideband quantizers also reduce feedback overhead maintain quantization performance less overhead compared narrowband quantizers vii onclusion paper advanced csi quantizers based codebook structure proposed systems using upas proposed quantizer designs focused detecting quantizing limited number dominant beams channel vectors exploiting dft codebooks lte setup scheme first narrowband rbs tone csis ninth narrowband tone csis codebook combiners designed cophase scale quantized dft beams furthermore analytically derived design guideline practical quantizers based systems predefined allocation scenarios developed csi quantizers taking predefined feedback scenarios account first narrowband quantizer proposed quantize combine one two dominant dft beams detect quantize beams properly also developed beam search approach scans vertical horizontal domains jointly moderate computational complexity reduce total feedback overhead also proposed wideband quantizer utilizes correlated information multiple frequency tones numerical simulations verified proposed narrowband quantizer gives better quantization performance previous csi quantization techniques proposed wideband quantizer improves quantization performance less feedback overhead compared narrowband quantizer wideband settings ppendix orrelation array response vector dft codeword discuss correlation array response vector domain selected dft codeword quantify quantization performance dft codebooks evaluating max ama max rewritten index selected codeword arg min complete lower bound first compute expectation squared effective channel vector khh cvn chn cvn chn expectation rewritten defining new random variable follows derived using correlation correlation computed channel selected dft codeword cvn chn discussed appendix next consider set dft vectors expectation squared approximated cos max max sin subject note computed sin cvn chn cvn chn cvn cvn chn chn ppendix ower bound normalized beamforming gain remark simplify analysis consider first order taylor expansion bivariate variables derived chosen assume beam directions uniformly distributed chosen one codewords dft codebook equal probabilities reason obtain cac cad computing arithmetic mean beamforming expectation bivariate variables approximated gain two different codewords cac cad ama beamforming gain gbq lemma lower bounded gbq max based expectation squared approximated inequality based holds derived based remark derived holds assuming correlation formula always positive cvc chc hhh cvd chd dmv dmv cvc dmv dmv dmv finally lower bound gbq approximated plugging derived formulas gbq ppendix case cac dma ovariance matrix effective channel vector entry rewritten note cac dma computed depending different cases follows case cac dma derived computing arithmetic mean derived appendix case cac dma chosen note derived computing arithmetic mean case cac dma dma derived dma ama note derived definition ppendix uantization performance zbc normalized beamforming gain effective channel vector selected combiner lower bounded cos sin tan cos based cos sin approximated tan approximated based remark appendix although gbc simplified still difficult solve cases special case equal gain vectors defined using beamforming gain derived cos cos cos cos sin sin note derived based definition follows arg min based assumption addition derived computing arithmetic mean equally probable derived cos arithmetic mean derived cos ppendix orrelation channel vector dft codeword derive correlation channel selected dft beam cvn chn dmv dmh dmh chn derived ama complete formula compute power largest channel gain without loss generality assume magnitude channel gains descending order channel gain follows characterized cumulative distribution function cdf consider order statistic smallest order statistic exponentially distributed random variables refer defining pdf yielding fak derived based binomial expansion formula expectation order statistic defined derived appendix note derived sin notice derived function derived based derived based compute largest channel gain derived finally correlation coefficient rewritten plugging cvn chn eferences song choi lee kim seol love advanced quantizer designs systems using uniform planar arrays proceedings ieee global telecommunications conference marzetta noncooperative cellular wireless unlimited numbers base station antennas ieee transactions wireless communications vol hansen phased array antennas hoboken wileyinterscience nam sayana zhang kim lee mimo next generation cellular technology ieee communications magazine vol jun discussions codebook enhancements tsg ran codebook enhancements tsg ran codebook antenna arrays tsg ran zeng zhao xiao codebook design uniform rectangular arrays massive antennas proceedings ieee vehicular technology conference jun choi lee love kim heath advanced limited feedback designs using uniform planar arrays proceedings ieee global telecommunications conference choi kim love seol exploiting preferred domain fdd massive mimo systems uniform planar arrays proceedings ieee international conference communications jun ngo larsson marzetta massive downlink tdd systems linear precoding downlink pilots proceedings allerton conference communication control computing rogalin bursalioglu papadopoulos caire molisch michaloliakos balan psounis scalable synchronization reciprocity calibration distributed multiuser mimo ieee transactions wireless communications vol mar hassibi hochwald much training needed wireless links ieee transactions information theory vol apr love performance random vector quantization limited feedback beamforming miso system ieee transactions wireless communications vol love sanayei trellis coded line packing large dimensional beamforming vector quantization feedback transmission ieee transactions wireless communications vol apr choi chance love madhow noncoherent trellis coded quantization practical limited feedback technique massive mimo systems ieee transactions communications vol choi love kim codebooks successive phase adjustment path fdd massive mimo systems ieee transactions wireless communications vol apr choi love bidigare downlink training techniques fdd massive mimo systems training memory ieee journal selected topics signal processing vol noh zoltowski sung love pilot beam pattern design channel estimation massive mimo systems ieee journal selected topics signal processing vol han lee love compressed downlink channel training fdd massive mimo systems appear ieee transactions communications mukkavilli sabharwal erkip aazhang beamforming finite rate feedback systems ieee transactions information theory vol love heath strohmer grassmannian beamforming wireless systems ieee transactions information theory vol love heath lau gesbert rao andrews overview limited feedback wireless communication systems ieee journal selected areas communications vol love heath grassmannian beamforming correlated mimo channels proceedings ieee global telecommunications conference limited feedback diversity techniques correlated channels ieee transactions vehicular technology vol mar xia georgios design analysis based feedback ieee transactions signal processing vol may raghavan sayeed veeravalli limited feedback precoder design spatially correlated mimo channels proceedings conference information sciences systems mar raghavan choi love design guidelines limited feedback spatially correlated broadcast channel ieee transactions communications vol jul study channel model lte spatial channel model mutiple input multiple output simulations performance evaluations feedback framework tsg ran may way forward codebook mimo tsg ran han jin huang jiang wang design double codebook based channel multiuser mimo system eurasip journal advances signal processing vol jul mailloux phased array antenna handbook artech house ying vook thomas love ghosh kronecker product correlation model limited feedback codebook design channel model proceedings ieee international conference communications jun ayach rajagopal heath spatially sparse precoding millimeter wave mimo systems ieee transactions wireless communications vol mar borga learning multidimensional signal processing dissertation university sweden stuart ord kendall advanced theory statistics distribution theory london arnold david order statistics new york john wiley sons
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fast distributed algorithms testing graph properties keren eldar gregory yadu may may abstract initiate thorough study distributed property testing producing algorithms approximation problems property testing congest model particular dense graph testing model emulate sequential tests nearly graph properties tests general sparse models obtain faster tests trianglefreeness bipartiteness respectively addition show logarithmic lower bound testing bipartiteness holds even stronger local model cases aided parallelism distributed algorithms much shorter running time compared counterparts sequential querying model traditional property testing simplest property testing algorithms allow relatively smooth transitioning distributed model complex tasks develop new machinery may independent interest technion israel institute technology department computer science ckeren eldar gregorys yaduvasudev supported part israel science foundation grant introduction performance many distributed algorithms naturally depends properties underlying network graph therefore inherent goal check whether graph given subgraph certain properties however cases known hard congest model model computation proceeds synchronous rounds every vertex send log message neighbors lower bounds number rounds type known verifying many global graph properties number vertices network diameter see overcome difficulties adopt relaxation used graph property testing first defined distributed setting rather aiming exact answer question whether graph satisfies certain property settle distinguishing case satisfying case appropriate measure far apart theoretical interest relaxation motivated common scenario distributed algorithms tasks perform better given certain property network topology given graph almost satisfies property example hirvonen show algorithm finding large cut graphs additional constraints finding fraction edges part triangle similarly pettie provide fast algorithms coloring graphs construct fast distributed algorithms testing various graph properties important byproduct study toolbox believe useful settings well contributions provide rigorous study property testing methods realm distributed computing congest model construct error distributed graph satisfies property vertices output accept satisfying property least one vertex outputs reject probability least using standard amplification method invoking test log times vertex output reject least one invocation output reject gives rejection higher probability price multiplicative log factor number rounds definition graph satisfying property roughly one following see section precise definitions changing entries adjacency matrix give graph satisfies property dense model changing max entries adjacency matrix give graph satisfies property number edges general model particular case degrees bounded constant resulting graph must comply restriction well sparse model sequential access input provided queries whose type depends model dense model asking whether two vertices neighbors general sparse models either asking degree vertex asking neighbor ordering neighbors arbitrary sequential touch small handful vertices queries distributed test lack ability communicate large distances offset vertices operating parallel first contribution general scheme emulation distributed hides factors polylogarithmic context originating dense graph model section makes use fact dense model sequential testing algorithms made roughly means queries depend responses previous queries see section definition fact tests made simple structure allowing vertices distributed model band together emulation test one additional technical condition define since distributed model handle properties whose split disjoint graphs example distributed model hope handle property graph disjoint union two triangles property exists test dense model theorem dense graph model property makes queries converted distributed takes communication rounds next move away dense graph model sparse general models sometimes considered realistic general model exists test property containing triangle makes number queries independent number graph vertices distributed model better reason deficiency addressed vertices operate concurrently section adapt interim lemmas used best testing algorithm constructed construct distributed algorithm whose number rounds independent theorem algorithm distributed general graph model property containing triangles requires rounds sparse general models inherently require adaptive property testing algorithms since way trace path given vertex forward follow neighborhood testing triangle freeness sequentially uses adaptivity small degree however problems sparse general models one explore next high degree adaptivity built sequential algorithms need take special care emulating distributed setting sparse model degrees bounded constant adapt ideas bipartiteness testing algorithm search cycles performance distributed algorithm surpasses test number rounds polylogarithmic number queries lower bound given following proved section theorem algorithm distributed bounded degree graph model property bipartite requires poly log rounds course proving theorem develop method consider independent algorithm works performing random walks concurrently two starting vertex parallel execution random walks despite congestion restriction achieved making sure walks uniform stationary distribution showing congestion close average uniform stationary distribution constant section show fast test makes use combinatorial lemma prove cycles remain graph removing edges independently probability following summarizes result testing technique recently independently concurrently devised different use theorem algorithm distributed general graph model property requires log rounds also prove lower bounds testing bipartiteness matching upper bound latter roughly speaking obtained using probabilistic method alterations construct graphs far bipartite cycles length least logarithmic technique bears similarity classic result showed existence graphs large girth large chromatic number following given section theorem distributed property bipartite requires log rounds communication theorem distributed property requires log rounds communication roadmap paper organized follows remainder section consists related work historical background property testing section contains formal definitions mathematical tools emulation sequential tests dense model given section section give distributed test section provide distributed test bipartiteness along new method executing many random walks section give test section gives logarithmic lower bounds testing bipartiteness conclude short discussion section related work previous work directly relates distributed setting due brakerski show tolerant property testing algorithm finding large linear size graph clique set vertices pairs vertices edge algorithm tolerant sense finds linear exists linear clique testing algorithm considers two thresholds close property case containing linear size clique unaware work property testing distributed setting testing different distributed setting considered arfaoui study testing setting vertex may collect information entire neighborhood distance send short string bits central authority decide whether graph related information sent received central authority concept schemes introduced korman extensions see baruch setting vertex given external label exchanging labels vertices need decide whether given property graph holds different setting information vertex ids available another setting related schemes differs model model foerster sequential property testing goal computing without processing entire input wider family local computation algorithms lca known connections distributed computing shown parnas ron later used others recent study proves conditions fact centralized algorithm query distant vertices help speeding computation however consider local model results apply certain properties influenced distances finding induced subgraphs crucial task studied several different distributed models see notice finding subgraphs many instances desired subgraph help speedup computation contrast algorithms perform faster instances explained test property rather property containing triangles notice fact every graph vertices triangle parallelizing many random walks addressed question graph covering via random walks discussed shown certain families graphs substantial speedup time takes walks starting vertex cover graph compared single walk edge congestion constraints taken account shown perform congestion single random walk length rounds random walks kld rounds diameter graph method dependence diameter allowing perform multitude short walks much faster historical overview first papers consider question property testing original motivations defining property testing connection computerized learning models ability leverage properties construct probabilistically checkable proofs pcps related property testing areas locally testable codes locally decodable codes ltcs ldcs motivations since entered fray foremost among algorithms considerations since virtually property decidable without reading entire input property testing introduces notion allowable approximation original problem general algorithm distinguish inputs satisfying property inputs information general scheme classical property testing consult surveys older graph testing models discussed dense model defined seminal work goldreich goldwasser ron dense graph model historically kickstarted combinatorial property testing earnest shortcomings main one distance function makes sense consider graphs many edges hence name dense model graph edges indistinguishable model empty graph stricter times plausible distance function one relative actual number edges rather maximum general model defined sparse model defined already main difference sparse general graph models former also guaranteed upper bound degrees vertices given algorithm advance query complexity may depend either explicitly commonly implicitly considering constant preliminaries additional background property testing introduction provided rough descriptions different property testing models provide formal definitions dense model property testing defined follows definition dense graph model dense graph model considers objects graphs given adjacency matrix hence defined following features distance two graphs vertices considered one obtained deleting inserting edges constant factor normalized hamming distance querying scheme single query algorithm consists asking whether two vertices form graph edge allowable properties properties invariant permutations input pertain graph isomorphisms prerequisite graph properties number vertices given algorithm advance discussed earlier sparse general models property testing relate distance function actual number edges graph formally defined follows definition sparse general graph models two models consider objects graphs given adjacency lists defined following features distance two graphs vertices edges defined denser two considered one obtained deleting inserting max querying scheme single query consists either asking degree vertex asking neighbor ordering neighbors arbitrary allowable properties properties invariant graph isomorphisms translate relabeling affects vertex order neighbor ids obtained neighbor queries reordering individual neighbor lists orderings considered arbitrary paper mainly refer distance functions models less querying scheme since latter replaced processing scheme provided distributed computation model note property testing models get one bit response query response edge dense graph model however sparse general models may receive log bits information one query neighbor vertex also degree vertex given answer query general model takes log bits since distributed congest model allows passing vertex vertex degree along edge rounds equally relate three graph models another important point difference testing algorithms difference adaptive algorithms sometimes sparse graph model allowed number changes relates maximum possible number edges held constant difference essential definition types algorithms property testing algorithm said error possibility error accepting satisfying inputs input satisfies property accepted probability input property rejected probability high enough traditionally means probability least error algorithm also allowed reject satisfying inputs long probability correct answer high enough traditionally least property testing algorithm said decides queries advance based internal coin tosses receiving results query output may depend actual input adaptive algorithm may make query turn based results previous queries possible internal coin tosses following address adaptive algorithms however restrict error algorithms since notion error good match distributed computation model mathematical background important role analyses played multiplicative chernoff bound see hence state completeness fact suppose independent random variables taking values let denote sum let denote expected value convenient variations bounds distributed emulation sequential tests dense model begin showing certain assumption define property sequential test dense model requires queries tested distributed setting within rounds prove constructing emulation translates sequential tests distributed ones first introduce definition witness graph adapt theorem restricted error tests terminology definition let property graphs vertices let graph vertices say witness induced subgraph graph satisfies notice induced subgraph witness definition also witness work transforms tests graphs dense model canonical form query scheme based vertex selection useful particular distributed model computational work essentially based vertices require following special case error tests lemma theorem let property graphs vertices exists error query complexity exists error uniformly selects set vertices accepts induced subgraph witness emulation leverages lemma assumption property define follows definition say property every graph satisfy induced subgraph witness connected component also witness call components witness components ready formally state main theorem section theorem dense graph model property makes queries converted distributed takes communication rounds following lemma essentially says satisfying property rely subgraphs connected exactly need forbid distributed setting lemma property property minimal witnesses induced subgraphs connected minimal refers standard terminology means proper induced subgraph witness proof first satisfy every subgraph witness witness component minimal must connected since otherwise contains connected component witness contradicts minimality direction minimal witnesses induced subgraphs connected every induced subgraph witness either minimal case connected minimal case subgraph connected minimal witness connected component contains witness otherwise witness hence follows next give distributed test algorithm test outer loop vertex picks probability collects neighborhood certain size edges picked vertices inner loop rejects identifies witness outer loop repeats two times sequential test error probability also small probability may randomly pick many enough vertices order emulate repeating main loop twice reduces error probability back inner loop vertex collects neighborhood picked vertices checks connected component witness limit communications done components picked vertices sufficiently small vertex detects part component many edges accepts participate next iteration outer loop algorithm emulation algorithm input property variables edges known edges update send temporary variables perform times reset state vertices vertex simultaneously vertex picks probability picked notify neighbors picked set picked perform times iteration subgraph connected component need recently discovered edges add operate many edges send picked neighbours propagate known edges wait time bound vertices finish iteration set union edge sets received neighbors witness vertex outputs reject ending operations else wait time bound vertices finish iteration outermost loop every vertex reject outputs accept analyze algorithm begin proving constant probability number picked vertices sufficient large lemma probability number vertices picked algorithm proof every denote indicator variable event vertex picked note independent random variables using notation gives vertex picked probability using chernoff bound fact bound probability picked vertices bounding probability many picked vertices use direction chernoff bound giving thus probability least holds use guarantees sequential test obtain guarantees algorithm lemma let graph property satisfies vertices output accept algorithm satisfying probability least exists vertex outputs reject proof first assume satisfies vertex outputs reject part witness definition component extended satisfies however every component induced subgraph satisfy thus every component extended implies vertex outputs reject assume satisfying since sequential test rejects probability least probability sample least vertices induces graph extended graph satisfies least induced subgraph must connected witness note sample vertices reduce rejection probability hence denote event subgraph induced picked vertices connected witness conditioned least vertices picked however sample large may cause vertex output accept collect neighborhood denote event number vertices sampled lemma probability least bound using bayes theorem obtaining since outer loop consists independent iterations gives probability least vertex outputs reject address round complexity vertex sends receives information edges chosen vertices many vertices chosen detect accept otherwise communicate chosen vertices edges requires communication rounds using standard pipelining together lemma proves theorem applications perfect graphs next provide examples usage theorem result alon shapira states graph properties closed induced subgraphs testable number queries depends note except certain specific properties proofs dependence usually tower function worse asymptotically larger together lemma theorem deduce property closed induced subgraphs testable every fixed constant number communication rounds pipelining means vertex buffer edge holds information edges chosen vertices case needs send edge vertex sends pieces information one example property testable distributed manner algorithm minimal graphs witnesses connected therefore according lemma property closed induced subgraphs exists error uniformly picks log vertices number queries square expression note polynomial dependency already known emulation implies distributed error requires poly rounds example perfect graphs graph said perfect every induced subgraph chromatic number equals size largest clique another characterization perfect graph via forbidden subgraphs graph perfect odd holes induced cycles odd length least odd complement graph odd hole odd holes odd connected graphs since minimal witnesses property according lemma property using result know property graph perfect testable emulation implies distributed error perfect graph requires number rounds depends distributed test section show distributed notice since property theorem gives distributed dense model number rounds number queries required sequential however known number queries tower function log leverage inherent parallelism obtain checking neighbors vertex show test requires rounds algorithm importantly algorithm works dense graph model general graph model distances relative actual number edges subsumes sequential setting test general model requires number queries constant power proof actually follows groundwork laid general graph model algorithm picks vertex checks two neighbors connected perform check vertices parallel theorem algorithm distributed general graph model property containing triangles requires rounds line proof follows distinguishing edges connect two vertices formally let number edges graph denote deg say edge light otherwise say heavy set heavy edges begin following simple claim number heavy edges claim number heavy edges proof number heavy edges since get gives algorithm triangle freeness test vertex simultaneously perform times pick uniformly random send ask neighbor foreach sent asked neighbor send yes else send received yes reject ending operations accept vertices reject next fix iteration algorithm every vertex chooses two neighbors let first two vertices chosen vertex let light edge triangle let say edge matched triangle matched triangle detected begin following lemma states iteration bound expected number matched edges let number matched edges lemma expected number matched edges single iteration algorithm greater proof every let random variable indicating whether matched giving following bound matched last inequality follows light edge chosen vertex degree hence third triangle vertex gets picked probability least next argue see every edge let random variable indicating whether let last inequality follows light edge least one endpoint degree hence edge gets picked probability least remains bound claim prove first notice since triangle free least triangle edges since otherwise remove make graph triangle free less edge changes claim number heavy edges satisfies subtracting number triangle edges gives least edges light triangle edges finally inequalities using iterated expectation get eat prove correctness algorithm follows lemma vertices output accept algorithm probability least exists vertex outputs reject proof triangle free iteration receives iterations returns accept assume let indicator variable event vertex detects triangle iteration first note indicators independent since vertex detecting triangle affect chance another vertex detecting triangle note graph fixed iterations done independently let number detections iterations notice total lemma implies fixed holds sums using chernoff bound fact gives hence probability least least one triangle detected associated vertex outputs reject completes proof every iteration vertex initiates two messages size log bits one sent one sent back since iterations implies number rounds well together lemma completes proof theorem distributed bipartiteness test bounded degree graphs section show distributed bipartite graphs degrees bounded test builds upon sequential test case triangle freeness takes advantage ability parallelize queries number queries sequential test number rounds distributed test polylogarithmic polynomial assume constant omit expressions implicit notation let first outline algorithm since distributed test borrows framework analysis part derived sequential test basically tries detect odd cycles consists iterations vertex selected uniformly random random walks length performed starting source iteration chosen source vertex reached even prefix random walk odd prefix random walk possibly walk algorithm rejects indicates existence odd cycle otherwise algorithm accepts obtain parameters chosen main approach distributed test similar except key ingredient afford perform much fewer random walks every vertex namely poly log run random walks parallel originating vertices however crucial challenge need address several random walks may collide edge violating congestion bound address issue central observation lazy random walks chosen uniform stationary distribution provide low probability many collisions main part analysis showing high probability never many walks concurrently vertex comply congestion bound begin formally defining lazy random walks use definition lazy random walk graph degree bound random walk sequence random variables taking values vertex set transition probability edge deg cases stationary distribution lazy random walk definition uniform section next describe procedure handle one iteration moving random walks algorithm followed distributed test bipartiteness using lazy random walks every vertex concurrently algorithm quite immediate algorithm takes communication rounds main result algorithm indeed distributed bipartiteness theorem algorithm distributed bounded degree graph model property bipartite requires poly log rounds number communication rounds immediate algorithm dominated calls algorithm making total rounds indeed poly log prove rest theorem need notation lemma bounds probabilities detecting odd cycles bipartite given source vertex reached even prefix random walk odd prefix random walk say walks detect violation let probability random walks length starting two detect violation using notation probability sequential algorithm outlined beginning rejects iteration chosen since interested walks length denote good vertex vertex probability bounded follows algorithm move random walks input variables walks residing multiset history walks input maximum congestion per vertex allowed walk characterized number actual moves origin vertex vertex simultaneously give exceeded maximum allowed every draw next destination according lazy walk scheme walk exits send remove wait maximum time vertices process walks add walks received walks entering algorithm distributed bipartiteness test variables walks residing multiset history walks perform log times vertex simultaneously initialize two copies walk perform times move walks using algorithm input vertex simultaneously contains even odd reject ending operations odd cycle found accept vertices reject definition vertex called good proved far bipartite implies many good vertices lemma bipartite least vertices good contrast perform random walks every vertex iteration rather hence need analysis bound end use parameter express terms lemma every vertex proof fix source vertex every let probability walks detecting violation different walks independent conclude every holds let event walks detecting violation implies using relationship prove algorithm first prove random walks ignoring possibility algorithm skip moving random walks due condition line lemma bipartite perform iterations starting random walks length every vertex probability violation detected bounded proof assume bipartite lemma least vertices good means vertices implies let random variable indicating whether two random walks starting detect violation let prove first bound fixed log holds using chernoff bound fact gives completes proof explained earlier main hurdle road prove theorem proving allowed congestion exceeded prove following general claim probability lazy random walks length vertex exceed maximum congestion factor walks allowed vertex beginning iteration iteration sequence rounds walks advanced one step whether actually switch vertices lemma probability least running lazy random walks length originating every vertex exceed maximum congestion factor walks allowed vertex beginning iteration show plugging lemma together lemma gives correctness algorithm prove lemma argue unlikely vertex walks iteration given indeed case every iteration lemma follows union bound denote random variable whose value number random walks vertex beginning iteration equal size set description algorithm lemma every vertex every iteration holds proof let first define random variables walks enumerating walks vertices arbitrarily let denote sequence corresponding walk yir vertex walk stationed beginning iteration particular yir let define new random variables zit following manner first choose uniformly random permutation set zit main thing note fixed random walk equal one random walks also every uniformly distributed vertex set started exactly random walks every vertex additionally since uniform distribution stationary lazy walks means unconditional distribution zit also uniform since permutation holds yir expectation linearity expectation thus prove lemma proof lemma first claim every iteration every vertex probability least holds show first fix let indicator variable event walk residing vertex beginning iteration variables independent use chernoff bound fact proven lemma obtaining applying union bound vertices iterations obtain probability least holds lemma bipartite vertices output accept algorithm bipartite probability least exists vertex outputs reject proof bipartite vertices output accept algorithm odd cycles thus violation detecting walks bipartite use lemma conjunction lemma parameters used algorithm union bound probability accept bounded assuming providing required bound rejection probability lemma communication complexity analysis algorithm gives theorem distributed test section give distributed algorithm test graph edges least edges removed make intuitively order search cycles one run search bfs vertex output reject two different paths reach downside exact solution running time depends diameter graph overcome basic approach would run bfs vertex graph shorter distances however running multiple bfss simultaneously expensive due congestion edges instead use simple prioritization rule drops bfs constructions lower priority makes sure one bfs remains instead technique consists three parts first make graph sparser removing edges independently probability denote sampled graph prove far particular contains cycle run partial bfs vertex prioritizing ids vertex keeps bfs originates vertex largest drops rest bfss length procedure according threshold log gives detection cycle contained component low diameter cycle exists since surviving bfs covers component cycle also cycle cycle exists component diameter larger large components take surviving bfs reached vertex certain distance run new partial bfs original graph bfss prioritized time according distance main tool proving claim says high probability shortest path length two vertices cycle length allows bfss find cycle start following combinatorial lemma shows claim lemma given graph let obtained deleting edge probability independently edges probability least every vertex vertex distance log closed path passing contains simple cycle length log proof first show every pair vertices distance log one shortest paths removed graph high probability pair vertices distance probability shortest path removed therefore union bound pairs vertices probability least least one edge removed least one shortest path every pair vertices distance log conditioned prove lemma suppose two vertices distance log let shortest path suppose shortest path log path longer present thus distinct closed path passing simple cycle length log log least two shortest paths length log one involved analysis multiple prioritized bfs executions used allowing bfs executions fully finish short time without much delay due congestion since require much weaker guarantee avoid strong prioritization algorithm settle simple rule keeps one bfs tree alive also multiple bfs construction fit demands may reach desired vertices within required distance case many vertices closer one removed choose therefore closed path passing length log hence contains simple cycle length log next prove indeed high probability contains cycle far claim probability least proof graph obtained deleting edge probability independently edges expected number edges deleted therefore chernoff bound fact probability least edges deleted exp claim follows describe algorithm takes input length priority condition vertices starts performing bfs vertex graph done steps vertex keeps bfs highest priority dropping rest vertex also maintains list bfss passed list list idu idp idu root bfs depth bfs tree idp parent bfs tree initially vertex sets include bfs starting continues bfs sending tuple idv idv neighbors idv identifier vertex intermediate step vertex may receive bfs tuple neighbors vertex adds bfs tuples list chooses one among according priority condition proceeding respective bfs discontinuing rest even bfs discontinued information bfs reached stored list algorithm gives formal description search use testing algorithm algorithm bfs priority condition input length priority condition variables list bfs tuples passing vertex simultaneously initialize idv idv send idv idv neighbors perform times times vertex simultaneously receives idur idpr neighbors add idur idpr select iduj idpj according idui send iduj idv neighbors except give informal details test lemma know vertex vertex distance log closed path starting contains cycle length log first part vertex gets name vertex performs bfs graph hope finding cycle bfs performed using algorithm priority condition intermediate steps selecting bfs lowest origin cycle present component diameter log discovered bfs check cycle one needs find appropriate tuples idu idp idu vertex cycle discovered step change ids vertices following way vertex tuple largest depth occurs bfs tree among searches reached perform bfs using algorithm priority condition pick bfs whose root lexicographically highest vertex log highest priority vertex vertex lemma bfs starting vertex detect cycle algorithm gives formal description tester algorithm test variables list bfs tuples passing vertex identifier idv construct deleting edges probability vertex simultaneously neighbor mark edge probability deletion send marked edge corresponding set idv vertex simultaneously delete edges incident marked deletion search cycles small diameter components use algorithm perform bfs log steps priority condition choosing bfs lowest root vertex simultaneously contains two tuples idu idp idu output reject set idv highest among tuples idui idpi use algorithm perform bfs log steps priority condition choosing bfs lexicographically highest root vertex simultaneously contains two tuples idu idp idu output reject vertex simultaneously output accept output reject yet prove correctness algorithm theorem algorithm distributed general graph model property requires log rounds proof notice vertex algorithm outputs reject detects cycle therefore every vertex outputs accept probability suppose notice probability least assertion lemma holds furthermore claim know probability hence contains least one cycle cycle could component diameter less log could component diameter least log analyse two cases separately suppose cycle component diameter log let vertex smallest algorithm bfs starting always propagated intermediate vertex due priority condition furthermore since diameter log bfs reaches vertices hence bfs detects cycle least one vertex outputs reject hand cycle present component diameter least log step algorithm vertex gets length longest path origin among bfss reached first component vertex gets lexicographically highest component vertex least log away since radius component least half diameter therefore lemma part cycle length log hence vertex highest priority bfs vertex vertex distance least log walk contain simple cycle length log least one vertex simple cycle output reject algorithm run number rounds log since algorithm performs two searches graph number rounds lower bounds testing bipartiteness section prove distributed algorithm bipartiteness graphs requires log rounds construct boundeddegree graphs bipartite cycles length log argue distributed algorithm runs log rounds detect witness nonbipartiteness also show construction proves every distributed algorithm requires log rounds communication formally prove following theorem theorem distributed property bipartite requires log rounds communication prove theorem show existence graph far bipartite cycles least logarithmic length since rounds distributed algorithm output every vertex depend vertices distance greater vertex detect cycle less log rounds proves theorem prove existence use probabilistic method alterations prove following lemma let random graph vertices edge present probability let obtained removing edges incident vertices degree greater one edge cycle length log log probability least bipartite lower bound applies even less restricted local model communication limit size messages since graph bipartite also immediately obtain lower bound testing follows theorem distributed property requires log rounds communication rest section devoted proving lemma need show three properties far bipartite small cycles maximum degree bounded begin following definition similar spirit far satisfying property assist proof definition graph bipartite least edges removed make bipartite note graph maximum degree bipartite bipartite let random graph vertices pair vertices edge present probability expected number edges graph since edges sampled independently probability chernoff bound fact probability least graph least edges show far bipartite high probability lemma far bipartite probability least bipartite proof fix bipartition vertex set pair vertices let random variable edge present otherwise expected value random variable counts number edges within linearity expectation since random variables independent chernoff bound fact exp therefore probability least exp least edges within total number bipartitions taking union bound bipartitions probability least one bipartitions contains less edges within side exp lemma follows expected degree vertex therefore chernoff bound fact probability degree greater exp show sufficiently high probability number edges incident high degree vertices small remove edges obtain graph still far bipartite lemma mostly bounded degrees probability least edges incident vertices degree greater proof pair vertices probability edge one degree greater degree exp therefore expected number edges incident vertex degree greater exp markov inequality probability least edges incident vertices degree greater exp completes proof lemma bound number cycles length log graph lemma small cycles probability least cycles length log log proof fixed vertices probability cycle among vertices therefore expected number cycles length log log means expected number cycles length log log therefore probability least cycles length log log ready prove lemma completing lower bounds intuitively since contain many high degree vertices many small cycles removing obtain changes distance bipartite small term proof probability least edges lemma bipartite lemma probability least edges incident vertices degree greater lemma probability least cycles length log log hence probability least graph degree bipartite therefore bipartite discussion paper initiates thorough study distributed property testing provides emulation technique dense graph model constructs fast distributed algorithms testing trianglefreeness bipartiteness also present lower bounds bipartiteness triangle freeness work raises many important open questions immediate devise fast distributed testing algorithms additional problems one example testing freeness small subgraphs ambitious goals handle dynamic graphs find general connections testability sequential model distributed model finally fertile ground obtaining additional lower bounds setting order fully understand complexity distributed property testing references noga alon chen avin michal gady kozma zvi lotker mark tuttle many random walks faster one combinatorics probability computing noga alon tali kaufman michael krivelevich dana ron testing general graphs siam discrete noga alon michael krivelevich testing siam discrete noga alon asaf shapira characterization natural graph properties testable error siam heger arfaoui pierre fraigniaud david ilcinkas fabien mathieu distributedly testing concepts computer science international workshop france june revised selected papers pages mor baruch pierre fraigniaud boaz randomized schemes proceedings acm symposium principles distributed computing podc spain july pages manuel blum michael luby ronitt rubinfeld applications numerical problems comput syst zvika brakerski boaz distributed discovery large distributed computing keren petteri kaski janne korhonen christoph lenzen ami paz jukka suomela algebraic methods congested clique proceedings acm symposium principles distributed computing podc spain july pages maria chudnovsky neil robertson paul seymour robin thomas strong perfect graph theorem annals mathematics danny dolev christoph lenzen shir peled tri tri finding triangles small subgraphs distributed setting extended abstract distributed computing international symposium disc salvador brazil october proceedings pages andrew drucker fabian kuhn rotem oshman communication complexity distributed task allocation acm symposium principles distributed computing podc funchal madeira portugal july pages paul graph theory probability canad math eldar fischer art uninformed decisions primer property testing current trends theoretical computer science challenge new century foerster thomas luedi jochen seidel roger wattenhofer local checkability strings attached proceedings international conference distributed computing networking icdcn singapore january jacob fox new proof graph removal lemma corr mohsen ghaffari fabian kuhn manuscript oded goldreich shafi goldwasser dana ron property testing connection learning approximation acm oded goldreich dana ron sublinear bipartiteness tester bounded degree graphs combinatorica oded goldreich dana ron property testing bounded degree graphs algorithmica oded goldreich dana ron algorithmic aspects property testing dense graphs model property testing current research surveys outgrow workshop institute computer science itcs tsinghua university january pages oded goldreich luca trevisan three theorems regarding testing graph properties random struct algorithms mika juho hirvonen reut levi moti medina jukka suomela probes help graph problems corr juho hirvonen joel rybicki stefan schmid jukka suomela large cuts local algorithms graphs corr stephan holzer roger wattenhofer optimal distributed pairs shortest paths applications proceedings acm symposium principles distributed computing podc pages new york usa acm jarkko kari matamala ivan rapaport ville salo solving induced subgraph problem randomized multiparty simultaneous messages model structural information communication complexity international colloquium sirocco montserrat spain july pages amos korman shay kutten david peleg proof labeling schemes distributed computing christoph lenzen david peleg efficient distributed source detection limited bandwidth acm symposium principles distributed computing podc montreal canada july pages michael mitzenmacher eli upfal probability computing randomized algorithms probabilistic analysis cambridge university press michal parnas dana ron approximating minimum vertex cover sublinear time connection distributed algorithms theor comput david peleg distributed computing approach society industrial applied mathematics seth pettie distributed coloring algorithms graphs inf dana ron property testing learning theory perspective foundations trends machine learning dana ron algorithmic analysis techniques property testing foundations trends theoretical computer science ronitt rubinfeld madhu sudan robust characterizations polynomials applications program testing siam atish das sarma stephan holzer liah kor amos korman danupon nanongkai gopal pandurangan david peleg roger wattenhofer distributed verification hardness distributed approximation siam atish das sarma danupon nanongkai gopal pandurangan prasad tetali distributed random walks acm
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feb multidimensional multiscale scanning exponential families limit theory statistical consequences claudia institute mathematical stochastics university axel munk munk institute mathematical stochastics university felix bernstein institute mathematical statistics bioscience university max planck institute biophysical chemistry germany frank institute mathematical stochastics university max planck institute biophysical chemistry germany paper consider problem finding anomalies field independent random variables tyi nud distributed according onedimensional natural exponential family given baseline parameter field scanned using local likelihood ratio tests detect large given system regions regions provide unified methodology controls overall family wise error fwer make wrong detection given error rate fundamental method gaussian approximation asymptotic distribution underlying multiscale scanning test statistic explicit rate convergence obtain weak limit theorem seen generalized weak invariance principle non identically distributed data independent interest furthermore give asymptotic expansion procedures power yields minimax optimality case gaussian observations keywords exponential families multiscale testing invariance principle scan statistic weak limit family wise error rate ams classification numbers primary secondary introduction suppose observe independent field random variables corresponding ind nud author observation drawn given natural exponential family model potentially different parameters prominent examples include varying normal means poisson field varying intensities given baseline parameter gaussian field consider problem finding anomalies hot spots field aim identify regions ind runs given family candidate regions power set set simplicity suppress subindex whenever clear context write follows problems occur numerous areas application ranging astronomy biophysics genetics engineering specific examples include detection radiographic images kazantsev genome screening jiang object detection astrophysical image analysis friedenberg genovese mention setting includes important special cases gaussian sharpnack bernoulli walther poisson random fields zhang extensions models without exponential family structure well replacing baseline parameter varying field known baseline intensities treated well remark keep presentation simple restrict afore mentioned setting inline mentioned references see also section problem finding hot spots regarded multiple testing problem many local tests regions performed simultaneously keeping overall error wrong detections controllable fixed region likelihood ratio test lrt testing problem powerful test general often known certain optimality properties depending structure see lehmann romano therefore lrt always considered throughout paper local test stress however methodology could also used systems local tests provided obey sufficiently well behaving asymptotic expansion lrt based test statistic ipr pyi log ipr pyi denotes density rejected large known priori regions might contain anomalies alternative might hold true great importance control family wise error arising multiple test decisions local tests based obviously without restriction complexity error controlled end assume regions represented sequence discretized regions ind ind system subsets hypercubes unit cube specified later gives rise sequence multiple testing problems simultaneously aim paper provide methodology control asymptotically family wise error rate fwer considered multiple testing problem provide multiple test see dickhaus sup phr false rejection rprn words ensures probability making wrong detection controlled given error level task focus several papers last decades detailed discussion see section contribute field providing general theory unifying method model includes gaussian sharpnack kou cheng schwartzman also bernoulli walther poissonian observations kulldorff rivera walther zhang view also observations exponential families discussed local tests always gaussian case emphasize local test type hence exploiting likelihood exponential family result improved power see frick main technical contribution prove weak limit theorem asymptotic distribution test statistic general exponential family models arbitrary dimension viewed multiscale weak invariance principle independent necessarily identically distributed provide asymptotic expansion test power leads minimax optimal detection test specific models multiscale testing test controlling fwer maximum local lrt statistics max rtr penv rprn denotes number points values penv prq plog pnd log denotes natural logarithm act scale penalization necessary guarantee optimal detection power scales simultaneously prevents smaller regions dominating overall test statistic noticed spokoiny others see walther walther frick constant upper bound given complexity measured terms packing number see remark whenever finite choose however test better detection properties small possible property see section hence point view advantageous know exactly complexity topic received less attention computing therefore compute packing numbers three important examples namely hyperrectangles hypercubes halfspaces explicitly appendix construct test controls fwer find sequence universal global thresholds rtn phi corresponds case anomaly present threshold suffices readily seen sup phr false rejection false rejection ind rprn given multiple test reject whenever local test rejects penv due rejections correct probability asymptotically obtain thresholds provide gaussian approximation scan statistic given prn max penv rprn standard normal ind thm also give rate convergence approximation thm determined smallest scale based results obtain distribution sup penv white noise slight abuse notation denotes lebesgue measure holds true soon finite complexity consists sets sufficiently regular boundary see assumption smallest scales system restricted suitably see discussion case subset hypercubes also give asymptotic expansion test power allows determine necessary average strength anomaly detected asymptotic probability possible due penalization otherwise asymptotic distribution finite anomaly sufficiently small show anomalies detected asymptotic power one described multiscale testing procedure oracle single scale test knows size scale anomaly advance generalizes findings sharpnack situations mean signal allowed change whole distribution furthermore observations gaussian test properly chosen achieves asymptotic optimal detection boundary test larger power minimax sense asymptotically note finally weak convergence viewed generalized weak invariance principle depend unknown quantity hence simulated generically advance given system soon bound complexity determined literature review connections existing work scan statistics procedures based maximum ensemble local tests received much attention literature past decades determine quantile common option approximate tails asymptotic distribution suitably done siegmund venkatraman siegmund yakir naus wallenstein pozdnyakov fang siegmund haiman preda jiang arbitrary dimensions random field sufficiently smooth contrast setting gaussian kinematic formula employed see taylor worsley adler also mention alm considers situation fixed rectangular scanning set two three dimensions papers penalization used automatically leads preference small scales order logpnq see kabluchko munk extreme value limit contrast study case unknown null distribution propose permutation based approximation shown perform well natural exponential family setting however conceptually related work weak limit theorems scale penalized scan statistics obtained frick sharpnack however results either limited special situations gaussian observations limit exists quantiles finite sample statistic bounded quantiles limiting ones done spokoiny rivera walther results interpreted ways provide gaussian approximation scan statistic obtain weak limit weak limit theorems immediately connected partial sum processes classical approximations see rio massart provide fact strong coupling whole process ptr qqrprn gaussian version results form employed previously frick proceeding like general restrict system scales logpnq prn unfeasibly large therefore take different route employ coupling maxima relies recent results chernozhukov see also proksch however contrast present paper consider local statistic require largest scale zero leads extreme value type limit contrast incorporates larger scales make use chernozhukov coupling results general setting provide symmetrizationlike upper bound expectation maximum partial sum process corresponding gaussian version proposition able approximate distribution soon restrict smallest scales need satisfy pnq prn compared allows considerably smaller scales whenever note lower scale restriction depend however consider sets ind corresponding lower bound sets pnq pan fact depends volume largest possible set standardized one see theorem coincides sampling rate contrast gives logpnq pan independent achieves sampling rate also obtain log pnq rate convergence approximation see also asymptotic power procedures discussed literature mention walther studies detection spacial clusters two dimensional bernoulli field kabluchko gives exact asymptotic expansions gaussian setting obtain optimal power scales proper penalization necessary stressed sharpnack provide optimality results twodimensional gaussian fields respectively butucea ingster kou general provide optimality scanning procedures gaussian fields able generalize results case set hypercubes exponential family model despite fact alternative whole distribution might change whereas gaussian fields typically mean changes obtain sharp detection thresholds known minimax gaussian situation parameter penalization chosen equal packing number system hypercubes contrast chosen detection power turns suboptimal emphasizes importance knowledge packing number explicitly illustrative example also given example potential alternative weaker error measures control fwer false discovery rate fdr could controlled see benjamini hochberg benjamini yekutieli different task beyond scope paper theory section summarize theoretical findings section give overview details precise setting present assumptions set candidate regions section provides validity gaussian approximation determines distribution section derive asymptotic expansion detection power setting assumptions following assume exponential family regular minimal lebesgue densities form pxq hpxq exp natural parameter space open cumulant transform strictly convex moment generating function exists random variables tails see casella berger brown details let pxq conjugate pxq lrt statistic written sup ipr ipr ipr note definition holds remark observations drawn exponential family replaced field qipind known baseline intensities representation lrt statistic valid anymore proofs rely taylor expansions tails random variables see theorem explicitly exponential family structure therefore general models corresponding assumptions posed see also sec results immediately generalize situation control supremum restrict system regions suitably end introduce notation set define inf denotes topological boundary furthermore define around boundary assumption complexity regularity set bounded exists constant lebesgue measure let briefly comment assumption remark assumption standard assumption control complexity set indexed process see massart van der vaart wellner spokoiny assumption immediately implies upper bound cardinality prn exist constants prn allow apply recent results chernozhukov couple process gaussian version following also need bound complexity terms packing number packing number subset metric given maximum number points largest number packed inside see van der vaart wellner def following consider symmetric difference corresponding metric suppose exists positive number constants assumption satisfied holds true basically follows relationship capacity covering numbers van der vaart wellner thm however might also satisfied considerably smaller numbers see examples stress assumption boundary smoothness satisfied whenever consists regular borel sets borel set example consider set hyperrectangles form according van der vaart wellner refined computations appendix lemma show arbitrary obviously corresponding discretization consists hyperrectangles ind determined upper left lower right corners psn consists regular borel sets assumption boundary smoothness also satisfied may also consider smaller set hypercubes form assumption satisfied precisely according despres refined computations appendix lemma show independent opposed let set see devroye lugosi cor proves assumption satisfied hand prove lemma appendix limit theory position show quantiles multiscale statistic approximated uniformly gaussian version furthermore prn converges limit former require lower bound smallest scale given given discretized set candidate regions introduce notation formulate main theorem theorem weak limit let ind field random variables satisfy assumption let prn sequence holds true holds fixed furthermore almost surely finite note depend unknown quantities simulated however practical purposes advantageous use finite sample gaussian approximation simulate quantiles justified following theorem theorem gaussian approximation let ind field random variables let set candidate regions satisfying assumption let prn sequence holds true let fixed holds pnq lim remark theorems compatible sense set candidate regions satisfying assumption satisfying holds example suppose set candidate regions satisfying assumption let discuss three important examples model gaussian fields let variance fixed case bernoulli fields let bin cases excluded obtain natural exponential family however cases one would screen field correctly anyway natural parameter exponential family log pqq using log exp compute log log poisson fields let excluded case trivial natural parameter exponential family log using exp compute log example distribution hyperrectangle hypercube case recall example let set hyperrectangles set hypercubes prn holds max rts penv spsn sup prs penv max rtq penv qpqn sup prt pen phnq white noise simulations means densities sides different values shown figure smallest possible values chosen according example given packing number respectively corresponding results depicted top row figure simplicity alternatively use respectively lead simulated distributions shown bottom row figure nicely illustrates using larger value lead larger quantiles hence loss detection power distributions pqn extremely close detecting system squares easier detecting system rectangles even though latter far bigger complex explanation penalization appropriate choice parameter tailored system asymptotic power section analyze power multiscale testing approach detection power clearly depends size strength anomaly describe latter frequently employ functions heuristics key point following power considerations observations approximated signal non zero anomaly aplus standardized noise component scaled factor case gaussian observations variance one recovers situation considered sharpnack whenever signal part strong enough anomaly detected following make statement mathematically precise also give comparison multiscale testing procedure oracle procedure pqn comparison pqn comparison figure simulated densities weak limits histograms obtained runs test statistic comparison corresponding densities estimated pqn top row optimal standard kernel estimator calibration covering number bottom row alternative calibration using considered alternatives consider given family qnpn hypercube anomalies lebesgue measure corresponding discretized anomalies ind ind size consider alternatives iqn parameters determine total strength anomaly given pqn clearly anomaly fixed size strength detected asymptotic probability therefore consider vanishing anomalies sense pqn furthermore restrict parameters yield uniformly bounded variances uniform tails rexp psy npn ind constants case gaussian observations variance obviously satisfied poisson field means intensities bounded away zero infinity oracle multiscale procedure recall set hypercubes example discretization size anomaly known position one would naturally still unknown restrict set candidate regions consequently scan true anomaly discretized gives versiono alsoosatisfies rise oracle test rejects whenever quantile rno similar theorem one show quantile sequence ensures oracle test asymptotic level asymptotic power oracle test seen benchmark multiscale test obtain competitive multiscale procedure let choose satisfying furthermore assume otherwise multiscale procedure never able detect true anomaly contained set candidate regions scan position size anomaly unknown consider sets candidate regions consequently scan rms clearly true anomaly satisfies discretized version also satisfies rms gives rise multiscale test rejects whenever prms rms quantile theorem ensures multiscale test asymptotic level whenever holds due theorem asymptotic power show multiscale procedure described requires priori knowledge scale anomaly asymptotically detects anomalies power oracle benchmark procedure known scale hence penalty choice calibrate scales renders adaptation scales free least asymptotically seen structural generalization sharpnack thms alternative whole distribution mean might change also power considerations proksch restrict simpler case theorem setting described let sequence scales plog pan denote survival function folded normal distribution parameters cumulative distribution function let furthermore satisfied following holds true single scale procedure asymptotic power log sufficiently small multiscale procedure asymptotic power rms log remark sharpnack similar result case gaussian observations shown note condition sufficiently small missing however necessary proof work proksch suffices assume maximum explicitly controlled due theorem allows explicitly describe anomalies detected asymptotic power corollary setting section assumptions theorem anomaly detected asymptotic power either single scale multiscale testing procedure log example case gaussian observations iqn variance baseline mean size anomaly yields detection log calibrate statistic packing number example coincides well known asymptotic detection boundary hypercubes see frick butucea ingster kou general bernoulli ber iqcn iqn condition reads follows log note minimax detection rate unknown case best knowledge poisson field poi iqcn iqn theorem corollary applied bounded sequence case reduces log minimax detection rate unknown case best knowledge auxiliary results results rely heavily coupling result allows replace maximum partial sums standardized nef maximum corresponding gaussian version obtained recent results chernozhukov soon certain moments controlled purpose following two lemmata proofs found section follows letter denotes constant might change line line start controlling maximum powers uniformly random variables lemma let independent random variables exist exists constant max plog bound improved random variables max log show maximum partial sum process independent random variables bounded maximum corresponding gaussian version latter controlled exploiting fact maximum dependent gaussian random variables always bounded maximum corresponding independent gaussian random variables see log piqq max ipi allows prove following ipi lemma let pzi independent random variables rzi denote arbitrary index set sets tiuipi exists constant independent ipi max log piqq max ipi help two lemmata following coupling result shown theorem coupling let ind independent rzi rzi satisfied uniform constants let furthermore ind independent inequality holds pnq max max rprn rprn ipr ipr proofs section give proofs following denote cardinality prn satisfies logppn log recall denotes generic constant might differ line line proof auxiliary results start proving auxiliary statements section proof lemma let hptq max max hptqq hptq let logpn max max hptq logpn plog last inequality follows integration parts proof lemma let rademacher random variables take values probability step since symmetric max max lemma ledoux talagrand choosing ptq pci itipiu scaled indicator function norm max obtain max max step let sequence independent copies pzi define equally symmetrized version metrized version ipi pzi using argument fubini theorem derive max max max max max last equality holds view symmetry use contraction principle theorem ledoux talagrand ptq conditionally independent pri choosing step get max multiplying sides max therefore max max max ipi max max max max max max max used second inequality statement follows proof theorem enumerate region define xij itiprj pxij sequence max xij satisfies max rprn ipr recall logppn logpnq according chernozhukov cor find every exists gaussian version max nij independent random vectors rpn rxi xit logpnq logpnq logpnq max pxij xik rxij xik max max max logppn controlled follows xij derive max ipr max pzi iprj xrk using restriction size rectangles find denote ind ipi max ipi using lemma obtain log piqq loooooomoooooon remains estimate max logpnq max max max max total get lemma logpnq log pnq logpn compute prn max prn log pnq max used lemma let fixed max plog tmax max log let plog hence max max plog plog plog large enough exppu holds exp plog exp plog furthermore plog log implies prn prn conclusion obtain log pnq log pnq log pnq logpnq yields claim proofs section let prove results section including theorems start taylor expansion allows apply theorem lemma let collection sets holds prn sequence plog suppose ind random variables recall denote ipr holds pnq max rprn proof independent gaussian random variables follows log max ipr hence max logpnq rprn ipr combining result theorem obtain max log rprn together follows max rprn therefore probability large uniformly enough position analyze jpy definition see sup pyi attained derive ipr therefore jpy note large enough latter open set taylor expansion around one second order around yields consequently max rprn max rprn max rprn ipr pyi used yields claim position prove theorem far shown maximum local likelihood ratio statistics approximated gaussian versions include scale penalization penv include approximation result slice maximum scales almost constant show may bound maximum scales sum maximum theses families price pay additional logpnq factor smallest scale proof theorem follows triangle inequality lemma max ptr penv rprn max rprn pnq penv define ipr ipr notation symmetry argument find proof theorem log pnq max max rprn rprn let pplog pnq define set candidate regions written log pnq logpnd jpj logpnd abbreviate penj penv pexp log slicing implies penj penv penj using get penj penj rlogpnd rlogpnd largest index logpnd therefore maximal value given logpnd logpnd therefore penj penj means penalty terms penv considered constant straight forward computations show max penv max penv rprn rprn max max max rprn jpj rprn claim follows logpn direct consequence continue proof theorem taking account result theorem prove remark done following lemma let satisfy assumption equipped canonical metric define furthermore let denote white noise define nud pid unit cube upper corner holds proof note totally bounded show assumptions kosorok thm tightness white noise tight totally boundedness markov inequality standard bounds modulus continuity obtain using assumption sup sup tends finite dimensional convergence convergence laws application central limit theorem random fields dedecker thm dedecker lemma shows regular borel sets consequently central limit theorem shows fixed similar computation shows cov pzn shows finite dimensional convergence want apply generalized version continuous mapping theorem see billingsley thm denote borel sets define penv pxq sup hcn pxq max rprn penv necessary conditions apply continuous mapping theorem given following lemma lemma consider hcn functions uniformly continuous phcn qnpn sequence functions uniformly pxn holds hcn pxn pxq proof let choose consider two functions dpx sup using max max find pxq hcn max rprn similar arguments yield uniform continuity let pxn since functions phcm qmpn equicontinuous find pxn hcm given pxn hcm choose pxn hcn let define pcnqd set compact set metric defined totally bounded furthermore finite subset fix introduce penv gpr holds pxq sup hcn pxq max pbn since subset straight forward computations show continuous pxq gprq implies compactness exists let sequence gprn gprq hence pxq hcn pxq gprn pxq consequently exists holds pxq hcn together implies pxn position prove remark proof remark proposition generalized version continuous mapping theorem see billingsley thm get hcn pzn functions hcn defined hcn pzn holds pcnqd since get lim lim pcnqd also readily seen definition lim inf let fixed assume pcnqd obtain altogether lim inf lim sup pcnqd yields claim proof theorem main statement follows theorem together remark remains show boundedness apply spokoiny thm let check three conditions theorem obviously fulfilled since since exp compute covpw covpw consequently find exp exp iii fulfilled assumption remark holds well hence get statistic nondegenerateness obvious always larger value local statistic one fixed scale proofs section let prove results section namely theorem corollary first introduce abbreviations ease notation let denote total signal pqq ipq brevity introduce gaussian process penv pqq pqq ipq let start analysis oracle procedure preparation require leave suitable subset hypercubes close true anomaly therefore choose sequence pqn denote set hypercubes close anomaly pan pqq pqn furthermore define extended neighborhood anomaly pan complement pan definition pqquqpt pqquqpun independent allow compute asymptotic power procedure sketch see figure figure exemplary elements sets anomaly shown red hatched cubes belong dotted cubes black cubes belong definition holds implies independence pqquqpt pqquqpun lemma consider setting section recall pqn pan max ipq qpu lim max pqq qpt proof starta bounding covering number canonical metric first note sets fixed size hence given one contains least many voxels consequently onlyp deal straight forward construct cardinality cubes follows dudley entropy integral see marcus rosen thm fixed max qpu ipq max log log markov inequality proves claim direct consequence max penv qpu furthermore note pqq consequently penv max pqq max qpt qpt penv max qpqn pan rmn pqn pan yields claim lemma hand position derive asymptotic power oracle procedure proof theorem analyze rtn pan start showing statement theorem lemma triangle inequality replace pan max penv qpqn pan furthermore theorem allow approximate latter sum gaussian version max pqq rtn pan qpq pan derive max qpq pan pqq max pqq max pqq qpt qpqn pan max pqq max pqq qpt qpqn pan max pqq max pqq max pqq qpt qpu qpt max pqq pqn max pqq qpt qpt max pqq pqn max pqq qpt qpt exploited independence pqquqpt pqn lemma states rmaxqpt pqq hence rtn pan pqn furthermore note pqn penv follows folded normal distribution parameters pqn ipqn pqn penv compute pqn ipqn penv log pan yields continuity pan proposed lower bound upper bound statement theorem proceed obtain rtn pan max pqq max pqq qpt qpu max pqq max pqq max pqq qpt qpu qpun max pqq max pqq max pqq qpt qpu qpun max pqq max pqq max pqq qpt qpun qpu zun max pqq max pqq qpun qpu zun used independence pqquqpt pqquqpun lemma obtain max max qpu qpu zun ipq ipq definition pqq pqn uzun exploiting implies max pqq qpu zun pqn log pqn log pqn pqn log pqn construction pqn log nothing shown altogether gives rtn pan max pqq qpun similar arguments lemma obtain max pqq pqn qpun hence claim proven turn multiscale procedure different scales considered set large enough especially construct subset pqquv pqn independent maxqpv pqq still negligible due corresponding proof sharpnack incomplete overcome difficulty follow idea distinguish anomaly asymptotically effect pqq whenever sufficiently large compared impact asymptotically negligible sequence introduce max pqn logpnq pqq pqn complement sketch see figure figure exemplary elements sets anomaly shown red hatched cubes belong dotted cubes however intersections marked black small enough asymptotically influence pqq opposed oracle procedure independence pqquqpt pqn however asymptotically similar property holds true shown following lemma lemma consider setting section recall pqn following statements hold true qqq max ipq qpv qpt ipqxqn pqq qpt proof let start canonical metric abounding covering number nnpv holds pqq pqn implies consequently contains cubes fixed scale contains set ofp tan constructed proof lemma elements gives help dudley entropy integral find max qpv log qqq used markov inequality gives claim holds pqq pqn hence consequently qpt max qpqn ipqxqn logpnq used side converges proves claim deduced follows holds pqq pqn hence max pqq penv qpt qpt ipqzqn pqq qpt ipqxqn pbq pqn qpt ipqxq last estimate follows pqn furthermore contains scales obtain max penv qpv ipq penv max qpv ipq paq qqq penv used sufficiently small consequently pqq qpt pbq penv qpt ipqzqn penv qpt ipq penv max qpqn ipq rmn pqn yields claim proof theorem multiscale procedure compute lower bound rtn similar proof theorem obtain rtn pqq pqn pqq qpt qpt lemma furthermore get pqq qpt penv pqq qpt ipq penv pqq qpt ipqzqn shows independence pqq pqn qpt pqq pqn qpt proof concluded one theorem proof corollary procedures asymptotic power log pan respectively estimate max shows case inserting values noting uniformly bounded gives claim appendix recall packing number remark appendix compute packing numbers given example done means covering number covering number subset metric given minimal number balls radius needed cover van der vaart wellner def immediately clear hence sufficed compute show following use notation example lemma exists constant depending dimension holds holds true proof approximate vertices lattice specified later set vertices size denoted denote edge lengths immediately clear exists approximating hence obtain choose compute cardinality first note number possible left bottom vertices bounded denote edge lengths find integers therefore obtain compute employ minkowski rem cassels sec ensures lebesgue volume comparable factor readily shown induction plog inserting obtain log log uqq used proves claim lemma exists constant depending dimension holds holds true proof proceed proof lemma contrast obtain instead better estimate furthermore cardinality edges length choose bounded number lower left vertices times number possibilities adjacent vertex gives therefore finally obtain proves claim lemma exists constant depending dimension holds holds true numbers proof let note definition pythagoras theorem convenient choose equidistant furthermore choose maximal system points paj implies spherical cap note psin psin small values given find split hai hai hai since hai hai space width hyperpyramids opening angle obtain hai union hai generic constant depending hence choose exists estimate elementary geometry follows furthermore boundary points sets side disjoint therefore obtain volumes implies consequently proves claim acknowledgements authors gratefully acknowledge financial support german research foundation dfg subproject crc also thank katharina proksch many helpful comments proof theorem references adler excursion sets tube formulas maxima random fields annals applied probability pages alm approximation simulation distributions scan statistics poisson processes higher dimensions extremes durand detection anomalous cluster network ann castro wang detection structured anomalies permutation scans journal american statistical association appear donoho huo detection geometric objects fast multiscale methods ieee trans inform theory benjamini hochberg controlling false discovery rate practical powerful approach multiple testing roy statist soc ser benjamini yekutieli control false discovery rate multiple testing dependency ann billingsley convergence probability measures john wiley sons brown fundamentals statistical exponential families applications statistical decision theory lecture notes butucea ingster detection sparse submatrix noisy matrix bernoulli casella berger statistical inference volume duxbury pacific grove cassels introduction geometry numbers classics mathematics berlin corrected reprint edition cheng schwartzman multiple testing local maxima detection peaks random fields ann chernozhukov chetverikov kato gaussian approximation suprema empirical processes ann dedecker central limit theorem stationary random fields probab theory related fields dedecker exponential inequalities functional central limit theorems random fields esaim probab despres dimension norms arxiv preprint devroye lugosi combinatorial methods density estimation springer series statistics new york dickhaus simultaneous statistical inference springer heidelberg applications life sciences spokoiny multiscale testing qualitative hypotheses ann walther multiscale inference density ann fang siegmund poisson approximation two scan statistics rates convergence ann appl frick munk sieling multiscale change point inference stat soc ser stat discussions authors rejoinder authors friedenberg genovese straight source detecting aggregate objects astronomical images proper error control amer statist haiman preda estimation distribution discrete scan statistics methodol comput appl jiang maxima partial sums indexed geometrical structures ann jiang qiu minn zhang assessing intratumor heterogeneity tracking longitudinal spatial clonal evolutionary history sequencing pnas kabluchko extremes standardized gaussian noise stochastic process kabluchko munk shao theorem maximum standardized random walk increments multidimensional arrays esaim probab kazantsev lemahieu salov denys statistical detection defects radiographic images nondestructive testing signal processing major approximation partial sums independent sample wahrscheinlichkeitstheorie und verw gebiete kosorok introduction empirical processes semiparametric inference springer series statistics springer new york kou identifying support rectangular signals gaussian noise arxiv preprint kulldorff heffernan hartman assuno mostashari spacetime permutation scan statistic disease outbreak detection plos medicine ledoux talagrand probability banach spaces volume ergebnisse der mathematik und ihrer grenzgebiete results mathematics related areas berlin isoperimetry processes lehmann romano testing statistical hypotheses springer texts statistics springer new york third edition munk sieling multiscale segmentation electron marcus rosen markov processes gaussian processes local times volume cambridge studies advanced mathematics cambridge university press cambridge massart strong approximation multivariate empirical related processes via kmt constructions ann naus wallenstein multiple window cluster size scan procedures methodol comput appl pozdnyakov glaz kulldorff steele martingale approach scan statistics ann inst statist proksch werner munk multiscale scanning inverse problems arxiv preprint appear ann rio strong approximation processes via kmt constructions ann rivera walther optimal detection jump intensity poisson process density likelihood ratio statistics scand munk multiscale methods shape constraints deconvolution confidence statements qualitative features ann sharpnack exact asymptotics scan statistic fast alternatives electron siegmund venkatraman using generalized likelihood ratio statistic sequential detection ann siegmund yakir tail probabilities null distribution scanning statistics bernoulli taylor worsley detecting sparse signals random fields application brain mapping journal american statistical association maximum ratchet scanning process poisson random field statist sinica van der vaart wellner weak convergence empirical processes springer series statistics new york applications statistics rectangular confidence regions means multivariate normal distributions amer statist walther optimal fast detection spatial clusters scan statistics ann zhang yakir xia siegmund scan statistics poisson random fields applications genomics ann appl
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latent contracts michael greenberg jun pomona college michael abstract standard contract monitoring breaks tail recursion leads space leaks change program asymptotic complexity restores tail recursion bounds amount space used contracts contract monitoring contracts enforcing simple type disciplines gradual typing well studied prior work establishes semantics manifest contracts without dependency adapt work latent calculus dependency guarantee space efficiency dependency used generally guarantee space efficiency dependency used instead offer framework making programs space efficient basis introduction findler felleisen brought world allowing programmers write functions checked runtime easy languages clear blame contract violated precondition fails blame caller fails blame callee languages however harder tell calls blame function fails example consider following contract pred int pred int pred int mod contract applies function call type takes another function call type input contract says called positives return naturals must return even number returns odd number blame returns negative number blame called number say blame findler felleisen insight even setting two parties blame given bad values given due nefarious action case generalizes negative positions contract blame caller positive positions blame callee dependent codomain contract refer function particularly useful example square root function greenberg sqrt satisfies contract pred real pred real abs sqrt takes real returns real within square root dependent variable bound codomain variable local domain predicate contracts leak space implementations contracts proven quite successful particularly racket problem contracts leak space default implementation contracts works wrapping function function proxy example check int satisfies contract pred int mod pred int mod monitor function wrapping function proxy monl proxy called input first check satisfies domain contract even run get result check satisfies codomain contract result even contract always fail blaming one always odd contracts leak space two ways first bound number function proxies appear given function grievously contracts break tail recursion demonstrate issue tail calls use simplest example mutual recursion detecting parity let odd int false even even int true odd functional programmers expect program run constant space tail recursive adding contract breaks tail recursion add contract odd call odd contract checks accumulate fig notice checks accumulate codomain even though mutually recursive calls even odd syntactically tail calls bound number codomain checks occur bound size stack tail recursion broken even though one function proxy odd contracts create space leak overview contributions space efficiency gradual types contracts constrained type tests well studied greenberg developed semantics general contracts used manifest calculus conflating contracts types however contracts typically implemented latent calculi contracts distinct whatever types may exist greenberg believe would easy design latent version eidetic following translations greenberg pierce weirich gpw paper readers may observe contract betrays deeper knowledge numbers functions offer example minimal naturally occurring latent contracts monlodd pred int pred bool mod int false even even int true odd let odd odd monlodd pred even monlodd pred monlodd pred odd monlodd pred int monlodd pred monlodd pred even monlodd pred monlodd pred monlodd pred odd monlodd pred int monlodd pred monlodd pred monlodd pred even fig contracts break tail recursion show belief well founded giving semantics dependent variant contract pcf cpcf rest paper discusses formulation contracts enjoys sound space efficiency slightly change implementation contracts programs observationally equivalent standard semantics contracts consume bounded amount space paper omitted detailed examples refer curious readers greenberg though intend paper follow greenberg general structure defining two forms dependent cpcf cpcfc classic semantics cpcfe follows eidetic semantics able prove space efficiency without dependency bounding amount space consumed contracts unable prove space efficiency general dependency instead offer framework allows dependent contracts made space efficient offer two primary contributions adapting greenberg work latent calculus extending possibility space efficiency dependent contracts smaller contributions well first adding nontermination moves beyond greenberg strongly normalizing calculi showing popl paper result artifact strong normalization theory bound size term evaluation advance contracts second simpler type system makes clear type system invariants necessary bookkeeping proving complicated manifest type system sound third separating contracts types give tighter space types function greenberg collects types never used contract collect exactly contracts finally explore space efficiency attained dependent contracts give guarantee dependent greenberg types terms bool int errl monl mon true false errl monl mon nil fig syntax classic cpcf contracts show possible achieve discuss different ways classic contract pcf present classic cpcf separate calculi sharing syntax typing rules fig fig single parameterized operational semantics rules held completely common fig others specialized system fig formal presentation modal two modes classic much shared two core syntax expressions typing use colors highlight parts belong one system classic cpcf typeset salmon cpcf periwinkle contract pcf cpcf plain cpcf extension plotkin pcf developed first dimoulas felleisen syntax fig simply typed language recursion typing rules straightforward fig operational semantics generic fragment also uses conventional rules fig dimoulas felleisen use evaluation contexts offer concise description system write relation full giving congruence rules eif error propagating rules raise need restrict congruence casts methods transparent written explicit congruence rules using subtly nested evaluation contexts herman error prone contracts cpcf distinguishing feature contracts installed via monitors written monl monitor says ensure satisfies contract blame lies label monitors apply appropriate types tmon two kinds contracts cpcf predicate contracts base type written function contracts written latent contracts predicate contracts two parts predicate base types identifies values satisfy contract closing substitution keeps track values substituted contract example identity substitution mapping variables int identifies positives int identifies numbers greater unspecified number pred int identifies numbers greater closing substitution identity mapping write pred instead cpcfc closing substitutions map variable either value substitution contracts without dependency contract closed introduced dependency use explicit closing substitutions rather direct substitution three reasons first simplifies space efficiency proof simple contracts sec second explicitness lets distinguish contract pred int pred int third emphasizes contracts another form closure predicates solely base types functions function contracts satisfied functions satisfying parts functions whose inputs satisfy whose outputs satisfy function contracts dependent codomain contract refer back input function example contract pred int pred int satisfied increasing functions positives note bound codomain function contracts dependent omit binder front pred int pred int means operators positives check contracts satisfied runtime use explicit delayed substitutions keep track values substituted predicate contracts help proof space efficiency track variables appear predicate otherwise alpha equivalence allows give fresh names variables domain consistently renaming variables inside predicate holding substitutions close free variables way modeling closures dependent predicate closes finite number variables compiled representation would generate closure corresponding number slots closing environment restricting substitutions exactly variables appearing free predicate serves another purpose easily recover bounds programs without dependent contracts sec greenberg typing rules tvar tconst tabs top bool monl tmon tpred tcnil tid tmonc tfun closing substitutions tapp tif mon nil trec contract typing tblame errl tcpred tcfun fig typing rules classic cpcf tmap latent contracts operational semantics ebeta eiftrue true eopl errl errl errl errl eif eapplraise eoplraise edelta efix eiffalse false eappl eappr errl errl errl errl errl errl eopr eifraise eapprraise eoprraise fig shared operational semantics cpcf classic contract pcf cpcfc classic cpcf gives straightforward semantics contracts figs largely following seminal work findler felleisen check predicate contract simply test emonpred returning either value appropriately labeled error function contracts deferred monl value called function proxy function proxy applied unwraps proxy monitoring argument domain contract running function monitoring return value codomain contract emonapp semantics may seem lax monitor applied dependent uses argument codomain monitor fact agnostic could picky requiring function contract monitors monl concrete syntax predicates written much nicely ignore concerns greenberg emonpred monl errl monl monl monl monl monl emon monl errl errl monl mon labell mon nil emoncnil mon mon mon emonc emonraise emonlabel mon mon errl mon mon mon emonapp emoncpred emoncapp mon errl errl mon mon mon join emoncraise emoncjoin fig operational semantics classic cpcf substitution monl throughout could indy monl throughout default lax rule make proof soundness easier corollary classic semantics yield result regardless closing substitutions codomain sec standard congruence rules allow evaluation inside monitors emon propagation errors emonraise metatheory prove cpcfc type system sound minimum fuss using usual syntactic methods subtlety must careful proving substitution property since slightly changed definition section consider typing rules cpcfc typeset white salmon lemma weakening latent contracts proof mutual induction terms contracts lemma substitution implies implies proof mutual induction terms contracts case analysis using weakening lemma appropriate immediate ihs ihs case analysis narrowing two equal case analysis narrowing two equal ihs errl immediate monl ihs must show inversion know base type two cases cases use tpred substitution actually stored find tmap assumption substitution ignored find assumption ihs corollary closing substitutions close exactly context lemma closing substitutions proof induction using substitution lemma lemma progress either value error errl proof induction typing derivation tvar well typed empty context tconst value tblame errl error tabs function abstractions values trec takes step efix top value error steps latter two cases step eoplraise eopl respectively value error steps latter two cases step eoprraise eopr values step edelta greenberg tapp value error steps latter two cases step eapplraise eappl respectively value error steps latter two cases step eapprraise eappr values know either function function proxy step either ebeta emonapp depending shape tif value error steps latter two cases step eifraise eif value must either true false step eiftrue eiffalse depending shape tmon either value error steps latter two cases step emonraise emon respectively value step emonpred emonapp depending shape contract lemma preservation proof induction typing derivation cases step taken tvar well typed empty context tconst normal forms tblame normal forms tabs normal forms trec must stepped efix substitution lemma tfix top stepped eopraise rule done tblame stepped congruence rule eop done top stepped edelta assumption operations sound denotations tapp stepped eappraise rule done stepped congruence rule eapp done tapp stepped ebeta substitution lemma stepped emonapp domain well typed tmon apply proxied function tapp codomain check well formed substitution lemma entire resulting term well typed tmon tif stepped eifraise tblame stepped eif tif stepped eiftrue eiffalse assumption tmon stepped emonraise resulting error well typed tblame stepped emon tmon stepped emonpred closing substitution lemma type condition tapp branch well typed assumption false branch well typed tblame whole lot well typed tif contract pcf cpcfe recover tail calls cpcf cpcfc happily wrap arbitrarily many function proxies around value bound number latent contracts labell labell labell labell join nil join drop join join join join wrap drop nil nil drop drop drop wrap wrap nil wrap wrap join mon otherwise nil wrap wrap wrap wrap fig contract labeling predicate stack management codomain contract checks accumulate key idea joining contracts make two changes language bound function proxies function one bound stacks avoid redundant checking ultimately show contracts without dependency use constant space story dependent functions complex sec fortuitously notion join solves problems working identically simple dependent contracts ensure function value one proxy change semantics monitoring monitoring proxied value join new monitor old one bound size stacks contract checks join pending contracts avoid redundancy join operation works labeled contracts move label monitor contract allow keep track many predicates fig concretely labeled contracts use metavariable opposed comprising function contracts usual predicate stacks fig predicate stack list labeled predicates predl nil empty stack join operation takes two labeled contracts combines eliminating redundant contracts goes join new old predicate stack keep new contracts eliminate redundant old ones recent contracts kept joining functions works contravariantly careful maintain correct substitution behavior using wrap greenberg finally establish mean redundant using predicate implication one contract imply another definition predicate implication let relation predicates reflexivity transitivity substitutivity adequacy true implies true decidability decidable whether entire development parameterized implication relation characterizing one contract subsumes another write negation relation total preference relation would total order may necessarily enjoy example could int int vice versa even though two predicates equal also view total order contextual equivalence least one workable implication relation syntactic equality say iff since careful store values actually referenced closure predicate steps determine equalities finite computable runtime example suppose wish show pred int pred int code predicate int implementation might observe function pointers equal environment one slot value implementation might compare two environments could given operational semantics behaves like implementation explicitly generating conditionals merge operations terms believe slightly abstract presentation digestible substitution codomain lax picky indy extend greenberg notion join account dependency new function wrap greenberg pierce weirich identified two variants latent contracts literature differing treatment dependent substitution arguments codomain picky monitor value substituted codomain domain contract lax actual parameter value substituted codomain unmonitored third variant indy applies monitor argument value uses different blame label different models substitution exhibit different behavior abusive contracts codomain contract violates domain contract latent contracts another source substitutions codomain multiple function proxies monitors unfold two function proxies classic lax semantics find leaving domain check unevaluated mon mon mon mon mon mon mon mon mon mon even though using lax semantics substitute contracts codomain semantics sound must behave exactly like classic semantics matter joins happen cpcfe must replicate contract substitutions done cpcfc construct abusive contract cpcfc though lax inner function proxy abuse outer one fig blame raised codomain contract abused domain contract even though cpcfc lax semantics wrapping multiple function proxies leads monitoring domains one contract codomain situation ripe abuse means joining contracts emulate classicsemantics substitution behavior use wrap function forcing substitution two function contracts joined keeping track substitutions every join joins happen future working contracts already appropriate substitutions cpcfe uses labeled contracts fig substitution labeled predicate contracts explicit delayed ordinary contracts otherwise nil nil joining substitution occurs see sec cpcfe closing substitutions map variable monitor mon value add evaluation rule taking ordinary contract monitors monl monitors mon means labeling function label emonlabel comes restricting congruence apply abutting monitors emonc cpcfe emon cpcfc two monitors collide join emoncjoin checking function contracts usual emoncapp emonapp latter works labeled contracts checking predicate stacks proceeds straightforwardly emoncnil emoncpred metatheory prove cpcfe type system sound proof follows cpcfc though evaluation rules consider consider typing rules cpcfe typeset white periwinkle greenberg pred int pred int pred int pred int true pred int true pred int referring domains codomains int int int int int int int int int int int pred int true int pred int true int int pred int true int int int pred int true pred int int pred int int int pred int true pred int int int int pred int true pred int int int pred int true int int int int int int int int pred int int pred int err int pred int int int err int pred int int err int pred int int false int pred int int fig abusive function proxies cpcfc latent contracts lemma weakening proof mutual induction terms contracts lemma substitution implies implies implies proof mutual induction terms contracts labeled contracts case analysis using weakening lemma appropriate immediate ihs ihs case analysis narrowing two equal case analysis narrowing two equal ihs errl immediate monl ihs must show inversion know base type two cases cases use tpred substitution actually stored find tmap assumption substitution ignored find assumption ihs nil immediate predicate contracts using show still well typed ihs corollary closing substitutions close exactly context lemma closing substitutions proof induction using substitution lemma lemma progress either value error errl proof induction typing derivation greenberg tvar well typed empty context tconst value tblame errl error tabs function abstractions values trec takes step efix top value error steps latter two cases step eoplraise eopl respectively value error steps latter two cases step eoprraise eopr values step edelta tapp value error steps latter two cases step eapplraise eappl respectively value error steps latter two cases step eapprraise eappr values know either function function proxy step either ebeta emoncapp depending shape tif value error steps latter two cases step eifraise eif value must either true false step eiftrue eiffalse depending shape tmon step emonlabel tmonc inner term monitor step emoncjoin otherwise inner term value error steps error step emoncraise steps step emonc knowing already monitor value either step emoncnil step emoncpred value already function lemma labell proof induction typing derivation using tcpred tcnil predicate case tcfun function case lemma join proof induction typing derivation lemma join proof induction typing derivation using lemma base case substitution lemma lemma preservation proof induction typing derivation cases step taken tvar well typed empty context tconst normal forms tblame normal forms tabs normal forms latent contracts trec must stepped efix substitution lemma tfix top stepped eopraise rule done tblame stepped congruence rule eop done top stepped edelta assumption operations sound denotations tapp stepped eappraise rule done stepped congruence rule eapp done tapp stepped ebeta substitution lemma stepped emoncapp domain well typed tmon apply proxied function tapp unmonitored codomain well formed substitution lemma entire resulting term well typed tmon tif stepped eifraise tblame stepped eif tif stepped eiftrue eiffalse assumption tmon immediate assumptions since labeling contracts preserves typing lemma tmonc stepped emoncraise resulting error well typed tblame stepped emonc tmon stepped emoncjoin lemma emoncnil assumptions stepped emoncpred closing substitution lemma type condition tapp branch well typed assumption false branch well typed tblame whole lot well typed tif lemma determinism proof induction first derivation recall exclude cpcfc rules emonlabel fires monitors emonc emoncjoin carefully avoid overlapping soundness space efficiency cpcfc cpcfe operationally equivalent even though cast semantics differ make connection formal proving every cpcf term either diverges cpcfc cpcfe reduces equivalent terms cpcfc cpcfe one minor technicality forms language necessary runtime appear one two calculi characterize source programs omit runtime terms definition source program well typed source program use tblame tmonc tcnil tcpred tcfun used lemma associativity join predicate stacks join join join join proof induction nil immediate greenberg predl join join join join calculate join join join join drop join join drop join join join drop join join join join join lemma associativity join join join join join proof induction predicate stack case lemma function case calculate join join join join join wrap join join join wrap join join wrap join join join wrap join wrap join join join wrap join join lemma idempotence predicate stacks true join mon join iff mon join drop proof induction length observing true redundantly successfully checked left right greenberg identified key property proving soundness space efficient semantics sound semantics must recover notion congruence checking manifest setting calls cast congruence since cpcf uses contract monitors call monitor congruence lemma monitor congruence single step mon iff mon proof cases step taken find easy case joining coercions rule apply derivations interesting case two contract monitors join either case suffices show terms ultimately confluent since determinism lemma rest latent contracts reductions efix give case explicitly original proof greenberg talk fixpoints must show mon iff mon former steps latter immediately emonc efix done determinism lemma following cases similar edelta ebeta eiftrue eiffalse eappl eappr eopl eopr eif emonlabel emoncapp raise rules word emoncapp one might think joining reduction fact unwrap inner step function proxy able join outer monitor joining reductions emoncnil must show mon mon nil mon terms confluent mon mon nil mon join nil mon emoncpred must show mon mon mon mon errl side steps mon join mon drop join emoncjoin turn steps mon drop join errl cases behavior true sides reduce eiftrue using emonc right left hand side reduces mon drop join right hand side mon join idempotence contracts lemma false sides reduce eiffalse using emonc right right reduces emonraise sides errl finally diverges terms diverge greenberg emonc must show mon mon mon mon given knowing another monitor sides step emoncjoin left steps emon find mon join emoncjoin must show mon mon mon mon mon join side joins mon join join side joins mon join join confluence associativity join lemma emoncraise must show mon mon errl mon errl steps emoncjoin sides step emoncraise errl lemma monitor congruence mon iff mon diagrammatically mon mon proof induction derivation using monitor congruence lemma particularly satisfying key property showing soundness space efficiency proved independently inefficient semantics implementors work entirely context semantics knowing congruence ensures soundness show observational equivalence cpcfc cpcfe logical relations fig gives contextual strongest equivalence could ask lemma similar contracts logically related monl monl latent contracts result rules errl errl term rules diverges diverges contract rules invariant relation closing substitutions open terms dom fig logical relation classic cpcf proof induction type index invariant relation let given must show monl monl side steps emonlabel mon labell side reduces errl emonpred left emoncpred right assumption invariant relation conditions two sides behave exactly conditions diverge raise blame return false return true return related values interesting case must show monl mon labell labell side value side may may value depending whether proxied sides values show logical relation let arguments given must show monl mon labell labell greenberg side unwraps emonapp emoncapp since closed evaluation use domain monitor reducing right convert monitor labeled contracts note codomain monitor unless closing substitution already mon brevity let labell side steps emoncjoin mon mon mon join mon join join let arguments given unwrap side emonapp emoncapp respectively must show monl monl mon join wrap mon join use monitor congruence resolve extra contracts spaceefficient side since closed evaluation step righthand side back separate inner monitors letting recover something behaves like show goal show monl monl mon mon mon break check apart order apply explain find relation proving two terms related sufficient show original terms related sufficient monitor congruence lemma allows consider domain codomain monitors separately knowing get identical behavior see wrap original term accounted codomain monitor appears know monl mon terms terminate diverge original side domain monitor join lemma case done terms yield values continue using monitor congruence step inside codomain check lemma mon mon mon mon mon must show monl mon mon latent contracts step right hand side mon mon mon monitor congruence know mon join mon coterminate either diverge blame label reduce common value done last case must consider mon mon recall mon mon mon wrap wrapping done join captures exactly substitution would occur evaluated layer function proxying step step patch derivation use substitutivity definition part see codomains merged substitution would also substitution modified term still lock step original sees value monitored assumed know mon side steps terms diverge errors together done otherwise evaluate values remains see monl mon return original term see given left joining wrap monitor congruence let know monitors evaluate merged unmerged lemma unwinding iff exists unrolling fixpoint times converges value proof induction evaluation derivation observing must finite number unrollings induction observing replace substitution finite unrolling theorem cpcfc cpcfe terms logically related source program source program proof mutual induction typing relations greenberg terms case proceeds letting given showing sides diverge raise blame return related values tvar definition tconst definition tabs ihs letting argument given show bodies behave similarly top ihs tapp ihs tmu give case full new proof let must show side reduces single step efix rearranged rearrange something looks like dual substitution unwinding lemma see fixpoint side either diverges converge related values finite unrolling replaced finite unrolling former case done terms diverge latter case apply tmon ihs lemma tmonc appear source programs tblame appear source programs contracts tpred tfun ihs bounds space efficiency seen cpcfe behaves cpcfc theorem cpcfe actually space efficient programs use dependency yes dependency story complicated simple case greenberg showed simple put bounds space recover result general framework observe given source program starts finite number predicate contracts runs new predicates appear dependent substitutions effect predicates may accumulate stacks worst case predicate stack could contain every predicate contract original program exactly joins remove redundancy function contracts also bounded starts function contracts latent contracts predicate extraction preds preds preds var preds preds preds preds preds preds preds preds preds preds preds monl preds preds preds mon preds preds preds nil preds preds preds preds preds preds preds preds preds preds preds preds preds preds preds preds preds preds errl contract size size preds size size size size fig predicate extraction contract size certain height evaluation shrink height leaves function contracts labeled predicate stacks largest contract could ever see maximum height maximal predicate stacks every leaf program runs abutting monitors joined giving bound total number monitors program one per ast node make ideas formal first defining mean predicates program showing evaluation introduce predicates lemma let preds set predicates term predicate represented pair term closing substitution lemma preds drop pred preds proof induction predicate stack nil proof trivial otherwise observe lose predicates lemma preds preds labell proof induction predicate case immediate arrow case uses ihs say program uses simple contracts predicates closed every predicate stack redundancies program reduces new contracts appear contracts may disappear concretely requirements restricting programs simple contracts lets prove substitution joining increase set predicates redundancy requirement greenberg means redundancy following theorems hold programs use simple contracts lemma preds preds preds proof induction using absorptive property set union predicate contract free variables assumption substitution hold anything lemma preds join preds preds proof induction predicate stack must nil proof immediate predicate stack lemma function contract case ihs since using simple contracts know wrap never anything predicates free variables lemma reduction simple predicates preds preds proof induction step taken edelta predicates ebeta substitution lemma efix observe fixpoint operator introduces preds body substitution lemma eiftrue definition preds preds true preds preds eiffalse definition emonpred definition side new predicates possibly one fewer emonapp substitution codomain ignored lemma side rearrangement left emonlabel definition fact labeling leaves predicate set alone lemma emoncnil definition emoncpred emonpred side new predicates possibly one fewer emoncapp emonapp rearrange constituent substitutions ignored lemma emoncjoin monotonicity join lemma knowing wrap nothing eif emon emonc raise predicates side compute concrete bounds define number distinct predicates base type represent predicate stack type bits number bits needed represent blame label latent contracts given well typed contract represented size bits predicate stacks represented bits function types represented trees predicate stacks finally since reduction lemma bound amount space used contract looking source program represent contracts program maxc size fixed source program readers familiar greenberg paper earlier work like herman notice bounds precise tracking number holes contracts per type size rather simply computing largest conceivable type dependent case dependent case generally bound number contracts size contracts used program consider following term let downto monl pred int pred int int downto many different contracts appear run program downto runs see different forms predicate int one first call second call magnitude affect measure size source program contracts number distinct contracts appear effectively unbounded simple case get bounds automatically using smallest possible implication equality dependent case programmer identify implications recover space efficiency recover space efficiency downto saying int int iff codomain checks recursive calls able join downto monl pred monl pred monl pred monl pred able recover space efficiency case come easily decidable implication rule specific predicates matching function checks narrower narrower properties recurses recall mutually recursive example fig monlodd pred int pred bool mod int false even even int true odd let odd greenberg make example adding implication bool mod bool mod iff suppose put contracts even odd let odd monlodd pred int pred bool mod int false even even monleven pred int pred bool mod int true odd trace contracts homogeneous eliding domain contracts odd monlodd pred even monlodd pred monleven pred odd monlodd pred monleven pred monlodd pred even monlodd pred monleven pred monlodd pred monleven pred odd monlodd pred monleven pred monlodd pred monleven pred monlodd pred even make checks space efficient need several implications write oddp bool mod evenp bool mod following table gives conditions implication relation row predicate imply column predicate oddp evenp oddp evenp four implications allows eliminate pair checks generated recursive calls odd even reducing codomain checking constant one check could define different implication relation say oddp oddp iff mod mod implication would apply generally table always obvious define implication relation usual time space possible write contracts necessary implication relation space efficiency amounts checking contracts consider following factorial function let int true let fact monl pred acc pred pred int int int acc acc latent contracts contract wrong strange call fact negative number program diverges indeed get value back contracts enforce partial correctness call fact yields monitors check outside inside say int int could check time cost like checking original contract starkly consider following function strange contract prime predicate identifying prime numbers let absurd monl pred int true pred int prime int absurd absurd run absurd monl pred int prime absurd monl pred int prime monl pred int prime implication relation could resolve two unrelated checks one could say int prime int prime iff prime savings would allow collapse tall stacks monitors implication relation come simplest option punt derive implication relation reflexive transitive closure programmer rules programmer might specify several different predicates interrelate follows int implies int int implies int int implies int default collection implications might come language library programmers able write well probably unwise allow programmers write arbitrary implications wrong good implementation would accept verified implications using theorem prover smt solver avoid bogus implications rather programmers write implications could try automatically derive implications given program fixed number predicates occur even unbounded number substitution pairings might occur runtime collect possible predicates source program consider pair predicates base type pred pred derive typing derivation shapes respective closing substitutions conditions looking property true true greenberg ideally also efficiently least efficiently deciding predicates problem finding reduced finding weakest precondition safety following function fun let bindings bindings else error else since would weakest precondition would know found general condition implication relation whether general condition best condition depends context also consider cost model programmers may want occasionally trade space time bothering join predicates would expensive test finding implication conditions resembles liquid type inference programmers get small control knob expressions smt solver rest settings different though liquid types verifying programs executing checks runtime implementation implementation issues abound free variables terms represented kind refactorings optimizations compiler might interfere set contracts appear program right moment compilation fix implication relation generally design space closure representations calling conventions languages contracts extensions generalizing semantics sums products seem particularly hard need contracts corresponding shapes join operation would push shapes recursive types datatypes interesting findler lazy contract checking keeps contracts changing asymptotic time complexity program may able adapt work avoid changes asymptotic space complexity well predicates range base types could also allow predicates types allow predicates higher types adequacy constraint predicate implication definition change impredicative polymorphism style system would require even technical changes introduction type variables would make reasoning names binders trickier order support predicates type variables need allow predicates higher notion adequacy would change order support predicates quantified types need change adequacy adequacy would end looking like logical relation used show relational parametricity would latent contracts substitute left right somehow related would technicalities tricky implementations would need careful manage closure representations correctly happens polymorphic code differs boxed unboxed types treat blame interesting algebraic enough proofs show always produce answer changing calculus interesting notion blame like indy semantics involutive blame labels would matter pushing shallow change semantics proofs finally would make sense substitution predicate stacks perform joins saying join nil substituting value predicate stack checks newly revealed redundancies proved change would sound would require changes type soundness particular need revise lemma simultaneously prove substitution joins preserve types lemmas related work technique space efficiency refer reader greenberg full description related work striven use greenberg notation exactly made changes adapting dependent contracts cons operator predicate stacks avoid ambiguity formerly two things named join one folded predicates closing substitutions account dependency place one requirement implication relation greenberg monotonicity substitution call substitutivity substitutions issue system must require join happen without value join happens know value cpcf first introduced several papers dimoulas later subject studies blame dependent function contracts static analysis exact behavioral equivalence means could use results static analysis terms cpcfc optimize space efficient programs cpcfe interestingly predicate implication relation seems work static analysis may deeper relationship thiemann introduces manifest calculus compiler optimizes casts time efficiency theorem prover uses delta types synthesize efficient checks deltas predicate implication relation similar uses separate logical language predicates restricts dependency codomains depend base values avoiding abusive contracts sekiyama also use delayed substitutions polymorphic manifest contract calculus different technical reasons delayed greenberg stitutions resemble explicit substitutions explicit bindings use delayed substitutions selectively resolve issues dependency manifest type system greenberg work somewhat disappointing compared type system given greenberg works much harder prove stronger type soundness theorem theorem enough help materially proving soundness space efficiency developing approach dependency used much easier latent calculus though several bugs along way would caught early stronger type system type system complexity sort old story racket implementation contracts leak space used effectively throughout plt racket racket designed avoid using contracts leaky ways racket contracts tend module boundaries calls inside module trigger contract recursive calls like example racket monitor recursive calls across module boundaries checks indeed lead space leaks terms racket tends implement contract checks recursive functions follows downto monl pred int pred int int note calling downto merely check final result less none intermediate values version downto puts contract inside recursive knot forcing checks every time sec racket also offers less thorough form space efficiencyvia construct program racket avoid redundant checks wrapping underlying function contract twice leads space leak figure use contract mechanisms highlight compromises racket makes space efficiency strik ing balance common ways tail recursion broken checking would expensive case redundant wrappers finally contracts racket monitors take two expressions one contract one monitored computing new contracts runtime breaks framing predetermine contracts arise runtime fix implication relation advance hope cpcfe close enough racket internal model provide insight achieve space efficiency least contracts racket conclusion translated greenberg original result manifest calculus latent one offered simpler explanation robby findler personal correspondence latent contracts define printf checking integer letrec contract lambda zero pos neg program letrec contract contract lambda zero pos neg pos neg program fig racket original result isolated parts type system required space bounds intermingled complexities conflating contracts types extended original result covering features dependency nontermination precisely bounding programs acknowledgments existence paper due comments sam david van horn chose interpret encouragement robby findler provided racket example helped correct clarify draft sam also offered corrections suggestions reviews offered helpful comments references abadi cardelli curien explicit substitutions journal functional programming jfp ahmed findler siek wadler blame principles programming languages popl dimoulas felleisen contract satisfaction world toplas nov dimoulas findler flanagan felleisen correct blame contracts scapegoating principles programming languages popl dimoulas felleisen complete monitors behavioral contracts seidl programming languages systems lncs vol springer berlin heidelberg findler felleisen contracts functions international conference functional programming icfp greenberg findler guo rogers lazy contract checking immutable data structures chitil eds implementation application functional languages berlin heidelberg flatt plt reference racket tech plt design http garcia calculating threesomes blame international conference functional programming icfp greenberg manifest contracts thesis university pennsylvania november greenberg manifest contracts principles programming languages popl greenberg pierce weirich contracts made manifest principles programming languages popl grossman morrisett zdancewic syntactic type abstraction toplas nov herman tomb flanagan gradual typing trends functional programming tfp apr herman tomb flanagan gradual typing higher order symbol comput jun jhala refinement types haskell programming languages meets program verification plpv acm new york usa meyer eiffel language plotkin lcf considered programming language theoretical computer science racket contract system rondon kawaguchi jhala liquid types programming language design implementation pldi sekiyama igarashi greenberg polymorphic manifest contracts revised resolved toplas accepted september appear sekiyama nishida igarashi manifest contracts datatypes principles programming languages popl acm new york usa siek thiemann wadler blame coercion threesomes together first time programming language design implementation pldi siek taha gradual typing functional languages scheme functional programming workshop september siek wadler threesomes without blame principles programming languages popl acm new york usa thiemann delta hybrid type checking wadler festschrift lncs springer switzerland van horn symbolic execution via contracts oopsla acm new york usa vazou rondon jhala abstract refinement types felleisen gardner eds european symposium programming esop springer berlin heidelberg berlin heidelberg wadler complement blame ball bodik krishnamurthi lerner morrisett eds snapl lipics vol schloss fuer informatik latent contracts wadler findler programs blamed european symposium programming esop
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expressing monitoring oscillatory dynamics petr brim david faculty informatics masaryk university brno czech republic safranek express temporal properties signals signal temporal logic stl defined maler work presented monitoring algorithm deciding satisfiability stl formulae finite discrete samples continuous signals logic used express analyse biological systems expressive enough sufficiently distinguish oscillatory properties important biology paper define extended logic stl augmented freezing operator allowing express distinguish detailed properties biological oscillations logic supported monitoring algorithm prototyped matlab monitoring procedure evaluated case study introduction paper deal automatic decision question given continuous signal satisfies given temporal property question originally arose domain analogous circuits verification procedure deciding question called monitoring monitor constructed given property allowing even decision continuous signal arising technical device real case numerical simulation procedure model case important fact monitoring procedure always time bounded validity given logic property discrete nature decided approximately signal continuous nature restrictions necessary limiting significantly help avoiding errors systems construction systems biology continuous signals typically represent model case theoretical behaviour mathematical models mimicking dynamics biological processes temporal properties employed express biological hypothesis monitoring procedure provides promising analysis tool many dynamical phenomena arise biological systems form oscillations physics term oscillation represents infinite behaviour periodically alternating certain identifying aspect oscillations fact certain states phase space dynamical system repeatedly biological example phenomena circadian rhythms interesting experiment achieved cyanobacterium cyanothece showed relation metabolic cycles circadian rhythms mathematical model oscillations gene regulation cyanobacteria provided models targeting oscillatory behaviour found contrast physics notion oscillation biology usually understood informally wider scope many experiments show oscillatory behaviour decreasing amplitude socalled damped oscillation dual notion oscillation increasing amplitude also relevant question targeting many oscillations strongly increase permanent oscillation work supported grant agency czech republic grant ezio bartocci luca bortolussi eds hsb eptcs dluhos work licensed creative commons attribution license expressing monitoring oscillations achieved comes significant interest tuning biological processes via mathematical models considering population behaviour signal domain need quantitatively express compactly encode mentioned types oscillatory phenomena logic constraints real values defined practically used express many dynamical phenomena including oscillations species concentration logic interpreted finite data obtained discretely sampled solutions differential equations odes series physical measurements formally technical problem may arise domain considered signals general particular interval time axis signal may exist infinitely many points considered property predicate changes truth value rigorous semantics treats problem given terms continuous signals required affine setting assumed exists finite interval covering signal end points sampled signal behaves reasonably individual time interval logic semantics called signal temporal logic stl based bounded metric interval temporal logic mitl neither stl temporal logic far know provide possibility express distinguish classes oscillations damped oscillations oscillations increasing amplitude reason impossibility globally referencing relatively comparing concrete signal values occurring time points local property satisfied course stl express directly single concrete signal universal property without references concrete values damped oscillation figure expressed stl sequence exactly reaching local extremes given order general appear number local extremes observed time interval general property expressed stl sin sin sin figure various types oscillations paper propose extension stl denoted based enriching logic freeze operator allows referencing signal value time point determined true means atomic propositions enriched variables form referring frozen values express damped oscillations following formula says always time instant near future local maximum future interval also another time instant near future local minimum future interval intervals another local maxima minima distant intervals occur implies signal values surrounded decreasing function top increasing function bottom constant sets extent damping dluhos concept freeze quantification introduced increase temporal logic expressiveness allowing every temporal operator bind time variable time refers context systems biology concept used biological oscillator synchronisation logic bosl reason oscillators synchronisation work shift notion temporal quantification bind signal variables temporal reference time level stl logic freezing variable values used formalisms employing local variables however formalisms support binding necessary note values derivatives signal values known sampled points oscillatory properties expressed using plain stl predicates derivatives however case needed derivatives required orders must included within signal getting data computationally hard even impossible allows express monitor oscillatory properties without need additional information supporting signals contribution paper extension stl logic signal value freeze quantification section motivated need express oscillations appearing biology paper primarily focused practical aspect give algorithm monitoring formulae bounded continuous signals section result supported prototype implementation evaluated biological case study coli repressilator section please note detailed discussion expressiveness provided master thesis makes preliminary version paper background first briefly recall signals continuous time purposes paper focus signals definition let finite interval time domain signal finite length signal function denotes length signal definition given signal denotes order signal use denote component signal canonical projection element signal order represents one finite run system variables variable represent quantity system concentration species derivative attribute signal measured quantified translate properties signal terms logic use function transforming real values signal components every time instant boolean set representing satisfaction dissatisfaction property signal truth values used propositional variables formulae logic unlike restrict allowed operations signal variables linear combinations simple comparisons restriction needed monitoring algorithm section limit expression properties interest definition linear predicate order function form real coefficients input variables boolean set predicate returns iff expression true otherwise however logic defined paper need work values variables acquired single time instant also second set values another time instant reason define additional function extracting truth values signal expressing monitoring oscillations definition atomic predicate function class signals order signal defined formula linear predicate order components signal values atomic predicate considered undefined atomic predicate signal describes boolean property signal constraint specified linear predicate satisfied respect two time instants example property condition difference values variables higher time time rearranged standard form linear predicate according definition instead usually write denoting variables signal time letters variables time avoid undesired cases atomic predicates output values predicate varying infinitely often assume deal signals respect every bounded variability every change detected sense every point limt included sampling logic oscillatory dynamics section describe syntax semantics stl based signal temporal logic stl extend freeze operator consider atomic predicates restricted linear defined previous section syntax formulae stl inductively defined following grammar atomic predicates definition standard propositional logic operators bounded operator constrained closed nonsingular time interval rational interpretation operator similar one used metric interval temporal logic mitl finally newly added unary freeze operator standard way derive additional temporal operators eventually true globally dluhos semantics formulae stl interpreted triplets signal two time instants write iff formula satisfied triplet satisfaction stl formula defined inductively structure models stl signals also speak satisfaction formula signal definition satisfaction formula signal defined meaning operator freeze values signal formula interpreted current time point use values inside subformula formula true time iff formula true time frozen time combination temporal operators interesting properties value variable expressed nondecreasing interval arbitrary constant future interval value increases value within two time units determine satisfaction formula signal signal sufficient length necessary length computed formula inductively structure freeze operator require additional length max max definition see formula contains atomic predicate operator wrapping predicate variables concerning frozen time instant handled definition semantics stl follows formula several nested operators meaning frozen variable local relates nearest operator content variable overwritten nested freeze operator implies single set frozen signal values accessible every place formula example formula first second occurrence variable relate first usage address time formula interpreted third occurrence contrast previous two expressing monitoring oscillations relates second operator addresses time instant subformula right side becomes true semantics restricted way excessively limit capability expressing various biological behaviour remaining computationally feasible algorithm section deal monitoring stl formula means determine satisfaction formula finite length signal monitoring procedure introduced paper inspired constrained ltl monitoring stl formulae needed extended space due freeze operator task solved simultaneously two time instants actual time frozen time idea monitoring procedure construct parse tree formula check satisfaction manner first step checking formula signal construct set time points formula satisfied definition let stl formula signal set called satisfaction set formula signal use notation whenever signal obvious context inductively construct satisfaction sets nodes higher levels parse tree last step procedure decide satisfaction set formula according definition equivalent checking whether satisfaction set contains point constructing satisfaction set general signal difficult task reason consider monitoring algorithm piecewise linear signals reasonable requirement cases deal time series produced numerical simulations modelled systems measurements series interpreted piecewise linear signals considering values changing linearly two adjacent points required points generated existing points make signal precise definition signal order called piecewise linear signal iff projections piecewise linear functions defined finite set intervals theorem inductive construction satisfaction set let piecewise linear signal formula stl satisfaction set formula signal constructed inductively respect structure formula proof equations follows directly definition semantics stl dluhos assumption piecewise linear signals combination linearity atomic predicates definitions enables construct satisfaction sets atomic predicates polynomial time easily compute sets belonging higher formulae formalism use representation satisfaction sets convex polytopes definition convex polytope set matrix column convex polytope called line segment convex polygon inequality definition identifies subspace convex polytope obtained intersection subspaces theorem assume piecewise linear signal set intervals defined let atomic predicate signal predicate linear function signal variables according definition denote set time instants atomic predicate satisfied signal rectangular area complement arrive two situations set makes line segment possibly degenerated single point empty set space divided line two subspaces figure might degenerated case line cross rectangle case vice versa proof described properties result linear form atomic predicates fact signal behaves linearly interval figure illustration set formula subspace cases set makes convex set easily describable set operations convex polytopes first case line segment convex polytope second case either set convex polytope convex polytope subtraction one bordering line segment polytope using algorithm express satisfaction set sets described constructed set operations polytopes set simplified expressing monitoring oscillations joining adjacent polytopes satisfaction sets atomic predicates could inductively construct satisfaction sets composite formulae theorem operations polytopes could achieved polynomial time however working precise satisfaction sets defined theorem quite difficult task work polytopes different dimensions polygons lines points operations sets objects unnecessarily demanding applications could end performing expensive monitoring noisy signals measured imperfect devices computed computers finite numerical precision hence think less precise yet faster monitoring algorithm imagine noisy signal signal measured computed finite precision actual values variables differ values presented signal error inaccuracy domain values implies also inaccuracy time domain investigating time instant event occurred sure exactly happened critical value burdened error possible also particular time instant imagine situation theorem working noisy signal sure border set due error condition defining points border could easily broken changing values signal arbitrary small values reason ignore border points count depending would computationally easier consequence first case theorem second case matter result simplification set allowed operators atomic predicates restricted remaining three operators lost sense signals burden error satisfaction atomic predicates meaningfully determined time instants inside satisfaction set hence satisfaction formula using operator determined every reference precise value replaced sufficiently large interval formula replaced based assumptions monitoring solved approximately every pair intervals satisfaction set area described single convex polygon whole satisfaction set get problem construction satisfaction sets composite formulae theorem reduced operations sets convex polytopes plane efficient algorithms exist algorithm approximative monitoring algorithm piecewise linear signals input piecewise linear signal stl formula output answer question algorithm inductively constructs satisfaction set computations performed parts signal sufficient length computed according signal sufficient length answer returned algorithm might wrong computation satisfaction sets individual intervals performed according theorem without determining satisfaction borders justified section dluhos result computed simple boolean operation polygons necessary ensure resulting set consists convex polygons could achieved example triangulation boolean polygonal operation satisfaction set freeze operation computed finding intersection line satisfaction set black line segments figure substituting values second component whole axis making projection first component cartesian product axis figure figure example computing satisfaction set formula satisfaction set represented set overlapping convex polygons different colors used depicting fact set consists convex polygons line regions intersection line satisfaction set depicted black color resulting set obtained projecting intervals intersection axis making cartesian product axis illustrative explain part algorithm geometrical point view identifying values horizontal axis values vertical axis figure first step computation find intersection convex polygon figure following procedure performed expressing monitoring oscillations ordered vertices rectangular polygons stripes figure next space divided horizontal stripes task solved stripe separately vertices polygons increasingly ordered second component duplidf cates removed denote ordered set figures every neighbouring pair vertices specifies rectangular polygon stripe figure form line segment considered empty care border points task solved every stripe separately every nonempty denote due way construction polygons single polygon lower border stripe vertices upper figure fact polygons take shape triangle trapezoid figure stripe points left side line rightmost polygon figure potential polygons identified solution placed time property moreover property must continuously hold time points satisfied get rid polygons lying right side points solution represent time past event isolate upper lower rightmost vertices polygon denote figure get new area lies left side line dluhos solution lie area restrict figure example stripe possible shapes polygons inside stripe adjoining polygons sharing edge connected form maximal seamless convex polygons green orange polygons figure let rightmost polygon figure even connecting polygons shape still triangular trapezoidal figure one rightmost polygon solution stripe figure operation defined equal minkowski sum final satisfaction set given application operator shifted polygon final solution stripe figure last step consider interval bounding operator result equivalent returned remark satisfaction set temporal operator future trueu hence computed easily according step obtained directly time space complexity time space complexity steps algorithm proportional number polygons working number polygons generated step atomic predicate number intervals signal defined number polygons asymptotically change computation operations described steps algorithm functions work slower see details output polygons also keep small number vertices process upper bound total computation time therefore number intervals signal defined size investigated formula expressing monitoring oscillations evaluation case study prototype approximative monitoring algorithm implemented matlab version toolbox mpt package version used polygonal operations focused efficiency implemented algorithm goal prove concept presented algorithm detailed description implementation results experiments performance analysis found studied biological system three transcriptional repressors designed built bacteria escherichia coli concentrations involved proteins periodically oscillating causes periodic production green fluorescent protein intensity fluorescence protein measured gives evidence ongoing activity network figure figure fluorescent intensity colony escherichia coli containing repressilator intensity increasing overall due increasing number individuals colony period oscillations lower time cell division whole repressilator transmitted generation generation figure taken network modelled system six differential equations dmi correspond activity genes lacl tetr concentrations corresponding proteins constants typical behaviour system depicted figure analysis system estimation parameters producing oscillatory behaviour originally done means manual qualitative analysis odes however perform analysis automatically using monitoring stl formulae specify desired oscillatory property formula test satisfaction different values parameters runs length least minutes expected period lower minutes start testing minutes later beginning avoid initial swing end testing minutes end measurement signal might cut middle period dluhos figure typical signal produced system variables depicted figure sample runs system different sets parameters initial values values depicted formula satisfied runs figure avoid case damped oscillations add formula ensures values reached period decreasing connecting formulae get formula express desired oscillatory behaviour figure satisfied signals type depicted figure another property repressilator expressed formula means values precede values sense every value reaches similar value short time figure full estimation parameters could performed due high time demands implemented monitoring algorithm single run monitoring algorithm formulae signal sampled points took several hours regular reduction number points describing signal would lead excessive loss information found expressing monitoring oscillations figure sample run system satisfying formula parameters initial values depicted variables red green bottleneck efficiency prototype implementation lies polyhedral operations performed mpt since work need small subset operations believe algorithm significantly accelerated optimal implementation employed focused demonstrating applicability leaving efficiency future work different task would ensure behaviour concentrations fluorescent protein bacteria figure correspondence model could done manually data would able test properties measurements identically properties signals produced odes conclusion proposed extended signal temporal logic motivated need express properties biological dynamical systems detail sufficient distinguish different shapes oscillation provided monitoring algorithm approximately computes truth value formula given continuous linear signal method prototyped matlab results achieved case study oscillatory behaviour repressilator showed method satisfactorily works signals generated numerical simulation however employed library mpt polyhedral operations appears efficient enough satisfy needs practical usage future work practical side plan implement efficient algorithms specific polyhedral operations use monitoring procedure another straightforward direction future development lifting robustness measure extended logic inspiring work logic tfl defined shifting stl semantics frequency domain tfl provides another way express permanent oscillations however nonpure oscillatory behaviour damped oscillations require specific elaboration joining tfl semantics concept freezing interesting step references accelera organization systemverilog language reference manual alur feder henzinger benefits relaxing punctuality acm alur henzinger really temporal logic acm dluhos ballarini guerriero verification qualitative trends oscillations biochemical systems theor comput sci barnat brim analysis biological systems dynamics divin model checker brief bioinformatics bartocci corradini merelli tesei detecting synchronisation biological oscillators model checking theoretical computer science batt ropers jong geiselmann mateescu page schneider validation qualitative models genetic regulatory networks model checking analysis nutritional stress response escherichia coli ismb supplement bioinformatics berg cheong van kreveld overmars computational geometry algorithms applications edition springer berlin calzone fages soliman machine learning biochemical networks temporal logic properties trans comput syst biol specification monitoring oscillation properties dynamical systems master thesis masaryk university available http maler robust satisfaction temporal logic signals formats berlin heidelberg maler bartocci nickovic grosu smolka temporal logic signal processing accepted atva eisner fisman augmenting regular temporal logic local variables fmcad ieee press elowitz leibler synthetic oscillatory network transcriptional regulators nature kholodenko negative feedback ultrasensitivity bring oscillations protein kinase cascades eur biochem kvasnica grieder toolbox mpt maler nickovic monitoring temporal properties continuous signals proc springer maler nickovic pnueli checking temporal properties discrete timed continuous behaviors pillars computer science lncs springer mateescu monteiro dumas jong ctrl extension ctl regular expressions fairness operators verify genetic regulatory networks theor comput sci miyoshi nakayama kaizu iwasaki tomita mathematical model chemical oscillator clock gene expression rhythms cyanobacteria journal biological rhythms nickovic maler amt monitoring tool analog systems proceedings international conference formal modeling analysis timed systems formats berlin heidelberg rizk batt fages soliman continuous degree satisfaction temporal logic formulae applications systems biology proc cmsb springer nedbal metabolic rhythms cyanobacterium cyanothece atcc correlate modeled dynamics circadian clock biol rhythms
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lottery ticket hypothesis training pruned neural networks mar jonathan frankle mit csail jfrankle michael carbin mit csail mcarbin abstract recent work neural network pruning indicates training time neural networks need significantly larger size necessary represent eventual functions learn paper articulates new hypothesis explain phenomenon conjecture term lottery ticket hypothesis proposes successful training depends lucky random initialization smaller subcomponent network larger networks lottery tickets meaning likely luck subcomponent initialized configuration amenable successful optimization paper conducts series experiments xor mnist support lottery ticket hypothesis particular identify subcomponents pruning weights trained networks demonstrate subcomponents successfully retrained isolation long subnetworks given initializations beginning training process initialized small networks reliably converge successfully often faster original network level accuracy however subcomponents randomly reinitialized rearranged perform worse original network words large networks train successfully contain small subnetworks initializations conducive optimization lottery ticket hypothesis connection pruning step toward developing architectures initializations training strategies make possible solve problems much smaller networks introduction recent work neural network pruning indicates neural networks dramatically simplified trained training complete upwards weights pruned without reducing accuracy network pruned function learned could represented far smaller network used training however researchers smaller networks trained readily larger counterparts spite fact demonstrably capable representing desired functions paper contend indeed possible train smaller networks directly fact small trainable networks embedded within larger models typically train paper articulates possible explanation disconnect neural network representation capacity trainability conjecture term lottery ticket hypothesis mentions cnns contain fragile features gradient descent able find good solution network initially trained layers retraining retrain pruned layers keep surviving parameters instead states training succeeds given network one subnetworks randomly initialized could trained rest high accuracy number iterations necessary train original network refer networks winning tickets subnetworks winning tickets perspective lottery ticket hypothesis network initialization procedure thought drawing many samples distribution initialized subnetworks ideally procedure manages draw subnetwork right architecture weight initializations optimization succeed winning ticket network size size measured units weights trained solve problem network size sufficient represent learned lottery ticket hypothesis views original network containing overlapping subnetworks larger network able train successfully one subnetworks lucked initialization amenable optimization metaphorically training network larger necessary represent function learned like buying many lottery tickets larger networks combinatorially subcomponents could facilitate successful training lottery tickets initialization strategy determines subcomponents optimization succeed tickets winners subcomponent initialized favorably network picked winning ticket training succeeds identifying winning tickets paper demonstrate possible automatically identify winning tickets making small critical modification experiment han prune trained neural network smallest weights measured magnitudes training manner han set connections survives pruning process architecture winning ticket anticipated lottery ticket hypothesis unique work winning ticket weights values connections initialized training began han aimed compress networks training process goal find small networks trained independently start show winning ticket extracted fashion initialized original weights training trained successfully isolation least fast typically faster full network methodology empirically assess lottery ticket hypothesis use following procedure extract winning tickets networks variety sizes mnist illustrative example small networks xor procedure identical han pruning process addition crucial last step resetting weights original values training randomly initialize neural network train network converges prune fraction network extract winning ticket reset weights remaining portion network values initializations received training began successful training really rely fortuitous initialization subcomponent network pruning really reveal winning ticket lottery ticket hypothesis predicts pruned reset original initializations train successfully sizes small network research questions test lottery ticket hypothesis evaluate following research questions effectively winning tickets train comparison original network randomly sampled networks similar size variety network configurations perturbations winning tickets measure convergence times test accuracy network converged big winning tickets relative size original network training networks various sizes explore whether size winning ticket remains constant particular learning problem grows proportion size larger network derived units units units units units figure success rates random xor networks specified number hidden units percent trials found correct decision boundary percent trials reached zero loss sensitive results particular pruning strategies test two broad classes strategies pruning hidden units incoming outgoing weights xor individually pruning weights mnist also study whether networks pruned single step whether must repeatedly pruned retrained reset iterative process results experimental results support lottery ticket hypothesis xor trained simple network one hidden layer learn xor minimal architecture capable representing xor hidden layer two units reached zero loss time contrast network ten hidden units reached zero loss iteratively pruned winning ticket winning ticket reached zero loss time trained original initializations mnist certain point winning tickets derived pruning converged faster least accurately original network point convergence times accuracy gradually rapidly dropped single step could prune networks still finding winning tickets average converged faster original network matched accuracy pruning iteratively time winning tickets smaller original network converged average faster networks iteratively pruned average still converged fast original network maintaining accuracy winning tickets randomly reinitialized weights randomly rearranged convergence times increased accuracy decreased compared original network depending metric winning ticket size winning tickets grew either gradually marginally network size contributions implications propose lottery ticket hypothesis new perspective neural network training posit pruning uncovers winning tickets lottery ticket hypothesis predicts leading algorithm extracting winning tickets trained networks apply algorithm empirically evaluate conjectures small networks evidence find supports lottery ticket hypothesis contention pruning extract winning tickets although paper focuses mainly measurement important implications understanding training increased representation power large networks necessarily required gradient descent learn functions small representations lurking within large networks small winning tickets efficient train product size faster converge product initialization examining initalizations architectures successful winning tickets might find new ways designing networks smaller superior learning xor function xor function among simplest examples distinguish neural networks linear classifiers presenting results mnist summarize lottery ticket hypothesis applies simple computation xor function four data points coordinates first last points placed class middle two points class geometrically problem requires nonlinear decision boundary experiment consider family fully connected networks xor two input units one hidden layer relu activation one output unit sigmoid activation pruning strategy units units pruned units pruned product input magnitude output magnitude product figure success rates different pruning strategies trials defined figure pruned columns include runs original network pruned winning ticket found right decision boundary reached zero loss first row table obtained pruning one shot subsequent rows involved pruning iteratively although network form two hidden units sufficient perfectly represent xor probability standard training randomly initializes network weights applies gradient learns xor network two hidden units low relative larger network figure contains overall success rates percent networks found right decision boundary reached zero loss training runs network two hidden units learned correct decision boundary trials loss reached meaning network learned output hard trials meanwhile otherwise identical network outfitted ten hidden units learned decision boundary trials reached loss trials figure charts loss hidden layer put central question paper concrete terms xor problem need start neural network ten hidden units ensure training succeeds much smaller neural network two hidden units represent xor function perfectly propose lottery ticket hypothesis explanation phenomenon lottery ticket hypothesis training succeeds given network one subnetworks winning ticket randomly initialized trained isolation high accuracy number iterations necessary train original network according lottery ticket hypothesis successful networks large number parameters xor network ten hidden units contain winning tickets comprising small number weights training still succeed methodology test lottery ticket hypothesis xor function instantiated experiment part following details randomly initialize network ten hidden units train iterations entire training set prune certain number hidden units according particular pruning heuristic extract winning ticket reset pruned network original initializations first three steps extract architecture winning ticket crucial final step extracts corresponding initializations ran experiment two different classes pruning strategies pruning involves pruning network single pass example pruning network would involve removing units trained contrast iterative pruning involves repeating steps several times removing small portion example satisfying weights first layer satisfying weights output unit satisfying bias output unit grows output approaches hard weights sampled normal distribution centered standard deviation values two standard deviations mean discarded resampled biases initialized network trained iterations units case two units iteration find iterative pruning effective extracting smaller winning tickets han found compressing large networks maintaining accuracy consider three different heuristics determining hidden units pruned input magnitude remove hidden unit smallest average input weight magnitudes output magnitude remove hidden unit smallest output weight magnitude magnitude product remove hidden unit smallest product magnitude output weight sum magnitudes input weights magnitude product heuristic achieved best results use unless otherwise stated results pruning generated networks ten hidden units pruned four two hidden units using magnitude product heuristic results appear first row figure winning tickets two hidden units found correct decision boundary time networks two hidden units reached zero loss time time random network iterative pruning conducted iterative version pruning experiment times starting networks containing ten hidden units eventually pruned two unit increments networks containing candidate winning ticket two hidden units ten hidden unit networks reached zero loss two hidden unit winning ticket also reached zero loss compared two hidden unit networks likewise ten hidden unit networks found correct decision boundary winning ticket compared two hidden unit networks four hidden unit winning tickets almost identically mirror performance original ten hidden unit network found correct decision boundary reached zero loss respectively cases ten hidden unit network pruned trials appear figure magnitude product row experiments indicate although iterative pruning computationally demanding pruning finds winning tickets higher rate pruning importantly also confirm networks ten hidden units pruned winning tickets two hidden units initialized values original network succeed training far frequently randomly initialized network two hidden units winning tickets four hidden units succeed nearly frequently ten unit networks derive results support lottery ticket large networks contain smaller winning tickets amenable successful optimization addition pruning heuristic also tested input magnitude output magnitude heuristics results appear figure magnitude product heuristic outperformed posit success due fact xor case input values either product input output weight magnitudes mimic activation unit therefore influence output mnist pruning section follow explore lottery ticket hypothesis applied mnist dataset analyze behavior pruning following section show additional power iterative pruning offers numbers derived last row figure networks ten hidden units reached zero loss networks started ten units reached zero loss pruned networks also reached zero loss methodology trained pruned network two layers used architecture input units corresponding pixels images mnist hidden layer units hidden layer units ten output units one class hidden units relu activation functions output units softmax activation functions default biases initialized weights randomly sampled normal distribution mean standard deviation values two standard deviations mean discarded resampled networks optimized using stochastic gradient descent learning rate section follows experimental template section randomly initialize network train iterations training data prune certain percentage weights within hidden layer removing lowest magnitudes extract winning ticket reset values weights pruned network original initializations training pruning strategy follow mnist removes individual weights rather entire units preliminary experiments found strategy effective srinivas babu explore pruning unit use simplest pruning heuristic possible remove weights lowest magnitudes within hidden layer weights connecting output layer pruned half percentage rest network pruned avoid severing connectivity output units results figure test set accuracy mnist training proceeds charts zoomed highest levels accuracy curve shows average progression five trials training specified pruning level percents percent weights layer remain pruning error bars show minimum maximum values one five trials dots signify moment corresponding colored line converged error bars showing earliest latest convergence times amongst five trials pruning substantial impact convergence times pruned size original network winning tickets converged average least faster accuracy remained average within original network accuracy winning ticket pruned original size converged average faster original network pruning caused convergence times slowly rise accuracy drop figure shows test set accuracy convergence behavior winning tickets pruned different levels curve average five different runs starting distinct define convergence moment moving average test accuracy changed less consecutive iterations measured test accuracy every iterations according definition convergence winning tickets improved test accuracy average standard deviation convergence acknowledge determining convergence times imprecise art metric seems adequately characterize behavior convergence purposes figure test set accuracy mnist training proceeds winning tickets various sizes winning tickets whose weights randomly reinitialized control experiment randomly initialized networks error bars indicate minimum maximum value run took point training process dots indicate average convergence times curve corresponding color error bars indicate minimum maximum convergence times left graph figure shows first pruning levels convergence times decrease accuracy increases winning ticket comprising weights original network converges slightly faster original network slower winning ticket original weights pattern continues network pruned original size right graph figure shows convergence times flatten increase winning ticket original size network returns performance unpruned network terms lottery ticket hypothesis attribute improving convergence times removal unnecessary noisy parts network pruning hones winning ticket convergence times reach tipping point pruning begins remove weights essential winning ticket convergence times increase accuracy decreases lottery ticket hypothesis also predicts behavior largely attributable confluence initialization architecture test conjecture ran two control experiments retain winning ticket architecture randomize weights retain winning ticket weights randomize architecture control experiment experiment evaluates extent initialization necessary component winning ticket figure shows experiment curves original network winning tickets original network size figure two curves added control experiments control experiment entailed training network used winning ticket architecture randomly reinitialized weights original initialization distribution trained three control experiments winning ticket control curves average experiments unlike winning tickets control experiments converged average slowly original network simultaneously achieving lower levels accuracy differences substantial average winning tickets converged times fast corresponding average controls error bars convergence times reflect control trials exhibited much wider variance behavior example control trials converged faster average unpruned network however average control trial convergence time converged slower average original network experiment supports lottery ticket hypothesis emphasis fortuitous initialization using pruned architecture original initialization withstood benefited pruning performance reinitialized network immediately suffered steadily diminished network pruned outcome mirrors larger scale result xor experiment networks many hidden units could pruned smaller winning tickets found right decision boundary much higher rate small networks figure provides broader perspective patterns across levels pruned left graph shows convergence time relation percentage network remaining pruning blue line average five winning ticket trials level convergence figure convergence times left accuracies right running mnist pruning experiment various degrees pruning blue line average five trials different starting initializations prune reuse original initialization multicolored lines represents three randomly reinitialized control trials one trial original initialization error bars minimum maximum value trial takes interval figure convergence times accuracy five winning tickets level pruning blue line trials winning ticket weights reinitialized orange line trials winning ticket weights maintained shuffled within layer green line time initially decreases leveling slowly climbing contrast multicolored lines represent groups control trials winning ticket steadly require longer converge network pruned control experiment error bars much larger suggesting wider variation convergence times compared consistent convergence times winning tickets right graph figure provides important context accurate networks moment converge average trial used original initialization blue line maintans accuracy within original network pruned accuracy drops contrast accuracy average control trial drops level network pruned falling precipitously pruned original network accuracy experiment supports lottery ticket hypothesis prediction fortuitous initialization necessary ingredient make winning ticket winning ticket structure alone insufficient explain success control experiment experiment evaluates extent architecture necessary component winning ticket winning ticket level pruning randomly shuffled locations weights hidden layer maintaining original initializations results appear figure figure blue line traces winning tickets pruned various sizes orange line average trials control experiment reinitializing winning tickets green line average trials control experiment shuffling winning tickets without reinitializing convergence times two control experiments similar start increasing immediately increase rapidly network gets smaller accuracy control experiment drops slightly earlier control experiment dropped winning ticket experiment supports lottery ticket hypothesis prediction winning tickets emerge combination initialization structure neither initialization control experiment structure control experiment alone sufficient explain better performance winning tickets figure convergence times accuracy winning tickets extracted networks mnist using pruning orange iterative pruning blue note logarithmic summary first notable result set experiments even pruned sizes much smaller original network winning tickets still able converge supports core prediction lottery ticket hypothesis pruning reveals smaller subcomponents originally initialized train successfully isolation networks train successfully converge faster maintain accuracy networks derive furthermore winning tickets emerge confluence fortuitous initalization structure mnist iterative pruning xor experiment section iterative pruning training pruning reinitializing pruning winning tickets likely train successfully section find iterative pruning makes possible extract winning tickets mnist network far smaller generated pruning methodology use experimental setup network architecture initialization strategy optimization strategy section follow similar procedure repetitively order iteratively prune randomly initialize network train iterations training data prune weights within hidden layer weights output layer removing lowest magnitudes reset weight values pruned network initializations training repeat steps network pruned desired size result last iteration winning ticket iteratively prune incoming weights first second layers network weights output layer start network two hidden layers hidden units prune network original weights remained comparison pruning figure shows difference convergence times accuracy pruning orange iterative pruning blue note logarithmic figure figures section average iteratively pruned winning tickets reach initially reach lower convergence times convergence times flatten original network pruned faster original network faster original network original network size compared faster original network faster mentioned section prune output layer lower rate reduce chances severing connectivity output units figure convergence times accuracy winning tickets extracted iteratively pruning control experiments blue line average five winning tickets orange line control experiment winning tickets reinitialized green line control experiment winning tickets whose weights randomly shuffled red line performance pruning locations control trials cut according metric longer converged original network pruning average iteratively pruned network returns original convergence time pruned compared pruning likewise accuracy actually increases slightly many winning tickets returning original network accuracy winning ticket size average contrast pruning begins drop winning ticket size original network although iterative pruning extract much smaller winning tickets pruning far costly find winning tickets extracting winning ticket pruning requires training original network single time regardless much network pruned contrast iteratively pruning network iteration original network size requires training network times however since goal understand behavior winning tickets rather find efficiently iterative pruning compelling advantage able extract smaller winning tickets maintain convergence accuracy performance placing tighter size network winning ticket results section control experiments section aim explore extent architecture initialization responsible winning ticket ability continue converge small sizes figure contains average results performing control experiment randomly reinitializing winning ticket weights orange control experiment randomly shuffling winning ticket weights green comparison curve red performance pruning control experiment pruning average convergence times control experiment begin increasing soon network pruned continue grow steady rate error bars figure reflect convergence times vary widely pruned networks reinitialized average control trial accuracy begins dropping lower original networks accuracy network pruned whereas average iteratively pruned network drops level pruned experiment control trial indicates initialization plays critical role making winning ticket control experiment average convergence times control trial increase steadily pattern similar control trial error bars indicate convergence times similarly vary widely accuracy begins dropping earlier steeply potentially suggesting architecture might important initialization summary control experiments iterative pruning put results section sharper relief iterative pruning makes possible extract smaller winning tickets pruning reach lower convergence times original network maintaining exceeding level accuracy control experiments show initialization network architecture figure distributions initializations weights survived iterative pruning across ten iterative pruning runs graphs contain initializations network pruned left rigth blue orange green lines distributions initial weights first hidden layer second hidden layer output layer respectively play factor creating winning ticket control trial suggesting network architecture might slightly important experiments xor mnist support lottery ticket hypothesis embedded within larger networks small subcomponents fortuitously initialized manner conducive successful training extracted winning ticket architectures pruning determined corresponding initializations resetting winning ticket connections original training networks trained successfully case mnist network converged faster accurately meanwhile neither architecture initialization alone could entirely account result next investigate architecture initializations small winning tickets section behavior winning tickets subjected wider variety parameters section examining winning tickets section briefly explore internal structure winning tickets result iterativelypruning mnist network already found evidence support claim winning tickets arise confluence architecture initialization exactly architectures initializations look like initializations figure shows initialization distributions winning tickets four different levels pruning note values winning ticket weights training graph upper left contains initial weights entire network pruning initialized according normal distribution mean standard deviation graph upper right contains weights iteratively pruning network original size blue orange green lines distriutions initial weights first hidden layer second hidden layer output layer respectively remaining weights already show impact pruning first second hidden layer distributions bimodal two peaks mirrored opposite since distributions plot original initializations weights survive pruning process weights training distributions created removing samples formerly normal distribution peaks graph appear left right tails original normal distribution missing weights pruned interestingly pruning occurs training graphs weights training words distributions emerge small weights training must remained small training second hidden layer orange retains center first hidden layer indicating weights likely moved training output distribution green closely resembles original normal distribution indicating weights probably moved significantly training one contributing factor output distribution prune slower rate meaning effects pruning make take longer appear figure unit current layer many units previous layer connect left graph first hidden layer middle graph second hidden layer right graph output layer blue orange green lines winning tickets iteratively pruned respectively point line represents single unit units sorted descending order number connections data points collected ten trials pattern pruning plays extreme form lower left graph lower right graph middles first second hidden layer distributions continue get hollowed happens albeit slowly output distribution even input distributions corresponding networks converged faster original network retained accuracy considering extent particular pruning strategy pursued left imprint winning tickets worth considering impact pruning strategies would broadly whether winning tickets found product pruning strategy pursued whether pruning strategy pursued happens exploit deeper reality way neural networks behave architecture network pruned becomes sparser figure shows distributions surviving connections aggregated across ten unit layer units layer network pruned blue orange green original size left middle right graphs show first hidden layer second hidden layer output layer pruned network remains almost slight differences units least connections network pruned units first hidden layer continue roughly equal number connections units input layer even network pruned small fraction hidden units first layer eliminated entirely second hidden layer becomes less evenly connected weights pruned time network pruned nearly third units second hidden layer fully disconnected steep decline units output layer shows less severe slope likely every output unit serves clear function prune output layer slower rate winning tickets quite sparse even network pruned nearly fraction units eliminated entirely units maintain large number connections pruning instead nearly units retain proportionally small number connections exploring mnist parameters section explores sensitivity mnist results parameters lottery ticket experiment namely explore role initialization network size play properties winning tickets emerge initialization although default network initialized normal distribution mean standard deviation experimented several standard deviations explore effect larger also removed edges path output unit figure convergence times accuracy groups five winning tickets initialized various standard deviations figure convergence times accuracy groups five winning tickets initialized various standard deviations smaller weights behavior winning tickets one might expect pruning strategy would especially vulnerable initializing network weights large selecting highestmagnitude weights might exacerbate exploding gradients likewise might resilient initializing network weights small since select largest weights training section present results using pruning strategy results iterative pruning similar figure shows convergence times accuracy winning tickets networks initialized standard deviations larger expected convergence times increase accuracy decreases standard deviations increase explore whether extent behavior resulted exploding gradients weaknesses pruning strategy figure contains information winning tickets networks initialized standard deviations smaller standard deviation produces fastest convergence times cedes certain amount accuracy contrast standard deviation causes winning tickets converge slowly optima behavior suggests sweet spots convergence times accuracy network size experimented increasing size default network layers hidden units order determine whether fixed winning ticket size particular learning problem whether larger networks naturally beget larger winning tickets consider two possible definitions size network winning ticket winning ticket minimal network minimizes convergence time since convergence times initially decrease pruning heuristic looks winning ticket lowest possible convergence time winning ticket minimal network retains accuracy original network accuracy remains relatively flat smaller smaller winning tickets created figure convergence times accuracy groups five winning tickets extracted networks various sizes pruning strategy error bars elided improve readability legend contains size network means network hidden layers units networks initialized standard deviation figure convergence times accuracy groups five winning tickets extracted iteratively networks various sizes error bars elided improve readability networks initialized standard deviation reaches tipping point drops rapidly definition considers winning ticket last moment accuracy takes place pruning trained networks whose sizes multiples original network size results applying pruning strategy appear figure plots convergence times accuracy according number weights winning ticket according definition winning ticket winning ticket sizes increase gradually size network architecture appears reach point weights weights pattern holds larger architectures larger networks capable representing sophisticated functions pruning larger networks may produce different network architectures exploit additional representation capacity converge faster indeed larger network lower convergence times winning tickets able achieve larger size reached definition winning ticket agreed bottom graph figure illustrates accuracy larger networks dropped steeply slightly earlier times accuracy smaller networks however differences quite order tens thousands weights although winning ticket size seem increase network size definition changes slight winning ticket sizes close uniform iterative pruning figure reflects convergence accuracy trends iteratively pruning larger networks remains case larger networks reach minimum convergence times gradually larger sizes accuracy plummets unison two key differences worth noting iterative case figure convergence times accuracy winning tickets extracted iteratively pruning different rates iteration error bars elided readability note logarithmic first minimum convergence times accuracy dropoffs occur much smaller network sizes experiments result coincides iterative experiments demonstrate iterative pruning creates winning tickets pruned much smaller sizes convergence times increase accuracy diminishes whereas accuracy dropoff took place networks weights experiments occurs winning tickets tens thousands weights second accuracy graphs small bulge upwards dropping indicating accuracy actually increases slightly winning tickets smallest bulges occur winning ticket size cases regardless initial size network summary analysis subsection leaves many open questions future research although undertake extensive analysis internal structure winning tickets study comparing winning tickets derived networks different sizes would shed light extent winning tickets similar different various initial network sizes exploring iterative pruning rates choosing exact rate prune iteration iterative pruning entails balancing performance resulting winning ticket number iterations necessary extract winning ticket figure shows convergence times accuracy architecture iteratively pruned different rates iteration note logarithmic experiment thought exploring middle grounds pruning iteratively pruning small rate although pruning larger percentage iteration reaches smaller winning tickets faster winning tickets pruned aggressively fail match convergence times accuracy winning tickets pruned slowly end spectrum iteratively pruning appears achieve best convergence times accuracy would require training network times extract winning ticket original network size experiments prune balances performance amount training required weight resetting training iteration iterative pruning approach reset weights unpruned connections original values training part experiment evaluate lottery ticket hypothesis exploring well winning tickets obtained pruning train isolation conjecture resetting training iteration makes easier find small winning tickets effect iteration recursive pruning problem subnetwork trains effectively starting original initializations must pruned slightly smaller network contrast han interleave training pruning without ever resetting weights round training weights pruned training continues based trained figure convergence times accuracy winning tickets extracted iteratively pruning using weight resetting iterations strategy blue continuing use trained weights pruning han strategy orange weights differences approaches reflect two different goals han want produce smallest possible trained network wish find pruned network trains successfully start figure shows convergence times accuracy achieved winning tickets extracted using two pruning strategies simulate han strategy iteratively trained network pruned weights continued training using trained weights iteration copied resulting network reset weights original initializations trained network obtain results figure figure shows han pruning strategy quite effective finding small networks rain successfully although strategy resetting weights iteration maintains lower convergence times higher accuracy slightly longer however since figure logarithmic scale differences appear small network sizes related work pruning lecun first explored pruning way reduce size neural networks pruned based second derivative loss function respect weight hassibi build approach recently han showed techniques could used substantially reduce size modern networks since rich variety neural network pruning approaches emerged pruning smallest weights pruning units bayesian fashion pruning entire convolutional filters fusing redundant units increase network diversity goal literature pruning compress trained neural networks reducing size large model run efficiently restricted computational platform mobile device without sacrificing accuracy contrast aim make possible train small neural networks start work network compression takes place three iterative steps first large network trained second weights units pruned according heuristic third network trained using weights han find without third retraining step network performance drops much earlier pruning process han also caution pruned network training consider reusing values surviving weights initialized original network work builds literature pruning shedding light mechanisms make pruning possible fact networks pruned maintaining accuracy indicates function learned represented much smaller network one used training aim understand pruning possible investigate whether small networks trained directly rather pruning large networks smaller sizes training lottery ticket hypothesis posits large networks small subnetworks facilitate successful training point view neural network pruning finds winning tickets evaluate lottery ticket hypothesis small networks leverage han experimental approach except make crucial modification pruning reset weight original value results explain complement han lottery ticket hypothesis offers insight han able prune networks many trends see accuracy winning tickets drops small winning ticket sizes original initializations pruned networks take bimodal distribution parallel han find continuing train pruned networks based trained weights dropout dropout creates smaller subnetwork training iteration randomly removing subset units weights unit activation reduced probability dropped intuitively dropout intended reduce overfitting improve generalization forcing units remain robust changes network work dropout characterized training dropout perform ing gradient descent respect ensemble possible subnetworks inference dropout approximately computing average ensemble terminology dropout experiment aims discover single particularly successful member ensemble subnetworks dropout heuristic training network without dropout drop lowest weights magnitude training probability weights probability words perform extremely aggressive form dropout based examining results training network without dropout however goal different dropout designed regularize network training process used produce sparse networks aim directly find small case networks found sparse networks trained start finish without removing weights broader formulation lottery ticket hypothesis closely relate dropout notion ensemble learning lottery ticket hypothesis views large network collection combinatorial number small networks lottery tickets one winning ticket must initialized fortuitously enable training succeed point view large network begins possibility coalescing toward one exponential number subnetworks gradient descent drives toward subnetwork comprising winning ticket find limitations work limited several ways examine networks two smallest possible examples xor mnist consider convolutional networks larger networks better reflect evidence lottery ticket hypothesis purely experimental offer theoretical analysis formally support claim finally although analyze structure initialization distributions winning tickets mnist yet devise way turn observations useful strategies training smaller networks anticipate exploring avenues future work updating paper conclusions future work paper proposes new hypothesis explain large neural networks amenable substantial pruning yet pruned networks trained effectively scratch conjecture known lottery ticket hypothesis holds training succeeds subcomponent larger network randomly initialized fashion suitable optimization furthermore conjectures pruning uncovers winning tickets empirically evaluate hypothesis devised experiment based work han pruning trained network remaining weights reset original initializations lottery ticket hypothesis holds pruning however preliminary experiments convolutional network reflect behavior described paper mnist uncovers winning tickets pruned networks train successfully isolation reset original initializations xor found winning tickets derived larger networks able learn decision boundary reach zero loss far frequently randomly initialized mnist winning tickets converged quickly reached higher accuracy original network control experiments supported claim winning tickets represent confluence fortuitious initialization network architecture paper articulates new perspective neural network training supports view empirically foundation laid numerous research directions evaluate lottery ticket hypothesis exploit perspective improve network design training larger examples largest network examine network mnist repeating experiments outlined paper convolutional network larger networks harder learning tasks would make possible understand whether lottery ticket hypothesis holds generally manifests settings understanding winning tickets paper focuses mainly behavioral properties lottery ticket hypothesis pruning winning tickets one logical next step systematically analyze architectures initializations lottery tickets extent winning tickets unique artifacts created randomly initializing large networks getting lucky extent common structure multiple winning tickets task winning tickets tell functions neural networks learn particular tasks lottery ticket networks lottery ticket hypothesis existence winning tickets demonstrate small networks trained start finish concrete work would exploit lessons learned leveraging winning tickets develop new network architectures initialization regimes allow smaller networks trained wider variety learning tasks could reduce amount computation needed train neural networks references pierre baldi peter sadowski understanding dropout advances neural information processing systems song han huizi mao william dally deep compression compressing deep neural network pruning trained quantization huffman coding corr http song han huizi mao william dally deep neural network compression pipeline pruning quantization huffman encoding arxiv preprint song han jeff pool john tran william dally learning weights connections efficient neural network advances neural information processing systems babak hassibi david stork gregory wolff optimal brain surgeon general network pruning neural networks ieee international conference ieee yann lecun bottou yoshua bengio patrick haffner learning applied document recognition proc ieee yann lecun john denker sara solla optimal brain damage advances neural information processing systems hao asim kadav igor durdanovic hanan samet hans peter graf pruning filters efficient convnets arxiv preprint christos louizos karen ullrich max welling bayesian compression deep learning advances neural information processing systems luo jianxin weiyao lin thinet filter level pruning method deep neural network compression arxiv preprint zelda mariet suvrit sra diversity networks neural network compression using determinantal point processes arxiv preprint suraj srinivas venkatesh babu parameter pruning deep neural networks arxiv preprint nitish srivastava geoffrey hinton alex krizhevsky ilya sutskever ruslan salakhutdinov dropout simple way prevent neural networks overfitting journal machine learning research
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infinitely generated semigroups polynomial complexity may birget abstract paper continues functional approach problem begun focus monoid morphisms free monoid polynomial input balance polynomial construct machine model functions evaluation functions prove finitely generated use show separation results introduction defined monoids partial functions question whether equivalent question whether monoids regular monoid consists partial functions computable deterministic turing machines polynomial time polynomial submonoid consists elements morphisms functions according exactly nonregular elements known functions according exist iff also regular iff regular hence iff regular iff regular refer background original motivation studying addition reminiscent groups monoids also quickly turned connection different properties regarding green relations actions see hard know whether approach contribute solution problem monoids interesting rest paper use following notation terminology alphabet unless contrary explicitly stated denotes set strings including empty string denotes length string partial function domain dom defined image dom say function mean partial function except explicitly say total function similarly deterministic turing machine alphabet domain machine set input words machine produces output set output words image machine function called polynomially balanced iff exists polynomials dom polynomial called input balance function said already set partial functions polynomially balanced dom computable deterministic turing machine hence dom hard show clearly monoid function composition function said respect complexity iff exists deterministic algorithm every input outputs always mean respect complexity hence functions cryptographic functions sense however important problem following folklore fact see functions exist iff easy prove see introduction iff regular definition element monoid regular iff exists case called inverse monoid called regular iff elements regular summary monoid regular iff let look detail monoid right ideal subset closed string two strings say prefix iff prefix code set word prefix another word right ideal exists unique prefix code say generates right ideal details see good reference prefix codes codes general morphism partial function dom case dom right ideals morphism let domc called domain code prefix code generates dom right ideal similarly let imc called image code prefix code generates morphism determined restriction domain code general imc domc happen imc domc define morphism prop regular iff regular monoid regular iff saw cor isomorphic group units trivial prop one prop see interesting actions interesting homomorphic images regular monoids regular iff overall seems structure proved section isomorphic submonoid prove use encoding alphabet words alphabet encoding also used first encode alphabet code code code word encoded code code fixed words encoded code code code function encoded defined domc code dom dom code dom code code dom equivalently code code every code prefix code belongs iff iff transformation isomorphic embedding moreover regular iff regular alphabet denoted always section introduced notion polynomial program turing machines polynomial counter input balance programs form machine model characterizes functions polynomial program let denote function computed program every polynomial form positive integers constructed evaluation map evc every polynomial program polynomial evc code code terminology varies depending field semigroup theory called numerical mathematics called generalized inverse ring theory category theory called weak inverse semigroup theory term inverse applied holds addition easy see satisfies dom dom evc code undefined used evq polynomial degree large enough coefficient prove following first finitely generated theorem second evc complete respect inversive polynomial reduction section later paper def following define completeness various reductions along lines note entirety evaluation maps belong respectively since maps would polynomially bounded complexity reason restrict evr complexity need precise machine models opposed intuitive models section define machine model characterizes functions large enough polynomial construct evaluation maps evrc functions evrq balance prove evrcc complete respect inversive turing reduction section prove finitely generated section show infinite generation complexity consequences infinite generation used argument machine model evaluation maps evaluation map evc code code constructed works particular provided evc morphism moreover evaluate functions want construct evaluation map belongs evaluates exactly elements balance complexity constructed machine model namely class turing machines polynomial counter controlling refine turing machines order obtain machine model accepting right ideals computing functions consider deterministic turing machines alphabet input tape output tape moreover assume input tape output tape head move right stay place move left assume input tape left endmarker right endmarker blank symbol beginning computation machine input input tape content input tape head initially tapes blank filled infinitely many copies letter output tape need endmarkers since assume special output state qout goes state qout output complete output state halting state transition state qout important convention turing machine function following input halts state qout output even output tape contains word case undefined content output tape considered unreadable hidden output state qout reached kind turing machine compute partial recursive function restrictions input output tapes limit machine compute function add polynomial used bound input balance see section order obtain machine model functions turing machines polynomial restricted compute morphisms done two steps first sequential functions sequential turing machines introduced easy obtain class turing machines compute morphisms special kind sequential functions recall function mean partial function definition function sequential iff dom prefix prefix obviously every morphism sequential function sequential turing machine deterministic turing machine special input tape special output tape output state according conventions function following holds every dom every word computation input head start reading written output tape read letter means make transition whose input letter input tape content head letter transition made letter yet output tape content course moment computation input necessarily finished state necessarily qout output might still grow qout might reached eventually qout never reached final output sequential turing machines form machine model partial recursive sequential functions let machines polynomial obtain machine model sequential functions finally obtain machine model functions take sequential turing machines polynomial following additional condition every dom every word computation input written output tape read input tape remaining input copied output tape point state qout reached call machine following shows function constructed provided let first consider right ideals rather functions polynomial program turing machine accepts language construct new polynomial program describing turing machine behaves follows input successively examines prefixes finds prefix say accepted read letter comes decided soon finds prefix accepts whole input accepts prefix rejects thus accepts right ideal generated right ideal polynomial let consider functions given polynomial program function construct new polynomial program input successively examines prefixes finds prefix dom let input machine outputs note since shortest prefix dom actually domc dom right ideal machine read letter comes prefix decided dom domc hence function computed construction describes transformation fpref fpref defined follows fpref shortest prefix belongs dom domc fpref thus every iff fpref based construct evaluation maps let polynomial integers define evrc follows evrc code code polynomial dom details construction evc see section although evrq belongs evaluates polynomial prove theorem complexity evrc higher following doubly coded evaluation function usually useful defined evrcc code code code code domc give relation evrc use following partial recursive evrq morphism defined every dom code code code shortest prefix dom equivalently domc dom code undefined essentially finds shortest prefix belongs dom equivalently domc function evaluated examining successively longer prefixes prefix dom fund computable recursive domain ranges fixed let restricted words dom code dom code code code dom dom opposed domc similarly define code dom code code code domc domc belong every fixed computable since work possible another restricted form belongs obtained choosing fixed polynomial defining restriction set code polynomial dom hence also define functions word denote code another important function decoding function defined decode code domc decode imc decode also define decoding function code code code domc imc formulate relation evrc evrq evrc evrq order show evrcc complete respect inversive reduction adapt padding unpadding functions defined section although keep names corresponding padding functions functions slightly different padding procedure begins function expand defined expand code code code code domc word form code word code also code word namely code since subset prefix codes code uniquely determined prefix code obtained code code domc moreover polynomial program polynomial max detailed justification numbers used definition expand well reexpand recontr contr given section important expand uses prefix padding format code domc whole input used computing amount padding expand would order isolate morphism reason introduce prefix domc iterate expansion padding applying following function reexpand code code code code even context reexpand used repeated contraction unpadding carried applying following function recontr code code code max code note max unpadding procedure ends application function contr code code code code functions expand reexpand recontr contr undefined cases output specified lemma let polynomial defined polynomial form positive integers domc decode contr recontr evrcc reexpand expand contr evrcc reexpand expand proof similar proof prop modifications domc expand code code code reexpand code code code induction string argument length much larger time takes simulate machine program input evrcc applied correctly continuing calculation evrcc code code recontr code code use recontr could much shorter polynomial input balance note input padding necessary harm recontracting unpadding needed effect definition recontr hence contr applied correctly complete calculation contr code code code lemma following infinite generating set decode contr recontr evrcc reexpand expand decode replaced yet another infinite generating set evrc polynomial form proof first infinite generating set follows lemma recall code second generating set follows straightforward way proof prop proposition generated set regular elements proof generators easily seen regular thus using second infinite generating set lemma enough factor evrc regular elements evrc defined follows every polynomial every domc code code code code code code code code code code functions undefined otherwise easy see inversion algorithms regular belong show evrcc complete respect certain inversive reduction need recall definitions concerning reductions functions particular reductions preserve inversive reductions definition let two polynomially balanced morphisms say simulates denoted iff exist turing simulation denoted iff computed oracle make oracle calls oracle calls particular calls membership problem dom definition need computable since prop every simulated every simulations definition inversive reduction simulation morphisms previous definition corresponding inversive reduction defined follows say inversively denoted iff every inverse exists inverse range polynomially balanced morphisms note prop apply inversive reduction since range one easily proves following polynomially balanced morphisms see section addition regular regular equivalently addition definition polynomially balanced morphism complete set morphisms respect inversive reduction iff see section details properties simulations reductions focus whereas concentrate simulations def similar standard notions reductions decision problems concept inversive reduction first introduced appropriate notion reduction functions preserved upward reduction regularity preserved downward reduction definitions refer polynomially balanced inverses justified following proposition according balanced functions balanced inverses proposition suppose morphism balance polynomial inverse inverse balance constant inverse chosen restriction proof let restriction set dom obviously balance note since inverse dom show inverse sufficient check domain contains let dom since inverse checking inequality holds since balance input checking since balance input since find bound first compute time thereby also verify dom check whether domain first compare time checking automatically compute time writing number binary evaluating see section similar computation check time checking done similar way time theorem map evrcc complete respect inversive turing reduction proof lemma provides following simulation evrcc decode contr evrcc reexpand expand obtain inversive turing simulation let inverse evrcc slightly modifying proof prop apply string form code code domc imc domc code code code code dom based construct inverse define decode contr recontr reexpandm expand defined code code similar except uses imc imc uses domc saw rmp unless rmp whereas general computed rmp makes oracle calls dom value follows input considers prefixes increasing lengths found since first prefix imc code code test whether pads produce code code code defined input thus code code dom hand code code dom let code code code code one oracle call yields hence use check whether dom holds iff way check whether thus find produces output matter since care defined outside known remaining simulation decode contr recontr reexpandm expand code code yields applied function inverse indeed dom imc code code reexpandm expand applying yields code code applying decode contr yields finally since show next evrcc complete proposition map evrcc complete respect proof prop evc complete inversive simulation prop maps moreover evrcc indeed since saw complete hence evrcc generation proved finitely generated left open question whether also finitely generated answer question negatively use following general compactness property semigroup finitely generated infinite generating set generated finite subset set theorem finitely generated proof saw generated infinite set contr recontr evrcc reexpand expand let assume contradiction finitely generated finite generating set extracted infinite generating set generated contr recontr evrcc reexpand expand finite set every word expresses finite sequence generators recall dom code dom dom code code code domc proof strategy consist showing infinitely many functions correct representation precisely domc code code code code hand show exist infinitely many every represents exist infinitely many domc code code code code thus obtain contradiction consider domc satisfies word domc contains subsegment domc domc domc integer exists domc length prefix equivalently domc domc picture path tree labeled ending vertex vertex along path distance vertex second path branches ends vertex length following family examples shows exist infinitely many satisfy properties examples parameterized domc code fixed word depending chosen domc domc thus property holds word long enough work indeed different words prefix codes different whereas finite property follows definition code namely code code property holds every code every take code code set code regular language regular expression code let representation family examples properties consider certain suffixes let shortest suffix domc code form code code exists since representing maps code code code inductively define shortest suffix strict suffix domc code code form code code theorem follows next lemma according infinitely many domc code code code hand represents hence definition every domc code code code empty thus assumption finite generating set represents leads contradiction lemma let domc code word chosen program satisfies properties let word represents let length let suffixes defined exist domc code code code moreover common suffix common suffix length least proof domc code code code want show domc sufficiently long suffix common number auxiliary parameter take form code use induction proof generators occur indeed generators namely contr recontr evrcc reexpand expand applicable inputs form code code would end generator contr recontr evrcc reexpand expand applied moreover start generator contr recontr evrcc reexpand expand indeed inputs code domc code contains generators defined element domc actions change input positions left end input actions preserve common suffix length thus consists instances lemma holds suppose contains instances transform input code code word code code applied action changes input positions left end input since assumed applicable must also domc output code code code code code thus common suffix could decrease length action let also one occurs since output form code code marks end action proves lemma inductive step induction assume domc code code code common suffix length let write definition also let claim contains generator contr recontr evrcc reexpand expand first rightmost letter occurs indeed applicable later output generator preceding would form code code would ended applied claim contains generator contr recontr evrcc reexpand expand last leftmost letter occurs indeed generator outputs word form code code ends generator consequence claims contains generator contr recontr evrcc reexpand expand consists generator form change assume remaining cases form generators let code code input also output common suffix length case generators changes input positions left end input affected case output form code common suffix preserved action containing generators affects positions near left side input changed case applications change fewer letters input near left end common suffix affected applied output produced form code code code domc affected pick case case handled combination previous two cases abovep cases constraints fulfilled pnfor code using fact note words depend choice input code whenever long enough indeed determine apply infinite word code notation given polynomial form integers let computed polynomial call iff balance polynomial let submonoid generated set obviously proposition set polynomials form nki sup sup generation result also holds proof similar need preliminary facts lemma every polynomial form every proof recall code code code domc input balance indeed input shorter output output length less compute code code input code proceed follows first machine reads outputs code runs program input simulates corresponding polynomial extra tape modifications searching prefix domc longest prefix examined far kept extra tape output written output tape prefix domc found written extra tape code appended output tape takes time lemma let polynomial larger certain polynomial degree generated contr recontr evrcc reexpand expand proof consequence lemma contr evrcc reexpand expand contr evrcc reexpand expand certain polynomial degree generating set indeed generate still need show generators belong functions contr recontr reexpand expand balance complexity lemma let verify evrcc balance complexity definition evrcc code code code code evrcc balance since output length input length computed polynomial evrcc code code computed time constant see proof prop since degree evrcc complexity thus exists degree generators belong theorem polynomial constant finitely generated proof proof similar proof theorem saw lemma generated infinite set contr recontr evrcc reexpand expand let assume contradiction finitely generated finite generating set extracted infinite generating set generated contr recontr evrcc reexpand expand finite set every let word expresses finite sequence generators proof identical proof theorem use fact domc code language program linear complexity computable mealy machine belongs complexity consequences generation hierarchy separation proposition let polynomial form set hence monoid contained finitely generated submonoid proof let evrc simulate directly without need padding unpadding domc evrc contained submonoid generated evrc compare lemma proof prop proof prop yields following chain submonoids generated finitely generated submonoids alternate corollary let sequence polynomials large enough evrc polynomial contains strict inclusion chain infinite upward direction evrc irmp proof strictness inclusions chain follows fact generation finite generation alternate theorem let polynomial form submonoid following properties contains elements arbitrarily high polynomial balance evrc theorem moreover evrc balance let polynomials form suppose also large enough evrc proof since contained finitely generated submonoid prop contained finitely generated submonoid inequality follows consider function indeed turing machine input read word times time turning input new letter time writing output tape produces output one copy made followed takes time composition instances complexity since output length high time must least much thus functions grows unbounded complexity degree coefficient contains functions arbitrarily high polynomial balance whereas contains functions balance complexity prop contained submonoid generated evrc easily see hence evrc belonged monoid would finitely generated contradicting theorem input balance evrc see lemma proved follows evrc otherwise would evrq follows since evrc evrq corollary strict complexity hierarchy submonoids exists infinite sequence polynomials form nki following holds moreover finitely generated union submonoids proof first statements follow theorem prop last statement follows cor since contains functions arbitrarily high polynomial complexity theorem monoids form strict complexity hierarchy new sort different usual complexity hierarchies fact could shown diagonal argument clear whether classical separation techniques complexity theory would show results theorem remark monoid finitely generated contain infinite strict complexity hierarchy monoids contain hierarchies sets indeed general fact finitely generated monoid contain infinite strict submonoids whose union indeed chain would exist contains finite set generators since contradicting strict hierarchy fact hold chains arbitrary order types holds limit ordinals generated monoid contains encoding fpc submonoid see section fpc finitely generated isomorphic copy fpc contains isomorphic copy leads strict chains submonoids irreducible functions another consequence generation irreducible elements elements expressed composition elements make precise next definitions subsection use evaluation maps use polynomials drop requirement integers allow real numbers definition inf complexity degree inf polynomial form also define inf complexity coefficient inf inf polynomial form ndf inf complexity polynomial polynomial given ndf since defined infimum might following definition inf proposition polynomial hand every polynomial ndf ndf definition let choose function called iff polynomial ndf called iff words iff composite elements ndf factored functions strictly lower complexity regarding degree coefficient note definition used def proposition polynomials exist functions proof contradiction assume exist every factored inf degree dfi inf coefficient cfi contradiction assumption among factors factored elements degree coefficient lower amount respectively factor factored dfi dfi cfi cfi hence dfi cfi repeating process keep reducing degree coefficient least respectively step finite number steps obtain factorization functions contradicting assumption remark finitely generated monoid like contain irreducible functions arbitrarily large complexity indeed elements expressible composite elements bounded complexity namely maximum complexity finitely many generators acknowledgement paper benefitted referee thoughtful reading advice references berstel perrin theory codes academic press birget semigroups functions international algebra computation birget morphisms congruences http birget monoid generalizations richard thompson groups pure applied algebra birget circuits groups richard thompson international algebra computation birget groups richard thompson complexity international algebra computation cannon floyd parry introductory notes richard thompson groups enseignement diffie hellman new directions cryptography ieee trans information theory theory computational complexity wiley goldreich foundations cryptography basic tools cambridge hemaspaandra ogihara complexity theory companion springer higman finitely presented infinite simple groups notes pure mathematics australian national university canberra levin tale functions problemy peredatshi informatsii mckenzie thompson elementary construction unsolvable word problems group theory word problems boone cannonito lyndon editors papadimitriou computational complexity richard thompson embeddings finitely generated simple groups preserve word problem word problems adian boone higman editors
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simple problems simplicial gluing structure pareto sets pareto fronts apr naoki hamada fujitsu laboratories kamikodanaka kawasaki japan abstract related studies fields quite studies applications optimization reported pareto sets pareto fronts form topological simplex class problems recently named simple problems pareto set pareto front observed gluing structure similar faces simplex paper gives theoretical observation proving gluing structure pareto subproblems simple problem simplicity standard benchmark problems studied theories concerning easiness covering solutions developed several well emo community studies topological properties solution sets contractibility earliest work found koopmans assertion applied linear programming economics pointed conditions making pareto front contractible peleg generalized result showed pareto front contractible feasible objective region convex set afterward study spread operations ccs concepts research closedness arcwise connectedness computing optimization decisioncontractibility pareto general settings making computing evolutionary algostudied linear programming quasi convex prorithms nonconvex optimization geometric topology gramming results collected luc section recently similar results obtained general probkeywords lem classes lexicographic quasiconvexity optimization continuous optimization problem class arcwise arcwise connectedness pareto set necessary condition homotopy method covers solutions acm reference format naoki hamada simple problems proceedings gecco companion berlin germany july pages doi http introduction motivation success evolutionary optimization emo widely spreading various academic industrial recent numerical studies showed emo algorithms awa ability approximate entire pareto set pareto front problems contrast abundance experimental successes theory shedding light work still developing especially problem class emo algorithms cover entire pareto understood paper discusses problem class solutions scalarization permission make digital hard copies part work personal classroom use granted without fee provided copies made distributed commercial advantage copies bear notice full citation page copyrights components work must honored uses contact gecco companion berlin germany copyright held doi http decomposition decomposition approach considers given problem also subproblems optimizing subset objective functions studies relation among solutions lowe showed weak pareto set convex programming union pareto sets subproblems malivert extended result explicitly quasiconvex upper semicontinuous functions popovici named property pareto reducible gave condition independent convexity ward showed strictly pareto solutions convex programming problem completely surrounded pareto solutions subproblems recent studies revealed pareto reducibility lexicographic quasiconvex programming problem closely related contractibility simply shadiness pareto front stratification pure mathematics singularity theory maps gives decomposition solutions smale applied theory economic problem stated pareto set pure exchange economy agents homeomorphic provided quasiconvexity monotonicity agents utility functions lovison pointed face simplex corresponds pareto set subproblem optimizing subset objective functions smale sequels discussed pareto critical points generic maps gecco companion july berlin germany hamada transversality rank assumption derivatives melo showed whose pareto critical points admit generic form dense subset space whitney topology recently lovison collected related works approach developed melo result showed local pareto sets proper maps admit whitney approach attempts going two courses analysis originated koopmans global analysis smale former seems much restrictive global optimization nature emo algorithms latter approach general currently hard compute need handy theory understanding behavior emo algorithms recently hamada class problems called simple problem pointed without rigorous proofs pareto set pareto front simple problem homeomorphic simplex faces simplex correspond pareto sets images subproblems also discussed property closely related scalarization paper gives rigorous proofs arguments contribution evaluation map depending context call empty set problem evaluation map empty map set problems sets say subproblem superproblem call set subproblems problem decomposition denote solutions problem satisfy conditions say denote denote solution problem called pareto solution set pareto solutions problem called pareto set denoted image map denoted especially image called pareto front paper abbreviate composition notation considered regarding map paper give proof boundary pareto set resp pareto front simple problem union interior pareto sets resp images subproblems property enables numerically compute pareto additionally investigate simplicity benchmark problems emo community problems zdt suite dtlz suite problems wfg suite simple restrictive situation med problem always simple simple problem first present simple problem graphical intuition contents rest paper organized follows section prepares basic notions notations used subsequent sections section gives properties solutions simple problems relation scalarization section discusses simplicity existing benchmark problems section gives conclusions remarks future work preliminaries paper considers following problem minimize call evaluation map objective function variable space feasible region solution objective space evaluation value make various problems removing objective functions discuss gluing structure solutions write arguments clearly problem set objective functions regard notation abuse denote problem equation section introduces simple problem shows solution structure section presents simple problem section shows inclusion properties solutions among subproblems section shows solutions gluing structure topological simplex section points gluing structure enables emo algorithms cover pareto set pareto front simple problem problem simple simplicity every subproblem following conditions embedding standard denote topological spaces homeomorphic continuous maps ida idb maps called homeomorphisms topology induced variable space open set written open set euclidean topology similarly spaces discussed paper implicitly equipped induced topologies either variable space objective space denote restriction map set composite inclusion map embedding simple problems gecco companion july berlin germany minimize rgb max min figure simple problem subproblem pareto set pareto sets colored converting rgb using equations cell homeomorphic point map set always special implication finally problem shown fig corresponds let cultivate intuitive understanding case objective function required formality fig considering simple problem also deal note conditions impose problem subproblems optimizing subsets given objective functions structure within pareto set admit arbitrary structure subproblem original problem optimizing pareto set contrastive conventional probthe objectives example lem classes programming probcondition imposes pareto set homeolems regulate structures entire domain morphic surface created bending stretching trisimplicity independent problem classes fact evangle without cutting connecting condition guarery class programming problems conantees restricted evaluation map tains simple problems problems ref inverse map sult simplicity characterizes new aspect easiness solve bijective continuous thus homeomorphisms implies inclusion properties pareto front also homeomorphic every point continuously corresponds section shows inclusion relations pareto sets unique solution vice versa next remove images subproblems simple problem isolate conone objectives resulting three subproblems sequence assuming simplicity general properties pareto sets homeomorphic begin without assumption curve without loops pareto fronts proposition problem possibly also curve point continuously corresponds subproblem following relations hold unique pareto solution removing objective get three subproblems pareto sets composite homeomorphism inclusion map gecco companion july berlin germany hamada weak pareto set set points satisfying proof inclusion fact see example miettinen section second relation directly follows simplicity conditions imposed subproblems well given problem means subproblems inherit simplicity superproblem proposition problem simple subproblem simple proof thus problems satisfy also implies simple however hold general example consider problem minimize clearly pareto sets implies contrast simple problems solutions proposition problem simple proof proposition prove contradiction suppose point exists since weakly condition holds hand since condition holds point satisfy means thus contradicts injection therefore simplicity ensures inclusion relationship pareto sets proposition simple problem subproblem holds proof combine proposition proposition therefore propositions hold simple problem also hold subproblems example actual assertion proposition subproblem well weakly point interpretation proposition bit complicated given simple problem assertion holds pair problems henceforth repeat property always valid propositions involve simple problems similarly empty set subset every set problem subproblem every problem therefore problems exist simple exist simple problem let check proposition exists problem unique simple proof existence uniqueness follow empty set empty map let check simplicity since subproblem show since problem show holds indeed restricted evaluation map evaluation map since decomposed homeomorphism inclusion map restriction embedding using fact see topology image pareto set although condition addresses topology pareto set combined show pareto front image superproblem topology subproblem pareto set image next section investigates solutions nicely glued together proposition simple problem subproblem goal section give proof solutions simple problem special gluing structure shown figs structure analogy faces simplex triangle spanned vertices boundary union three edges edge boundary consisting two points boundary vertex empty set expand relations using int see boundary simplex expressed disjoint union proof evident show nontrivial part show restriction embedding remember restriction embedding proposition general restriction embedding embedding gluing properties simple problems gecco companion july berlin germany open faces int int int int int int generally pareto sets images may complex topological structure however simple problem seen proposition form topological manifolds boundary hereafter simply call manifolds homeomorphic simplex point open neighborhood homeomorphic called interior point set interior points called interior denoted int points boundary points open neighborhood homeomorphic set boundary points called boundary denoted simple problem similar relation holds among pareto sets shown fig int int int int int int figure simple problem gluing structure pareto sets subproblems boundary pareto set subproblem consists pareto sets subproblems relation holds images shown fig int int int int int int known gluing structure solutions commonly appears facility location problems studied long time see example references therein also seen applications exploited heuristic practitioners start show arbitrary number objectives first let see basic properties pareto front hold class problems lemma possibly problem whose pareto front forms projection restricted int embedding proof generally projection restriction open set continuous open injective continuous open map embedding thus show injective restricted int injective int contains two points coordinates except value means one point another contradicting pareto front lemma asserts interior pareto front keeping topology ordering shown fig projection induces coordinate neighborhood point int points holds property key investigate interaction topology interior pareto front figure simple problem gluing structure pareto set images subproblems although shape fig topology lemma consider possibly problem whose pareto front forms following statement holds int neighborhood point dominated proof lemma projection restricted embedding thus neighborhood point take centered write vertices small positive number signs run possible combinations among vertices let one coordinates related related thus holds especially implies implies otherwise means contradicting repeating argument complete proof gecco companion july berlin germany figure possibly problem pareto front projections projection restricted int injective thus embedding fact interior point neighborhood mapped interior point neighborhood pareto front lemmas extend pareto set image subproblem answer general problems projection embedding int example consider problem example minimize pareto set subproblem image interior int projected int implies injective thus embedding furthermore since see set weakly existence weak pareto optima disrupts injectivity contrary problem simple solutions weakly enable extend lemmas image corollary consider simple problem subproblem restriction projection int embedding proof injective int contains two points coordinates except value means weak pareto solution nonpareto contradicting proposition corollary consider simple problem subproblem int following statement holds neighborhood exists point hamada proof chose objective function let remaining set corollary mapped homeomorphically next chose another let remainder corollary mapped homeomorphically though repeated application projections long embedding original pareto set image mapped let composite used projections generally composite embeddings embedding thus point int neighborhood mapped homeomorphically point int neighborhood proposition asserts together lemma completes proof property see simplicity ensures pareto set images subproblems located boundary superproblems lemma simple problem subproblems following relation holds proof suppose exists point interior point int holds thus corollary neighborhood point contradicts next question whether similar relation holds pareto set mapped check need following lemma lemma simple problem subproblem map commutes boundary interior int int proof generally embedding maps boundary boundary interior interior embedding thus holds int int using fact show simplicity pareto sets subproblems relation images corollary simple problem proper subproblem following relation holds proof since embedding inverse map lemma converts lemma assertion follows last key main theorem sphere embedding lemma every embedding surjective thus homeomorphism simple problems gecco companion july berlin germany proof suppose surjective exists point stereographic projection north pole denoted generally stereographic projection embedding composite embeddings embedding therefore embedding contradicts fact embedded remark keep proof elementary assumed known derived surjective alternative proof deriving exact sequence fashion consult hatcher two paragraphs proof proposition show goal section theorem simple problem subproblem holds int int proof first show proven int int int holds lemma holds inverse case thus consider case inclusion relation assume exists point int int let neighborhood exists point hold implying contradicts exist dimension must equal contradicts dim consequently exist since lemma states second third lines used general property map relation scalarization equations together gluing structure pareto sets images subproblems simple problem structure induces natural pareto set resp pareto front stratum interior pareto set resp image subproblem therefore numerically compute solving subproblem points spreading strata good covering pareto see structure enables emo algorithms cover pareto consider weighted scalarization minimize max get converting lamma int int int int int int thus inclusion map int generally inclusion map embedding combining lemma therefore int union manifolds homeomorphic simplex glued faces simplex fact sures int contrary implies lemma ensures surjective inclusion map surjective implying int holds weight chosen utopian point inf let standard base whose coordinate one coordinates zero standard rewritten using notation correspondence fact optima written choice arbitrary number indices problem simple proposition extends corollary therefore weight face gives boundary point stratum corresponding indices unfortunately well existing scalarization methods including weighted sum augmented chebyshevnorm pbi ipbi give correspondence interiors int int nevertheless boundary points stratum obtained new weights corresponding interior points stratum interpolating weights used boundary points thus grid arrangement generation weights practically often hit interior points smoothness determined evaluation map gecco companion july berlin germany simplicity benchmarks section investigates simplicity benchmark problems emo community zdt suite dtlz suite wfg suite med problem hamada proof every problem objective function ignoring variables follows zdt suite zdt suite six problems named decision variables split position variables distance variables problems following format minimize therefore furthermore huband table vii shows injective disconnected pareto front another evidence wfg suite wfg suite contains nine problems form minimize otherwise users make problems changing placeholder functions concrete see zitzler general formulas enough show problems theorem independent choice variable dimension dimension additionally suite create simple problems matter unless domains proof first exclude following analysis since problem clearly function depends single variable variables take arbitrary value optima thus means contradicts simplicity condition consequently reason suite simple depends generally problem objective function independent variables pareto set extends higher dimensions usual contradicting simplicity condition existence unused variables quick test dtlz suite dtlz suite consists nine problems named decision variables split position variables distance variables zdt number objectives set arbitrarily see deb theorem independent choice variable dimension objective dimension positionvariable dimension functions placeholders suite variables mapped transformation functions position variables distance variable passed objective functions reason called variables distancerelated variables respectively concrete see huband theorem always simple dimension variables one set number objectives huband table xiv shows problems require mod proof first consider huband table xiv shows disconnected pareto front front homeomorphic contradicts property simple problem shown proposition following discussion treats rest problems let check properties pareto set transformed variable space map described huband properties shape functions shown huband table embedding upon opposed huband actually pareto set conditioned avoid complication caused except disconnected used simple problems gecco companion july berlin germany med holds thus injective since transformation functions surjective described huband composite evaluation map single problem med follows minimize injective pareto set untransformed variable space meet simplicity condition case transformation function involves reduction weighted sum reduction decrease dimension examining huband table see functions injective therefore maps two points pareto solution contradicting simplicity condition remains case problems may simple first let consider subproblems huband table implies simplicity condition problems equivalent following criterion point since automatically follows problem criterion necessary condition simplicity see huband table always point depends case whether point introduce shift deceptive shift making point thus problems becomes point subproblems simple next let consider subproblems holds embedding point therefore equivalent condition simplicity follows embedding transformation function form thus inverse embedding checked simple note huband table xiv describes degenerate pareto front seems evidence always however degeneracy actually occurs analysis pareto front forms line segment disrupt simplicity parameters determine convexity pareto front parameters well variable dimension number objectives theorem med always simple independent choice parameters additionally changing individual optima break simplicity long independent proof first consider case corresponds facility location problem pareto set thus problem known convex hull holds independent indeed convex hull spanned problem independent thus pareto set ensures problem simplicity condition analyzing gradient see map embedding argument applies subproblems satisfy problem simple case considered composite case power since positive power homeomorphism composition preserves simplicity facility location problem conclusions paper discussed simple problem showed pareto sets subproblems resp images constitute pareto set resp pareto front topological property gives theoretical guarantee decompositionbased emo algorithms obtain entire approximation pareto set well pareto front also investigated simplicity benchmark problems emo community problems zdt dtlz suites wfg suite contains simple problems restrictive situation usually whereas med problem always simple believe absence simple problems standard benchmark suites considerable gap benchmark since many evidences large portion nowadays applications seems simple additionally applications involving simulations would important develop estimation method simplicity problems set approximate solutions gecco companion july berlin germany references benoist contractibility frontier simply shaded sets journal global optimization doi http melo structure pareto set generic mappings boletim sociedade brasileira bulletin brazilian mathematical society doi http deb jain evolutionary optimization algorithm using nondominated sorting approach part solving problems box constraints ieee transactions evolutionary computation aug doi http deb thiele laumanns zitzler scalable test problems evolutionary multiobjective optimization evolutionary multiobjective optimization abraham jain goldberg springer london doi http everson walker fieldsend edges mutually nondominating sets proceedings annual conference genetic evolutionary computation gecco acm new york usa doi http hamada nagata kobayashi ono adaptive weighted aggregation multiobjective function optimization framework taking account spread evenness approximate solutions proceedings ieee congress evolutionary computation cec hamada nagata kobayashi ono adaptive weighted aggregation scalable awa multiobjective function optimization proceedings ieee congress evolutionary computation cec hamada nagata kobayashi ono scalability adaptive weighted aggregation multiobjective function optimization proceedings ieee congress evolutionary computation cec hatcher algebraic topology cambridge university press cambridge new york http huband hingston barone review multiobjective test problems scalable test problem toolkit ieee transactions evolutionary computation doi http koopmans analysis production combination activities activity analysis production allocation proceedings conference cowles commission monograph koopmans john wiley sons new york torre popovici arcwise multicriteria optimization operations research letters doi http lovison pecci hierarchical pareto sets arxiv july arxiv http lowe thisse ward wendell solutions multiple objective mathematical programs management science doi http luc theory vector optimization lecture notes economics mathematical systems vol malivert boissard structure sets strictly quasi convex objectives journal convex analysis miettinen nonlinear multiobjective optimization international series operations research management science vol gmbh peleg topological properties point set proc amer math soc http popovici pareto reducible multicriteria optimization problems optimization doi http popovici structure sets lexicographic quasiconvex multicriteria optimization operations research letters doi http popovici explicitly quasiconvex optimization journal global optimization doi http puerto geometrical description weakly solution set multicriteria location problems annals operations research doi http sato inverted pbi impact search performance multi optimization proceedings annual conference genetic evolutionary computation gecco acm new york usa doi http shioda ono adaptive weighted aggregation enhanced relocation performance evaluation transaction japanese society evolutionary computation doi hamada http shioda ono adaptive weighted aggregation step size control weight adaptation multiobjective continuous function optimization sice journal control measurement system integration doi http smale global analysis economics pareto optimum generalization morse theory dynamical systems peixoto academic press doi http smale global analysis economics iia extension theorem debreu journal mathematical economics doi http smale global analysis economics iii pareto optima price equilibria journal mathematical economics doi http smale global analysis economics finiteness stability equilibria general consumption sets production journal mathematical economics doi http smale global analysis economics pareto theory constraints journal mathematical economics doi http smale global analysis economics geometric analysis pareto optima price equilibria classical hypotheses journal mathematical economics doi http ward structure sets convex objectives mathematics operations research zhang multiobjective evolutionary algorithm based decomposition ieee transactions evolutionary computation december doi http zitzler deb thiele comparison multiobjective evolutionary algorithms empirical results ieee transaction evolutionary computation doi http
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distributed spanner approximation keren michal feb february abstract address fundamental network design problem constructing approximate minimum spanners contributions distributed setting providing algorithmic hardness results main hardness result shows minimum directed kspanner requires log rounds using deterministic algorithms log rounds using randomized ones congest model distributed computing combined algorithm local model barenboim elkin gavoille well algorithm local model show lower bounds congest model imply strict separation local congest models notably best knowledge first separation models local approximation problem similarly separation directed undirected cases implied also prove minimum weighted problem requires number rounds congest model directed undirected graphs addition show lower bounds minimum weighted problem congest local models algorithmic side apart aforementioned algorithm minimum main contribution new distributed construction minimum uses polynomial local computations algorithm guaranteed approximation ratio log graph vertices edges matches best known ratio polynomial time sequential algorithms kortsarz peleg tight restrict polynomial local computations algorithm approximation factor previously known distributed setting number rounds required algorithm log log maximum degree graph approach allows extend algorithm work also directed weighted variants problem also provides congest algorithm minimum dominating set problem guaranteed log approximation ratio technion department computer science ckeren smichald supported part israel science foundation grant introduction graph sparse subgraph preserves distances multiplicative factor first introduced late spanners central numerous applications synchronization compact routing tables distance oracles approximate shortest paths due prominence spanners many distributed applications vital distributed algorithms constructing indeed many efficient distributed algorithms finding sparse spanners undirected graphs give global guarantee size spanner prime example algorithms construct edges graph vertices optimal worst case assuming girth conjecture opposed finding spanners best sparsity paper focuses network design problem approximating minimum fundamental optimization problem particularly crucial cases sparsity complete bipartite graphs directed spanners spanner approximation heart rich line recent work sequential setting presenting approximation algorithms well hardness approximation results distributed spanner approximation algorithms known date distributed algorithm expected approximation ratio log minimum problem given recently extended achieving approximation ratio directed matches best approximation known sequential setting yet distributed setting possible obtain better approximations local computation polynomially bounded constant time algorithm directed undirected minimum takes exp rounds constant positive integer given addition show polylogarithmic time algorithm problems following framework recent algorithm covering problems see section approximation much better best approximation acheived sequential setting due hardness results algorithms work classic local model distributed computing vertices exchange messages unbounded size synchronous rounds natural question whether obtain good approximations efficiently also congest model messages exchanged bounded log bits undirected case efficient constructions edges congest model imply since spanner connected graph least edges however directed graphs efficient algorithms congest model contribution paper twofold provide first hardness approximation results minimum distributed setting main hardness result shows efficient approximation algorithms directed problem congest model explains current approximation algorithms problem require large messages also creates strict separation directed undirected variants problem latter admits efficient approximations congest model addition provide new distributed algorithms approximating minimum problem several variants local model main algorithmic contributaion algorithm minimum uses polynomial local computations guarantees approximation ratio log matches best known approximation polynomial sequential algorithms way obtaining results develop new techniques algorithmically obtaining lower bounds potentially find use studying various related problems contributions hardness approximation show several negative results implying hardness approximating various spanner problems local congest models many recent hardness approximation results spanner problems sequential setting best knowledge first distributed setting directed congest model perhaps main negative result proof hardness approximating directed problem congest model theorem perhaps randomized distributed algorithm congest model directed problem takes rounds restricting attention deterministic algorithms prove stronger lower bound constant example gives constant polylogarithmic approximation ratio directed rounds using randomized algorithms problem congest model requires rounds using deterministic algorithms even approximation ratio hard rounds using randomized algorithms rounds using deterministic requires ones moreover deterministic case even approximation ratio rounds contrasted approximation appropriate values requires obtained without communication taking entire graph since least edges local congest major implication strict separation local congest models since former admits algorithm polylogarithmic algorithm see section directed separation previously known global problems problems subject lower bound diameter graph local decision problems determining whether graph contains best knowledge first separation local approximation problem directed undirected lower bound also separates undirected directed kspanner problems since efficient algorithms congest model constructing edges imply best randomized algorithm task takes rounds best deterministic algorithm recent algorithm takes rounds constant even achieving rounds using randomization approximation directed graphs necessitates rounds using deterministic algorithms weighted congest model addition main result consider weighted show weighted rounds undirected problem requires rounds needed weighted directed problem constant time randomized algorithm directed presented however deterministic network decomposition presented gives polylogarithmic deterministic approximation directed well shows separation also deterministic case weighted unweighted lower bounds hold also randomized algorithms obtain yet another separation weighted unweighted variants problem since aforementioned constructions imply approximation unweighted case local congest since algorithm approximating kspanners within factor algorithm give section suitable weighted case hardness result weighted case implies separation local congest models also weights holds also undirected weighted case iii weighted local congest models finally show lower bounds weighted problem nutshell obtained reduction captures intuition approximating minimum weight least hard approximating minimum vertex cover mvc emphasize reduction set cover problem unweighted problem given inherently sequential requiring addition vertex connected vertices graph hence unsuitable distributed setting reduction implies logloglogn rounds required logarithmic approximation ratio weighted local model plugging lower bounds lower bound exact solution mvc given addition reduction implies weighted congest model using lower bound exact mvc given recently tight logarithmic factors since rounds allow learning entire graph topology solving essentially natural graph problems distributed approximation algorithms show new distributed algorithms approximating minimum main algorithmic contribution new algorithm minimum problem uses polynomial local computations see section addition show local computation polynomially bounded possible achieve minimum see section distributed minimum section present algorithms spanner problems following framework recent algorithm covering problems show following theorem randomized algorithm complexity poly log local model computes minimum constant algorithm quite general adapted similarly additional variants theorem shows although spanner problems hard approximate sequential setting possible achieve extremely strong approximations efficiently local model demonstrates power local model however algorithm based learning neighborhoods polylogarithmic size solving problems finding optimal spanners desirable design also algorithms work realistic assumptions next focus problem show new algorithm uses polynomial local computations uses power local model learning neighborhoods diameter distributed approximation minimum restrict polynomial local computations best algorithm minimum problem log log expectation dinitz krauthgamer solves even general problem finding spanners however still leaves several open questions regarding minimum first best approximation problem sequential setting log number edges graph achieve approximation also distributed setting second approximation ratio holds expectation design algorithm guarantees approximation ratio third algorithm requires learning neighborhoods logarithmic radius hence direct implementation congest model efficient design efficient algorithm congest model design new algorithm minimum problem answering questions algorithm obtains approximation ratio log always within log log rounds maximum vertex degree summarized follows theorem distributed algorithm minimum problem local model guarantees approximation ratio log takes log log rounds approximation ratio log matches best approximation sequential setting constant factor tight restrict polynomial local computations addition approximation ratio algorithm guaranteed rather holding expectation crucial distributed setting since opposed sequential setting running algorithm several times choosing best solution completely blows complexity learning cost solution requires collecting global information note although algorithm converted algorithm guaranteed polylogarithmic time complexity approximation ratio holds expectation opposite hold another feature algorithm uses power local model learning vertices direct implementation algorithm congest model yields overhead rounds efficient small values address issue section iii distributed approximation additional techniques develop constructing analyzing spanner advantage allowing easily extend construction directed weighted variants problem obtain approximation ratio directed case undirected case weighted case give approximation ratio log improving upon log approximation expectation case best knowledge first distributed approximation obtain approximation ratio matches sequential algorithm distributed approximation mds finally technique also gives efficient algorithm minimum dominating set mds problem obtains approximation ratio log always algorithm mds works even congest model takes log log rounds mds problem studied extensively distributed time complexity rounds claimed however algorithm based sampling certain decomposition log times independently takes log rounds time independence decompositions computations parallelized local model achieving time complexity log rounds see also standard setting high probability refers probability least constant computing community several efficient algorithms mds congest obtaining approximation ratio log expectation best knowledge algorithm first guarantees approximation ratio always technical overview hardness approximation prove theorem reduction communication problems proven fruitful various lower bounds congest model principle family graphs constructed depending input strings two players solution required congest problem uniquely determines whether input strings players satisfy certain boolean predicate common usage although communication problems used well two players simulate distributed algorithm solving congest problem deduce output communication problem accordingly yields lower bound congest problem based known lower bounds communication complexity problem incorporating cost simulation prime caveat using framework approximation problems examples modification single input bit slight influence graph example showing lower bound computing diameter bit input affects distance one pair vertices sufficient computing global property graph indeed distance single pair vertices change diameter graph challenge designing construction approximating single bit needs affect drastically size minimum detail least edges hence meaningful lower bound input bit must affect least edges manage overcome challenge constructing graph captures requirement allows reduction main technical ingredient dense component many edges affected single input bits component resides entirety within set vertices simulated single player two thus resulting graph construction crucial proof otherwise density component would imply dense cut two sets vertices simulated players turn would nullify achievable lower bound property believe construction may give rise lower bound constructions additional local approximation problems graph construction designed using several parameters allows show time complexity algorithm approximation ratio gives lower bounds even large values stronger lower bounds deterministic case obtained using gapdisjointness problem rather common problem since allows slack obtain stronger lower bounds price holding deterministic algorithms believe flexibility problem may useful showing additional strong lower bounds approximation problems stronger lower bounds weighted case obtained assigning weights edges graph manner allows shave certain edges affect bound distributed approximation minimum algorithm approximating minimum inspired sequential greedy algorithm kortsarz peleg dense stars added spanner one one obtaining approximation ratio log star subset edges vertex neighbors density star ratio number edges star size star edge star includes path length two roughly intuition greedy algorithm dense star adding edges spanner allows many edges adding small number edges spanner direct implementation greedy approach distributed setting highly expensive since deciding upon densest star inherently requires collecting global information moreover one would like leverage ability distributed setting add multiple stars spanner simultaneously address sources inefficiency rather computing star densest entire graph compute stars densest local greatly speeding running time adding locally densest stars spanner extreme results poor approximation ratio instead consider stars candidates added spanner key challenge break symmetry among candidates balancing need choose many stars parallel fast running time need bound overlap spanned edges among candidates small approximation ratio tackle conflict constructing voting scheme breaking symmetry choosing among stars based random permutation interestingly approach inspired parallel algorithm set cover let edge vote first candidate according random permutation candidate receives number votes least edges added spanner continue process iteratively since add spanner stars receiving many votes approach guarantees much overlap edges different stars eventually culminates proof approximation ratio log matches one obtained greedy approach tricky obstacle lies showing algorithm completes log log rounds opposed set cover case may many different stars centered vertex vertex may required add candidate stars multiple times execution algorithm turns arbitrary choice candidate among densest stars centered vertex incapable providing efficient time complexity overcome issue design subtle mechanism proposing candidate star pair proof algorithm indeed completes claimed number rounds discussion results paper significantly advance distributed approximation minimum intriguing questions remain open first landscape approximation ratio running time distributed minimum algorithms yet fully mapped example log factor running time approximation algorithm weighted tight log log factor due reduction mvc known lower bounds however remains open whether log factor necessary additional gaps remain open various approximation ratios particular interesting question show lower bound approximating undirected unweighted minimum problem curious question whether algorithm efficiently made work congest model direct implementation would yield overhead running time computing densities stars sending candidate stars emphasize knowing density neighborhood vertices crucial additional algorithms algorithm harris another interesting question design efficient deterministic algorithm achieving approximation ratio larger values stretch lower bounds imply strict separation local congest models number rounds required approximating directed minimum separation previously known global problems problems subject lower bound diameter graph local decision problems determining whether graph contains interestingly first separation local approximation problem central open question whether separations hold also local symmetry breaking problems interestingly algorithm well distributed approximation algorithms minimum local model work also directed graphs achieving approximation ratio round complexity however hardness results create strict separation undirected directed variants congest model interesting show separations problems additional related work spanners studied extensively distributed setting producing many efficient algorithms finding sparse spanners undirected graphs algorithms construct edges fixed fastest completing rounds tight many additional works construct various spanners distributed setting excellent overview within many recent studies address spanner approximations sequential setting greedy algorithm achieves approximation ratio log minimum problem extended weighted directed cases approximation algorithms directed problem given best approximation ratio log approximation ratio approximation ratios matched recent distributed log algorithm uses polynomial local computations approximation algorithms given also pairwise spanners distance preservers spanners lowest maximum degree spanners hardness approximation results sequential setting give polynomial algorithm gives approximation ratio better log shows sequential greedy algorithm optimal problem even harder constant algorithms approximate problem within factor better log directed problem within factor better log similar results known additional variants spanner problems closely related covering problems set cover minimum dominating set mds minimum vertex cover indeed ingredients algorithms borrow ideas distributed parallel algorithms problems symmetry breaking scheme inspired parallel algorithm set cover rajagopalan vazirani however general structure algorithm requires global coordination hence suitable distributed setting also several ideas inspired distributed mds algorithm jia rounding densities comparing densesties however breaks symmetry candidates different way results approximation ratio log expectation connection spanners set cover used also show covering edges graph stars also useful approximating directed problem context also mention distributed algorithm minimum connected dominating set problem also uses stars main component construction work however incomparable especially since minimum connected dominating set problem global problem admitting lower bound even local model preliminaries let connected undirected graph vertices maximum degree let subset edges let say edge covered path length subgraph covers edges subgraph subgraph covers edges directed graph say directed edge covered subset edges includes directed path length define directed graph accordingly minimum problem input connected undirected graph goal find minimum size directed problem defined accordingly respect directed graphs weighted problem edge weight goal find minimum cost cost spanner problem introduced input connected undirected graph edges divided two types clients servers may edges goal find minimum size includes edges distributed setting input problem communication graph vertex initially knows identities neighbors needs output subset edges union outputs communication network bidirectional even solving directed problem roadmap section present hardness approximation results directed weighted congest model section provide hardness approximation results weighted section present algorithm minimum problem show extensions variants section describe mds algorithm finally section show minimum hardness approximation congest model section prove hardness approximation results approximating congest model explained section build upon previous used framework reducing communication problems distributed problems congest model key technical challenge overcome plant dense subgraph construction another variant weighted problem weight edge represents length emphasize case edges length without inducing large cut vertices simulated two players still choice edges taken dense subgraph spanner depend inputs describe graph construction allows provide reduction problems communication latter setting two players alice bob receive input strings respectively size goal solve problem related inputs communicating minimum number bits example set disjointness problem requires players decide input strings represent disjoint subsets need decide bit communication complexity set disjointness known linear length strings lemma solving set disjointness problem input strings size requires exchanging bits even using randomized protocols start showing approximating directed problem congest model hard modify construction provide hardness results weighted case general approach build dense graph edges depend inputs alice bob inputs alice bob disjoint sparse also otherwise many edges simulating distributed approximation algorithm problem alice bob solve set disjointness hence depending parameters graph construction communication lower bound latter would imply lower bound number rounds required former reduction set disjointness used order show lower bound computing diameter graph main idea bit inputs affects distance two vertices graph distance pairs vertices long affects diameter graph idea useful also showing lower bound spanner problems indeed one elements construction similar constructions however main difference case distance one pair vertices graph affect significantly size minimum spanner order overcome suggest following construction graph consists two subgraphs one depends inputs one complete bipartite graph sides divided blocks size connect two subgraphs way bit inputs affects edges must added spanner let positive integers construct graph according parameters later different values order obtain several graph directed graph xij yij see figure illustration set edges consists matching includes directed edges addition complete bipartite graph vertices includes directed edges xij yrs vertex xij edge xij vertex yij edge yij addition graph includes edges addition two input strings length bits denoted aij bij affect following way edge aij edge bij figure graph edges omitted clarity red dashed edges examples optional edges depend input strings note number vertices consists edges recall goal constructing sparse since dense subgraph taking edges spanner would expensive however order avoid taking edges spanner spanner must include directed path length every pair vertices xij yrs include edges existence path depends input strings following way claim one edges directed path length vertices xij yrs contain edges otherwise directed path xij yrs path consists edge xij yrs proof note directed path xij yrs include edges must begin edge xij must end two edges yrs hence existence path depends whether directed path show directed path length least one edges otherwise directed path length let directed path path must cross cut either edge edge since path length cross cut edge edge form however reachable crosses edge way reach edge second case edge must first edge path conclusion one edges directed path length vertices xij yrs contain edges otherwise directed path length case directed path xij yrs path consists edge xij yrs claim captures essence construction suitable approximation problem next use graph construction claim order show hardness results randomized directed section address problem show obtaining requires log rounds congest model even using randomized algorithms lemma let let let input strings disjoint size otherwise includes least edges proof input strings disjoint every pair indexes least one edges hence claim directed path length every two vertices xij yrs contain edges gives size edges taking edges since edges since also input strings disjoint pair indexes neither edges hence claim directed path vertices xij yrs except path includes edge xij yrs therefore need take edges xij yrs spanner values means adding edges spanner let let distributed algorithm minimum problem denote time complexity graph vertices approximation ratio algorithm may depend assume monotonic increasing function goal show used solve set disjointness lemma algorithm gives protocol set disjointness case show lower bound log time complexity stated following lemma lemma let threshold input strings disjoint optimal spanner edges otherwise spanner includes edges log proof use solve set disjointness input strings length following way let two input strings length given alice bob respectively take graph define since input strings affect edges vertices within within respectively holds alice knows edges adjacent vertices bob knows edges adjacent vertices cut consists edges edges matching edges alice bob simulate follows alice simulates vertices bob simulates vertices round alice bob exchange messages going cut either direction messages sent vertices vertices simulated locally alice bob without communication since size messages log bits size cut simulate one round exchanging log bits therefore simulate entire execution exchanging log bits end simulation alice knows edges taken spanner edges spanner alice concludes input strings disjoint otherwise concludes disjoint show produces correct output recall condition lemma input strings disjoint size optimal spanner otherwise therefore input strings disjoint since algorithm constructs spanner edges case alice indeed outputs input strings disjoint otherwise input strings disjoint size spanner edges case alice indeed outputs input strings disjoint hence alice bob solve set disjointness exchanging log bits however perhaps randomized protocol solves disjointness inputs size requires exchanging bits lemma gives log using lemma lemma prove following main theorem theorem perhaps randomized distributed algorithm congest model directed problem takes rounds proof show threshold distinguishes whether inputs disjoint using lemma get lower bound round complexity define following choice parameters let positive integer let let let ensures let requirement shows positive number vertices addition note since number vertices gives ncq let lemma inputs disjoint edges otherwise includes least edges definition holds gives since holds gives hence satisfies conditions lemma gives log since holds log theorem shows achieving constant polylogarithmic approximation ratio rounds even achieving directed problem congest model requires rounds approximation ratio hard requiring proves strict separation local congest models since constant round algorithm polylogarithmic algorithm see section directed local model also separates undirected directed problems since randomized algorithms congest model constructing edges algorithms obtain approximation ratio undirected minimum problem rounds achieving approximation directed rounds according theorem problem requires deterministic directed next show deterministic algorithm solving directed problem requires rounds trick allows stronger lower bound use different problem communication complexity refer gap disjointness problem problem also mentioned gap disjointness problem alice bob receive input strings respectively goal distinguish whether input strings disjoint far disjoint inputs far disjoint least indexes inputs neither disjoint far disjoint output alice bob valid gap disjointness problem easily solved randomized protocols exchanging bits however solving problem deterministically requires exchanging bits lemma solving gap disjointness problem deterministically input strings size requires exchanging bits proof lemma see example shown approximating size intersection requires exchanging bits proof relies showing distinguishing disjoint inputs inputs intersection bits difficult note inputs intersection size least hence exact proof shows solving gap disjointness requires exchanging bits using deterministic protocol order use set disjointness proof theorem necessary devise construction bit input affects many edges spanner order argue even one index players correctly decide whether inputs disjoint checking size spanner however use gap disjointness players need distinguish case inputs disjoint case far disjoint allows much flexibility gives stronger lower bounds deterministic case lemma let let let input strings disjoint size input strings far disjoint includes least edges proof input strings disjoint taking edges shown proof lemma edges since also input strings far disjoint least pairs none edges hence claim least pairs directed path vertices xij yrs except path consists edge xij yrs pair need take directed edges xij yrs spanner values means adding edges spanner summing pairs get must include least edges let let deterministic distributed algorithm minimum problem denote round complexity graph vertices following lemma adapts lemma gap disjointness problem proof proof lemma difference alice concludes input strings far disjoint constructed spanner edges also lower bound holds deterministic case since relies communication complexity gap disjointness lemma let threshold input strings disjoint optimal edges input strings far disjoint includes edges log using lemma lemma show following theorem deterministic distributed algorithm congest model directed problem takes rounds constant proof construct graph withp following choice parameters let positive integer let let let number vertices addition holds since number vertices gives order use lemma need verify note constant follows since constant choose get gives needed define lemma input strings disjoint size otherwise input strings far disjoint includes least edges choice since holds gives shows satisfies conditions lemma using lemma get log note shows theorem shows achieving constant polylogarithmic approximation ratio directed problem congest model requires rounds deterministic rounds algorithm addition even approximation ratio hard requiring notably even approximation ratio appropriate values hard rounds contrasted fact obtaining approximation requiring ratio requires communication since least edges theorem separates local congest models since deterministic network decomposition described gives deterministic directed constant polylogarithmic time local model also separates undirected directed problems deterministic algorithms currently best deterministic algorithm congest model undirected problem recent algorithm constructs size rounds constant even local nodel deterministic algorithm problem gives undirected achieving rounds according theorem approximation directed problem requires weighted extend construction weighted case showing approximation weighted congest model takes rounds even randomized algorithms similar result holds weighted undirected case weighted case rather guaranteeing input bit affects many edges spanner simply assign weight edges weight edges hence taking even single edge expensive avoid allows show simpler construction obtaining stronger lower bound weighted case follows build graph except following differences see figure define change set vertices since vertices form change names respectively replace two edges edge since size cut rest graph still figure graph edges omitted clarity red dashed edges examples optional edges depend input strings following theorem states lower bound weighted directed case theorem perhaps randomized distributed algorithm congest model weighted directed problem takes logn rounds proof let positive integer let note number vertices exactly cost path length edges weight every pair vertices path length includes edges weight must start edge must end edge following proof claim argue path one edges otherwise directed path weight follows every cost inputs disjoint hence distributed algorithm weighted problem used solve set disjointness define let alice bob simulate algorithm end simulation alice concludes inputs disjoint none edges taken spanner inputs disjoint spanner cost hence must return spanner cost exists otherwise spanner must include least one edges proves output alice indeed correct proof lemma get log since gives logn prove similar bound weighted undirected problem undirected case would like construct similar graph modifying edges undirected would still hold path length edges weight vertices one edges following proof however since edges undirected may path length longer edges weight vertices even none edges requires modify construction order bounds apply also change construction follows replace edge path length adding graph vertices required edges constructing path edges path weight path length edges weight must start edge must end path length added hence path length edges weight path length happen one edges graph rest proof exactly directed case however added vertices graph hence number vertices graph undirected gives allows prove lower bound problem still small values theorem perhaps randomized distributed algorithm congest model weighted undirected problem takes rounds hardness approximation weighted section show approximating weighted problem least hard approximating unweighted minimum vertex cover mvc problem therefore known lower bounds mvc translate directly lower bounds weighted problem mvc problem input graph goal find minimum set vertices covers edges required edge least one let input graph mvc problem construct new graph following way see figure vertex vertices connect vertices triangle edge weight edges weight addition edge edges weight one edges according order ids weight figure vertex corresponding triangle vertices edge corresponding edges show solution weighted problem gives solution mvc claim cost minimum exactly size minimum vertex cover proof let minimum vertex cover construct follows first includes edges weight addition every add edge note edges weight edges add weight hence cost exactly show edges weight added spanner hence covered edges weight covered edges weight since edge covered path let edge weight let corresponding edge since vertex cover least one vertices former case add hence edge covered path note weight included latter case covered path hence cost direction let minimum cost cost construct vertex cover size start converting cost first contains edges weight edges weight addition includes edge weight replace two edges weight transformation clearly increase cost next show still since includes edges weight covers edges weight edges weight explained addition edge weight covered path length includes edges weight covered way let edge weight covered path length includes edge weight may different holds since weight hence added edges since length follows first case path covers added since also weight second case path length covers therefore cost define size exactly since edges exactly edges weight includes edges weight addition claim vertex cover let one edges assume note since weight since includes path form covers must hold example edge weight hence least one edges means least one needed conclusion cost minimum exactly size minimum vertex cover relate number rounds required distributed algorithms solve approximate two problems lemma let distributed algorithm weighted problem takes rounds graph vertices algorithm mvc takes rounds graph vertices proof describe algorithm approximates mvc let input graph mvc algorithm simulates graph following way vertex simulates vertices time message sent one edges corresponding edge send message edge since may need send different messages edge round simulated three rounds finishes convert solution vertex cover described proof claim without communication claim follows weighted problem mvc let number vertices number vertices definition hence time complexity simulating lemma shows works congest model works congest model well hence lower bounds approximating mvc local congest models give lower bounds weighted problem gives following results theorem obtain constant polylogarithmic approximation ratio weighted problem even local model graphs every distributed algorithm log requires least log log rounds logloglogn rounds theorem follows theorem lemma note number vertices maximum degree equal constant factor number vertices maximum degree addition theorem allows show time complexity distributed algorithm weighted approximation ratio gets theorem every integer graphs communication rounds local model every distributed algorithm weighted problem approximation ratios least congest model solving mvc optimally takes carries exact spanners follows rounds see theorem theorem distributed congest model solves weighted algorithm problem optimally requires rounds lower bounds hold also randomized algorithms remarks reduction mvc adapted obtain additional bounds first changing weights edges weight weight obtain weighted problem gives mvc implies lower bounds theorem theorem also graphs weights viewed lower bounds augmentation problem given initial set edges need augment minimal number edges induces lower bounds hold directed weighted case modify construction edges triangle vertex edge includes directed edges one edges weights edges remain undirected case local model send messages one round however spend three different rounds order simulation work also congest model distributed approximation problems present distributed approximation algorithm minimum problem need following terminology notation subset edges subset neighbors density star respect subset edges denoted equals set edges star edge includes edges note covers edges also edges edges densest respect maximal density respect density vertex respect denoted density densest clear context refer density density denote respectively rounded density star respect denoted obtained rounding closest power greater similarly rounded density vertex respect denoted obtained rounding closest power greater full star includes edges neighbors vertex consists vertices distance algorithm vertex maintains set includes edges full still covered edges added spanner algorithm proceeds iterations iteration following computed vertex computes rounded density sends vertex candidate let density least chosen according section choice central analysis carry vertex informs neighbors let edges candidate chooses random number sends uncovered edge least one candidates votes first candidate according order values one candidate minimum value votes one minimum star receives least spanner votes edges added vertex updates set removing edges covered maximal density adds spanner edges adjacent still covered outputs edges adjacent added spanner algorithm end algorithm edges covered spanner edges since add spanner edges algorithm knowing exact value unnecessary typical assumption vertices know polynomial upper bound suffices since candidates maximal rounded density follows candidates cover edge rounded density crucial analysis addition rounding densities guarantees log possible values maximal rounded density allows show efficient time complexity iteration takes constant number rounds local model example calculate vertex learns edges neighbors still uncovered vertex send neighbors list neighbors edges still covered next show algorithm requires polynomial local computations compute densest polynomial time sequential algorithm see lemma maximal density problem solved polynomial time using flow techniques allows compute rounded density vertex next explain choose star polynomial time computations algorithm clearly polynomial choosing star step iteration candidate vertex chooses density least however may multiple density choosing arbitrary star meet claimed round complexity crucial choose stars certain way addition find star using polynomial local computations next describe choose star let hvi subset beginning iteration holds hvi let hvi star svi chooses iteration defined follows first iteration candidate rounded density svi chosen follows first computes densest denote edge hvi adds edge otherwise disjoint hvi adds edges continues manner edge disjoint star add without decreasing density resulting star svi already candidate rounded density iteration hvi define svi otherwise contains star density least respect hvi define svi follows starts computing densest contained svi adds edges disjoint stars however considers adding edges disjoint stars guarantees svi contain star density least respect hvi chooses arbitrary rounded density later show never happens computations polynomial adds edges times time adds edges following computation checks edge hvi since optional edges computation polynomial also checks disjoint star density least compute computes densest star disjoint analysis section present analysis distributed approximation algorithm minimum problem prove following theorem theorem distributed algorithm minimum problem local model guarantees approximation ratio log takes log log rounds let set edges spanner produced algorithm algorithm ends edges covered hence show size log set edges minimum afterwards show time complexity algorithm log log rounds approximation ratio start showing algorithm guarantees approximation ratio log analysis sequential algorithm obtains approximation ratio strongly depends facts stars added spanner one one star added step maximal rounded density allow dividing edges several subsets according order covered algorithm bounding number edges subset analysis borrows ideas analysis requires sophisticated accounting since algorithm adds multiple stars iteration varying densities addition overcoming uncertainties compelling aspect approach easily extends variants problem problem show approximation ratio assign edge value cost sum costs edges closely related satisfying cost log implies claimed approximation ratio write edges added spanner algorithm edges added spanner end algorithm maximal density vertex edge set cost edge let iteration first covered algorithm edge may covered candidate star votes added spanner iteration case set cost density star chooses iteration another option covered result adding stars spanner iteration may covered either different star one votes path length created edges added spanner iteration together edges added previous iterations cases set cost first show left inequality lemma cost proof topprove cost enough show thatp cost cost second inequality follows since cost next prove first inequality let stars set stars added algorithm holds since edge included least one star let star added iteration density iteration recall add spanner since gets least votes edges denote otes set edges vote iteration otes defined cost gives otes cost hence stars otes cost edge one star stars otes since edge votes one star iteration covered addition edge otes algorithm means hence get cost cost otes completes proof lemma bound cost let dlog divide edges subsets according costs show sum costs edges since log subsets conclude cost log let cost note edges edges algorithm candidate star vote divide edges subsets follows let cost let cost edge holds cost since density stars added algorithm least since defined cost edges hence edge cost gives lemma every cost proof claim holds trivially holds cost last equality follows fact least edges since connected let set edges minimum vertex let full define starsj next show cost prove write cost cost cost since cost get cost show cost consider specific star starsj let edges according order algorithm breaking ties arbitrarily note edges algorithm explained density beginning iteration least since may additional edges since candidates edge rounded density maximal rounded density holds density star least chooses star density least hence cost gives cost edge cost therefore cost last inequality follows since note let set edges star since every edge starsj least one star starsj summing stars starsj gives cost cost note since edge included exactly two stars starsj addition since covers edges particular edges minimum gives cost completes proof define starsj let starsj let edges according order added algorithm breaking ties arbitrarily must hold otherwise density greater one iteration added contradicts algorithm gives cost following arguments case get cost completes proof lemmas give cost log proves following claimed approximation ratio lemma approximation ratio algorithm log time complexity show algorithm completes log log rounds potential function argument given analyzing set cover minimum dominating set problems addressed analyze algorithm along similar argument algorithm necessitates intricate analysis mainly due fact vertex may center multiple stars added algorithm rather chosen dominating set latter may contain vertices spanner constructed algorithm initially possible stars may constitute nevertheless show get time complexity log log rounds minimum algorithm matches time complexity set cover dominating set algorithms crucial component proving small time complexity showing long rounded density change iterations always chooses star equal contained star chooses previous iteration explained section rounded density iterations tries choose star contained svi show always case following useful analysis observation let numbers let positive numbers min max addition inequalities become equalities yjj mini xyii maxi xyii observation follows writing prove following claim let candidate star iteration hvi chooses star contained svi iteration proof assume contrary iteration hvi star contained svi density least respect let first iteration let first iteration hvi star contained svi density least respect hvi let densest respect hvi hvi since hvi let full let sequence stars chosen iteration iteration order chosen holds since first iteration hold next show induction particular give hence iteration star contained svi density least contradiction definition claim holds trivially since full assume assume contrary note contained induction hypothesis let iteration chosen since write holds hvj hvi write hvi edges hvi edges edges one endpoint one endpoint since densest star respect hvi follows hvi otherwise observation shows denser star shows least one first case hvj hvi least shows disjoint star density least contained second case edge edges denote endpoint follows edge observation since density least get edges either way get contradiction definition completes proof rest analysis based potential function argument described set cover minimum dominating set problems let beginning iteration define potential function set edges star chooses iteration note potential function may increase iterations value changes however since round values powers two may log different values obstacle may increase iterations even value change vertex might change stars different iterations however claim long rounded density vertex remains among iterations always chooses star contained star chooses previous iteration hence size set edges decrease end last iteration beginning next one follows long change value decrease iterations goal show value change iterations potential function decreases multiplicative factor iterations expectation get time complexity log log rounds following lemma shows value change iterations potential function decreases multiplicative factor iterations expectation proof follows lines proofs included completeness say iteration legal random numbers chosen candidates iteration different lemma potentials beginning end legal iteration positive constant order prove lemma need following definitions let number candidates edge candidate sort edges according order let sets first edges last edges sorted order respectively indeed odd sets share edge pair candidate star say good next show chooses legal iteration star added spanner constant probability claim let legal iteration iteration chooses chooses proof let number candidates respectively choose chooses chooses chooses holds since gives chooses chooses claim good pair legal iteration chooses proof assume chooses denote number edges choose let note since good therefore edge claim edge chooses probability least hence equivalently using markov inequality get hence get since holds probability least case least edges choose added spanner completes proof bound value potential function proving lemma proof lemma let potential function beginning end pvalues legal iteration holds sum edges candidates rounded density pairs candidate rounded density note rounded density candidates edge since maximal rounded density edge chooses star star added spanner decreases ascribe decrease pair since chooses one candidate decrease ascribed one pair hence get chooses chosen chooses chooses good good good since least half pairs good get equivalently completes proof conclusion get following lemma time complexity algorithm log log rounds proof holds maximum density star size algorithm terminates maximum density since densities rounded powers may obtain log values maximum degree addition lemma value iterations legal iteration value decreases iterations factor least expectation since random numbers chosen different giving value two consecutive iterations value decreases iterations constant factor expectation since log log iterations expectation value must decrease shows time complexity log log rounds expectation chernoff bound gives also holds lemma lemma complete proof theorem additional approximations show algorithm extends easily following variants directed problem weighted problem problem describe differences algorithm analysis cases directed approximation directed case consider directed stars directed edge includes directed edges may include ingoing outgoing edges definition densities follows definition undirected case order give algorithm requires polynomial local computations directed variant show approximate rounded density densest star directed case rest analysis follows undirected case gives following theorem distributed algorithm directed problem local model guarantees approximation ratio log takes log log rounds compute approximation densest directed look edges neighbors remove directed edges edge two directed edges exist graph ignore directions edges compute densest undirected case let star computed let undirected density ignoring edges directed path let directed density show shows gives densest directed computing replace undirected edge two directed edges exist graph one exists otherwise claim proof let edges undirected case since densest undirected directed star edges may counted twice directed case increases density contains twice edges replaced undirected edge two edges gives claim proof claim next show let densest directed let directed edges write edge edge also write edge edge also look undirected density two directed edges replaced one undirected edge min min second inequality follows observation last equality follows since densest directed directed density equals next show since get observation note directed star directed edges may additional edges edges appear directions means paths also appear directions means edges paths shows directed star density greater contradiction definition conclusion shows min claims show approximate directed density densest directed star using polynomial local computations adapt algorithm directed problem approximate directed density gives round value closest power two greater denote rounded value rounded density value remains choose stars similarly undirected case difference look dense disjoint star necessarily find densest directed disjoint star analysis work need look since compute approximation density value may increase iterations avoid cases always define minimum value last iteration value computed current iteration always rounded density since density decrease iterations disjoint stars density least undirected case add edges disjoint stars star choose long density least rest analysis similar constants change sightly since choose stars less dense work approximation density weighted approximation weighted case cost spanner rather unweighted case requires several changes algorithm analysis let ratio maximum minimum positive weights edge show following theorem distributed algorithm weighted problem local model guarantees approximation ratio log takes log log rounds next describe differences weighted case star define define set edges star define emphasize take number potentially edges sum weights since intuitively edges need covered opposed taking sum weights edges star due need optimize cost spanner beginning algorithm add edges weight spanner edges covered stars weight already covered hence algorithm consider stars round densities closest power two include also negative powers two since density star may smaller depending weights slight difference vertex terminates density wmax wmax maximal weight edge adjacent vertex case adds spanner edges adjacent still covered denote edges rest algorithm according new definition observed sequential algorithm weighted case find densest star weighted case using flow techniques well next describe differences analysis cost solution obtained algorithm cost optimal solution give edges cost unweighted case depending new definition density addition edges define cost goal show cost log proof cost proof lemma minor changes note cost definition reason unweighted case addition new definition gives otes cost rest proof follows however difference unweighted case longer show approximation ratio log density stars added algorithm may smaller weight optimal may smaller still show approximation ratio log elements analysis similar analogues classic analysis greedy set cover algorithm first instead lemma show following cost log proof first show cost log edges cost edges clearly affect cost edges algorithm unweighted case vertex let full define consider star let sequence edges according order algorithm assume first density beginning iteration candidates rounded density since maximal rounded density particular density star least chooses star density lemma least hence cost iteration cost similarly density beginning least gives cost gives log log last equality number edges star star note cost edges since covered beginning algorithm without voting candidate hence get case log cost cost write cost cost holds cost next bound cost let set edges star since every edge least one star summing stars get cost cost log conclusion cost log holds since edge included exactly two stars gives cost log complete proof bound cost let optimal spanner define respect let let sequence edges according order added algorithm definition must hold wmax wmax maximal weight particular contains star edges otherwise added givesp cost wmax following since pthe arguments gives cost get cost cost cost log completes proof conclusion get cost log completes proof log ratio giving following lemma lemma approximation ratio algorithm log prove round complexity minor changes proof claim first replace size star cost order work new definition note adding edges weight star increase density shows chosen algorithm particular star contain edges weight adjacent shows star includes edges positive weight proof carries edges already beginning algorithm shows hvi needed addition second case proof instead showing edge show number possible densities depends weights following way let wmax wmin max maximum minimum positive weights edge recall wmin maximum density star since star edges addition algorithm terminates maximum density wmax since round densities powers two may log different values densities rest proof exactly unweighted case approximation recall problem edges graph divided two types clients servers goal cover client edges server edges let set client edges let vertices touch client edges let maximum degree subgraph includes server edges show following theorem distributed algorithm problem local model guarantees approximation ratio min log log takes log log rounds slight differences algorithm first throughout algorithm analysis consider stars composed server edges star define set client edges star set edges vertex maintains consists client edges star includes server edges adjacent terminates maximal density since client edges server edges perhaps best way cover client edge take path length covers density corresponding star cost changes slightly constants analysis terminates adds uncovered edge spanner client server edge edges edges note since edges server edges may client edges covered server edges case solution problem algorithm covers edges may covered server edges analyze algorithm assume solution problem otherwise defined cases restrict client edges edges covered server edges get new problem optimal solution approximation ratio get analysis slight differences follows first give costs client edges since edges need cover give costs minimum algorithm particular cost goal show cost log proof cost exactly proof lemma next show cost log let dlog define sets according new definition let cost let cost define since stars added algorithm density least cost edge gives next show following lemma every cost proof proof follows cases proof lemma holds cost first inequality use fact give costs edges last edges prove next equality follows fact includes least let let connected components note connected component includes least two vertices since includes least one edge means number connected components connected component denote number vertices connected component denote edges cover edges holds since connected edges need connect vertices otherwise edge covered addition edge least one vertices otherwise cover edge follows edge two different subsets gives completes proof lemma get cost cost log since cost cost log shows approximation ratio log addition show cost log following proof lemma replacing shows approximation ratio log problem note minimum problem half average degree maximum degree hence approximation ratio log better log however variant may case depending client server edges time analysis minimum problem note may log different values consider stars composed server edges completes proof theorem distributed approximation mds section show algorithm modified give efficient algorithm minimum dominating set mds problem guaranteeing approximation ratio log mds problem goal find minimum set vertices vertex either neighbor algorithm mds structure algorithm jia differs mechanism symmetry breaking approach guarantees approximation ratio log log ratio holds expectation following states results mds theorem distributed algorithm minimum dominating set problem congest model guarantees approximation ratio log takes log log rounds mds define star centered vertex set vertices contains neighbors note one star centered vertex simplifies algorithm analysis density star respect subset vertices denoted defined density vertex respect denoted defined definition rounded density algorithm minimum problem vertex maintains set contains vertices still covered vertices already added dominating set vertex covered set set neighbor set algorithm proceeds iterations iteration following computed vertex computes rounded density sends vertex candidate vertex informs neighbors candidate let candidate chooses random number sends neighbors uncovered vertex covered least one candidates votes first candidate covers according order values one candidate minimum value votes one minimum receives least votes vertices covers added dominating set vertex updates set removing vertices covered outputs added dominating set previous step crucial difference spanner approximation algorithm densities based number uncovered neighbors number uncovered edges potentially covered star reason computations algorithm implemented efficiently congest model analysis mds algorithm follows lines analysis minimum algorithm denote dominating set produced algorithm minimum dominating set assign vertex value cost equals covered first timep candidate density votes otherwise cost show cost log implies claimed approximation ratio lemma cost proof proof similar proof lemma vertex denote otes vertices vote added holds least vertices vote density definition hence vertex holds otes cost since vertex otes summing vertices gives pin one set otes cost cost proof cost log similar proof lemma ispreplaced edges replaced vertices replaced note equality replaced together lemma proves approximation ratio log time analysis main difference vertex one star simplifies proof claim longer required let beginning iteration define potential function value change iterations value decrease iterations definition densities density vertex since round densities may log different values following analysis analysis minimum algorithm difference edges replaced vertices candidate vertex star show value change iterations potential function decreases multiplicative factor iterations expectation gives time complexity log log rounds together approximation ratio proves theorem cost vertices distributed spanner problems section show distributed algorithms spanner problems following framework recent algorithm covering problems see section nutshell vertices invoke network decomposition algorithm graph value log computed vertices locally given polynomial bound decomposes graph clusters logarithmic diameter colored logarithmic number colors finally increasing order colors vertices color select edges spanner show indeed clusters color make choices parallel method choosing edges spanner results approximation factor giving following theorem randomized algorithm complexity poly log local model computes minimum constant proof start describing sequential algorithm explain implement local model using network decomposition algorithm vertices start adding edges spanner initialized empty keeping track edges covered edges beginning edges uncovered describe done need following notation given integer denote subgraph vertices within distance edges vertex let size optimal spanner uncovered edges notice spanner use covered uncovered edges whole graph process vertices according given order step look smallest radius since optimal spanner size increasing radius without condition met happen log times add optimal spanner uncovered edges bri mark presentation framework slightly different goes intermediate slocal model edges covered new edges covered particular edges bri covered step note optimal spanner bri contained bri shows step depends polylogarithmic neighborhood around next prove approximation ratio algorithm denote edges bri uncovered step since edges bri covered step follows distance least let optimal spanner let minimum set edges covers definition contained bri shows subsets disjoint step added edges inequality follows since size optimal spanner spanner since subsets disjoint summing gives completes approximation ratio proof show implement algorithm local model see also proposition let log consider graph set vertices two vertices connected distance network graph notice local model algorithm simulated vertices overhead rounds vertices invoke randomized network decomposition algorithm linial saks graph algorithm decomposes graph clusters diameter log colored log colors within rounds invoked completes poly log rounds assign vertex label idv color cluster idv lexicographic increasing order labels provides order vertices distributed algorithm runs log phases phase vertices color active collect information cluster neighbors since diameter cluster log completes poly log rounds vertex cluster locally simulates sequential algorithm vertices cluster according order since sequential algorithm depends vertices every two vertices neighbors means either cluster two clusters different colors guarantees algorithm indeed executed parallel vertices color completes proof correctness algorithm relies fact definition local optimal spanner contained hence algorithm adapted similarly weighted directed variants weighted case complexity poly log ratio maximum minimum positive weights edge acknowledgment would like thank seri khoury fruitful discussions references amir abboud keren seri khoury lower bounds distributed distance computations even sparse networks proceedings international symposium distributed computing disc pages baruch awerbuch boaz david peleg michael saks adapting asynchronous dynamic networks proceedings annual acm symposium theory computing stoc pages baruch awerbuch david peleg network synchronization polylogarithmic overhead proceedings annual symposium foundations computer science focs pages baruch awerbuch david peleg routing polynomial siam journal discrete mathematics leonid barenboim michael elkin cyril gavoille fast algorithm applications distributed computation theoretical computer science surender baswana sandeep sen approximate distance oracles unweighted graphs expected time acm transactions algorithms talg surender baswana sandeep simple linear time randomized algorithm computing sparse spanners weighted graphs random structures algorithms piotr berman arnab bhattacharyya konstantin makarychev sofya raskhodnikova grigory yaroslavtsev approximation algorithms spanner problems directed steiner forest information computation piotr berman sofya raskhodnikova ruan finding sparser directed spanners iarcs annual conference foundations software technology theoretical computer science fsttcs pages keren telikepalli kavitha ami paz amir yehudayoff distributed construction purely additive spanners international symposium distributed computing disc pages keren seri khoury ami paz quadratic lower bounds congest model international symposium distributed computing disc october vienna austria pages shiri chechik compact routing schemes improved stretch proceedings acm symposium principles distributed computing podc pages eden michael dinitz lowest degree approximation hardness approximation randomization combinatorial optimization algorithms techniques pages eden michael dinitz guy kortsarz bundit laekhanukit approximating spanners directed steiner forest upper lower bounds proceedings twentyeighth annual symposium discrete algorithms soda pages eden chlamtac michael dinitz robert krauthgamer spanners via dense subgraphs ieee annual symposium foundations computer science focs pages vasek chvatal greedy heuristic problem mathematics operations research bilel derbel cyril gavoille david peleg laurent viennot locality distributed sparse spanner construction proceedings acm symposium principles distributed computing podc pages bilel derbel mohamed mosbah akka zemmari sublinear fully distributed partition applications theory computing systems michael dinitz guy kortsarz ran raz label cover instances large girth hardness approximating basic acm transactions algorithms talg michael dinitz robert krauthgamer directed spanners via linear programs proceedings annual acm symposium theory computing stoc pages michael dinitz robert krauthgamer spanners better simpler proceedings annual acm symposium principles distributed computing podc pages michael dinitz yasamin nazari distributed network design distributed convex programming proceedings international conference principles distributed systems opodis michael dinitz zeyu zhang approximating spanners proceedings annual symposium discrete algorithms soda pages andrew drucker fabian kuhn rotem oshman power congested clique model proceedings acm symposium principles distributed computing podc pages michael elkin computing almost shortest paths acm transactions algorithms talg michael elkin unconditional lower bound distributed minimum spanning tree problem siam michael elkin distributed fully dynamic algorithm maintaining sparse spanners proceedings annual acm symposium principles distributed computing podc pages michael elkin ofer neiman efficient algorithms constructing sparse spanners emulators proceedings annual symposium discrete algorithms soda pages michael elkin david peleg problem applications network design international colloquium structural information communication complexity sirocco pages michael elkin david peleg approximating problems theoretical computer science michael elkin david peleg hardness approximating spanner problems theory computing systems michael elkin jian zhang efficient algorithms constructing distributed streaming models proceedings annual acm symposium principles distributed computing podc pages paul extremal problems graph theory theory graphs applications proceedings symposium smolenice pages publ house cszechoslovak acad prague orr fischer tzlil gonen rotem oshman distributed property testing subgraphfreeness revisited corr silvio frischknecht stephan holzer roger wattenhofer networks compute diameter sublinear time proceedings annual symposium discrete algorithms soda pages giorgio gallo michael grigoriadis robert tarjan fast parametric maximum flow algorithm applications siam journal computing mohsen ghaffari distributed approximation connected dominating set proceedings international colloquium automata languages programming icalp pages mohsen ghaffari david harris fabian kuhn derandomizing local distributed algorithms arxiv preprint mohsen ghaffari fabian kuhn yannic maus complexity local distributed graph problems proceedings annual acm sigact symposium theory computing stoc pages acm ofer grossman merav parter improved deterministic distributed construction spanners international symposium distributed computing disc october vienna austria pages david harris johannes schneider distributed sublogarithmic rounds proceedings annual acm sigact symposium theory computing stoc pages stephan holzer roger wattenhofer optimal distributed pairs shortest paths applications proceedings acm symposium principles distributed computing podc pages acm lujun jia rajmohan rajaraman torsten suel efficient distributed algorithm constructing small dominating sets distributed computing david johnson approximation algorithms combinatorial problems journal computer system sciences guy kortsarz hardness approximating spanners algorithmica guy kortsarz david peleg generating sparse journal algorithms guy kortsarz david peleg generating siam journal computing fabian kuhn thomas moscibroda roger wattenhofer local computation lower upper bounds journal acm jacm fabian kuhn rogert wattenhofer distributed dominating set approximation proceedings annual symposium principles distributed computing podc pages eyal kushilevitz noam nisan communication complexity cambridge university press new york usa nathan linial locality distributed graph algorithms siam nathan linial michael saks low diameter graph decompositions combinatorica ratio optimal integral fractional covers discrete mathematics david peleg distributed computing approach siam david peleg vitaly rubinovich lower bound time complexity distributed spanning tree construction siam david peleg alejandro graph spanners journal graph theory david peleg jeffrey ullman optimal synchronizer hypercube siam journal computing david peleg eli upfal space efficiency routing tables journal acm jacm seth pettie distributed algorithms ultrasparse spanners linear size skeletons distributed computing sridhar rajagopalan vijay vazirani rnc approximation algorithms set cover covering integer programs siam journal computing alexander razborov distributional complexity disjointness theoretical computer science liam roditty mikkel thorup uri zwick deterministic constructions approximate distance oracles spanners international colloquium automata languages programming icalp pages atish das sarma stephan holzer liah kor amos korman danupon nanongkai gopal pandurangan david peleg roger wattenhofer distributed verification hardness distributed approximation siam journal computing mikkel thorup uri zwick compact routing schemes proceedings thirteenth annual acm symposium parallel algorithms architectures spaa pages mikkel thorup uri zwick approximate distance oracles journal acm jacm
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apr security monitoring framework virtualization based hep infrastructures gomez martinez alice collaboration infrastructure computer systems data processing iri frankfurt cern geneva switzerland abstract high energy physics hep distributed computing infrastructures require automatic tools monitor analyze react potential security incidents tools collect inspect data resource consumption logs sequence system calls detecting anomalies indicate presence malicious agent also able perform automated reactions attacks without administrator intervention describe novel framework accomplishes requirements proof concept implementation alice experiment cern show achieve fully virtualized environment improves security isolating services jobs without significant performance impact also describe collected dataset machine learning based intrusion prevention detection systems grid computing dataset composed resource consumption measurements cpu ram network traffic logfiles operating system services system call data collected production jobs running alice grid test site big set malware malware collected security research sites based dataset proceed develop machine learning algorithms able detect malicious jobs introduction frequently hep computing also general purpose grid computing user supplied code data deployed executed farms around world exact location normally irrelevant allows scientists many areas beyond physics use huge computational power solve complicated scientific problems weather modeling brain simulation among others however also creates challenges operators administrators need tools monitor security incidents user code data isolated different users also physical computers networks order restrict access sensitive elements organizations propose novel paradigm developed framework focus protecting monitoring user payload execution moreover enforces isolation environment way jobs access sensitive resources tool enables job behavior analysis order detect possible intrusions accomplished collecting processing data generated jobs logs system calls resource consumption data traditional intrusion detection prevention systems idps perform attack detection using fixed rules based signatures identical traditional monitoring systems therefore employ machine learning overcome mentioned drawbacks achieving generalization among attack variants currently tool provides isolation monitoring security incidents algorithms grid computing authors defined threat model guides design implementation described framework devised detect attackers protected system trying actions like following exploit unknown unfixed vulnerabilities listen user network traffic gather sensitive clear text information perform man middle attack tamper user jobs escalate privileges access sensitive server configuration data proof concept implementing described framework alice grid cern alice large ion collider experiment dedicated detector designed exploit physics potential interactions large hadron collider cern alice experiment developed alice production environment alien implements many components grid technologies needed analyze hep data alien computing centers participate alice grid seen used single entity available node executes jobs file access transparent user wherever world file might figure shows picture alice grid figure alice grid computing farms aroud world document organized follows section introduces security isolation strategy distributed system section explains method collecting relevant data isolated infrastructure section shows collected data used determine security status system section detail intrusion detection prevention model section summarizes current state project challenges faced design implementation desired methodology finally section gives conclusions work done security isolation security isolation enforces application space separation idea one process compromised utilized attack entire system components stay untouched several technologies provide secure isolation virtual machines linux containers popular examples linux containers extension virtual memory concept allow isolation network interfaces pid tree mount points separation containers rest system enforced kernel affect host containers technology uses namespaces cgroups private view system limited resource assignment containers provide set advantages lightweight fast boot milliseconds intrinsic disk memory usage shown provide better performance vms commonly used grid cloud computing achieve isolation however performance comparable security features make suitable alternative proposed isolation architecture propose usage enforce hep grid site user isolation also extensible broader scientific computing clouds achieve require batch job orchestrator allowing execution user processes containers computing clusters section gives details requirement selected solution shown figure switch environment without isolation jobs access server user jobs environment jobs run one process space without access jobs sensitive components pilot job bob grid alice grid job job bob grid job container pilot job alice grid job pilot job container figure desired isolation scenario isolated environment job execution enough jobs could still perform several kind attacks allowed activities distributed denegation service ddos bitcoin mining hosting among others consequently need monitor activity detect incidents inside described next sections monitoring data mining hep distributed computing systems use continuous automated monitoring help administrators find fix situations affecting normal operation resulting monitoring data used find even predict software hardware failures valuable source security information well document focus measurement metrics related batch jobs submitted distributed system several relevant metrics collect instance job system logs system call sequence resource usage data cpu ram network traffic furthermore goal chose best information job behavior without affecting habitual performance decided employ data mining intelligent algorithms given ability find correlations analyze trends big datasets order provide better understanding security related events machine learning based security monitoring machine learning set mathematical models simulate human learning abilities context intrusion detection helps analyzing big amounts data learning expected behavior identifying abnormal situations traditional industrial ids use rather fixed rules search known attack signatures however problems unknown slightly different intrusion methods used selected supervised training analyze collected data supervised training set already classified tagged data training dataset used model function example neural network order make able classify new unseen data test dataset figure monitored data gathering processing training dataset collected training testing dataset gathering monitoring data production alice grid jobs figure additionally executed big set linux malware samples data first part tagged normal data second malicious dataset utilized compare several machine learning algorithms find one gives best accuracy following list algorithms selected tested define one gives best accurate results dataset support vector machines multilayer neural networks recurrent neural networks figure shows scheme proposed architecture usage figure proposed architecture intrusion prevention detection analyzing job monitoring data goal finding security incidents security incidents found analyzing anomalies system things beyond common state probably caused malicious software besides even execution environment sandboxed many possible attacks still affect distributed infrastructure user job misbehaving proposed framework raise alarm perform predefined actions instance terminate malicious processes figure shows desired implementation proposed system regarding intrusion detection intrusion detection prevention alerts admins automated actions figure proposed intrusion detection worker nodes challenges improving provided security impact system performance especially important hep computing hand also need innovative ways analyze trace log data efficient way another important challenge reduce amount false positives false negatives since system administrators rely accuracy security monitoring framework proof concept testing environtment far already deployed testing alice grid site based alien local linux cluster five ubuntu nodes order orchestrate run jobs inside linux containers tested three different tools offer functionality kubernetes apache mesos docker swarm end decided work docker swarm allows carry simplest deployment important requirement research environment use docker engine centos container images developed alien interfaces mentioned batch systems cvmfs installed hosts shared volume inside alien container allow access hep libraries currently execute one job per container useful increase traceability different jobs also natural micro service model monitoring infrastructure collecting data normal grid jobs prometheus sysdig prometheus allows take resource usage data directly containers collect via restful interface sysdig enables capture system calls linux fast reliable way develop custom python library integrate tools make fit needs infrastructure utilized execution measurement alice production jobs tagged normal jobs network isolated machine used malware data collection machine setup grid worker nodes downloaded set linux malware samples security research web site ran samples collected information normal jobs logs sequence system calls resource usage data finally obtained combined dataset allows train test selected machine learning algorithms representation implemented components shown figure conclusions distributed computing fundamental component high energy physics collaborations improving security kind infrastructures requires innovative tools automatically detect security related incidents security isolation also necessary protect sensitive components allowing traceability job activities propose usage linux containers order provide isolation without highly decreasing expected performance use machine learning techniques provide generalization overcoming common ids difficulties finding even slightly different threats describe ongoing development process new security monitoring framework linux containers based hep infrastructures tested proof concept alice experiment cern collected dataset normal malicious monitoring information grid jobs malware samples utilized train test algorithms algorithms enable autonomous intrusion detection prevention important component proposed new framework future work plan explore approach used detect anomalies beyond security scope instance find hardware failures even human mistakes acknowledgments authors acknowledge assistance cern security department specially stefan lueders romain wartel work supported german federal ministry education research sysdig system calls connections resource usage figure proof concept implementation references gomez ramirez lara kebschull jpcs intrusion prevention detection grid computing alice case alice collaboration alice technical proposal large ion collider experiment cern lhc cern geneve alice collaboration jinst alice experiment cern lhc vol begnasco jpcs alien alice environment grid iop publishing christoph technical design report lcg lhc computing grid technical design report version jun cern geneve available https abed clancy levy security trust management lecture notes computer science intrusion detection system applications using linux containers springer international publishing modi jnca survey intrusion detection techniques cloud jan computer fraud security security isolation graber ubuntu foundations team lxc blog post series updated jan cited may available https menage proceedings linux symposium adding generic process containers linux kernel june shiseki international conference computing high energy nuclear physics chep cloud computing scientific technical application development execution platform october lara jpcs autonomous system management alice cluster using sysmes framework azad ijitcs data mining intrusion detection comparative study methods types data sets yuping ijact intrusion detection approach using svm multiple kernel method jan gascon acm press structural detection android malware using embedded call graphs cited nov available http bishop library congress isbn pattern recognition machine learning new york springer google kubernetes updated dec cited available http apache foundation mesos updated dec cited available https docker docker swarm updated dec cited available https docker docker updated dec cited available https centos project updated dec cited available https buncic proceedings xii international workshop advanced computing analysis techniques physics research cernvm virtual appliance lhc applications erice pos prometheus updated dec cited available https draios sysdig updated dec cited available http virusshare updated dec cited available https
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machines think radio language december yujian college computer science beijing university technology beijing china email liyujian abstract people think auditory visual tactile forms language machines principally possible think radio language according first principle presented general intelligence principle language relativity answer may give exceptional solution robot astronauts talk space exploration keywords principle language relativity first principle radio language space exploration figure two robonauts talking earth radio language moon one says earth home truly beautiful potential application morse code artificial intelligence future beyond classical transmission text information people computer science one biggest unsolved problems develop intelligent machines since seminal paper turing topic artificial intelligence central question machines think began excite interest building systems learn think like people fascinating dream recently interest renewed impressive progress deep learning spite great difficulties character challenge frostbite challenge perform variety tasks rapidly flexibly people mean system learn think like person lake argued system build causal models world ground learning intuitive theories physics psychology harness compositionality claimed key ideas core ingredients would play active important role producing learning thought undoubtedly claim attractive promising ultimate dream implementing machines general intelligence however claim says little person ability communicate think natural language clearly vital human intelligence question develop capacity language machines language basic tool human society playing essential role communication thought people accustomed thinking sound language sound thinking different countries people generally speak different languages languages spoken world used less people estimated unesco united nations educational scientific cultural organization widely spoken languages mandarin chinese english spanish hindi arabic bengali russian portuguese japanese german french practice language usually takes forms speech text encoded whistle sign braille may lead interesting question machines think language forms radio appearance chinese think chinese american think english spanish think spanish viewpoint daily life forms language even including forms whistle sign braille equivalent people think world opinion quite common point generalized first principle establish theory mind intelligence broadly termed principle language relativity principle symbolic relativity principle described follows admissible forms language equivalent intelligent system think world principle language relativity admissible form means system use thinking formulation thoughts world therefore principle stated words admissible forms language equivalent respect formulation thoughts world note principle named inspiration principle relativity physics namely admissible frames reference equivalent respect formulation fundamental laws physics physic laws reference frames inertial analogy stated thoughts world language forms speech text whistle sign braille therefore sense language form also regarded reference frame formulate thoughts taken first postulate intelligence theory principle language relativity implies language independent modality explains human language encoded lot different media using auditory visual tactile stimuli moreover give profound original insights guide engineering future generations intelligent machines example principally robots think radio language robots would tremendously useful space exploration radio language much convenient talk sound language lack air since person inborn ability receive send radio waves radio form language admissible human thus radio language novel creative idea robots think although radio certainly ordinary information transmission remote control clearly thinking radio language radio thinking different way implement intelligence people one may argue even without language artificial intelligence residual network deep alphogo could equal even beat human intelligence deep learning performance tasks object recognition video games board games practical realization still reasonable require intelligent machine able communicate language goes without saying language essential ability general intelligence yet nobody quite sure intelligence perhaps general purpose intelligence measures agent ability achieve goals wide range environments nonetheless informal definition together mathematic description plays limited role design intelligent machines albeit bringing together key features many expert definitions human intelligence understand nature intelligence definition expected also comprehensive theory theory look like least contain first principles system level principles must fundamental independent phenomena intelligence deduced principles physics chemistry biology although principles may lead anything like maxwell equations capture essential characteristics intelligence comprehensively perspectives science philosophy importantly able make guide engineering intelligent machines especially different intelligence human implicates imitation grasping nature intelligence clearly implicates creation understanding genuinely one principles principle language relativity others certain obviously principle independent physics chemistry biology furthermore account modalityindependence language give rise revolutionary idea radio thinking intelligence envisaged future robot astronauts importance radio thinking emphasized may overturn public view robonauts talk space exploration imagine two robonauts talking earth moon see figure traditionally people think would talk sound language could seen science fiction films nevertheless reality without air alternatively talk radio language lacks neural mechanisms excellent thought experiment show although human language ability developed basis neural mechanisms brain intelligent machines may capacity radio language based mechanisms thereby intelligence may intelligence tries achieve intelligence demonstrated brains preferably highly evolved creatures nature intelligence understood way like secret fly indeed without flapping wings airplane fly way based theory aerodynamics bird brain space exploration autonomous robonauts would extremely helpful moon planets since environments change lot barriers implement robonauts require certain kinds intelligence autonomy actions example radio thinking helps plan radio talking helps collaborate kirobo world first talking robot sent space however kirobo tasked companion explorer could talk sound language inside spacecraft radio talking may exceptional solution outside terrific endeavor engineer machines general intelligence theory intelligence requires first principles example principle language relativity principles universe implications science play roles technology questions challenges set ahead remarkable years discovery references turing computing machinery intelligence mind lecun bengio hinton deep learning nature lake ullman tenenbaum gershaman building machines learn think like people behavioral brain sciences accepted publication https mikolov joulin baroni roadmap towards machine intelligence arxiv preprint yujian reveal secrets consciousness also theory cognitive relativity interdisciplinary science puzzles century science press beijing edited xixian https zhang ren deep residual learning image recognition ieee conference computer vision pattern recognition cvpr mnih kavukcuoglu silver control deep reinforcement learning nature silver huang maddison mastering game deep neural networks tree search nature legg hutter universal intelligence definition machine intelligence minds machines sendhoff sporns creating intelligence lanai http
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interpreting deep visual representations via network dissection nov bolei david aude oliva antonio torralba success recent deep convolutional neural networks cnns depends learning hidden representations summarize important factors variation behind data however cnns often criticized black boxes lack interpretability since millions unexplained model parameters work describe network dissection method interprets networks providing labels units deep visual representations proposed method quantifies interpretability cnn representations evaluating alignment individual hidden units set visual semantic concepts identifying best alignments units given human interpretable labels across range objects parts scenes textures materials colors method reveals deep representations transparent interpretable expected find representations significantly interpretable would random equivalently powerful basis apply method interpret compare latent representations various network architectures trained solve different supervised training tasks examine factors affecting network interpretability number training iterations regularizations different initializations network depth width finally show interpreted units used provide explicit explanations prediction given cnn image results highlight interpretability important property deep neural networks provides new insights hierarchical structure index neural networks network interpretability visual recognition interpretable machine learning ntroduction bservations hidden units large deep neural networks revealed concepts sometimes emerge individual latent variables within networks example object detector units emerge within networks trained recognize places part detectors emerge object classifiers object detectors emerge generative video networks internal structure appeared situations networks constrained decompose problems interpretable way emergence interpretable structure suggests deep networks may learning disentangled representations spontaneously commonly understood network learn efficient encoding makes economical use hidden variables distinguish input appearance disentangled representation well understood disentangled representation aligns variables meaningful factorization underlying problem structure encouraging disentangled representations significant area research internal representation deep network partly disentangled one possible path understanding mechanisms detect disentangled structure simply read human interpretable factors address following three key issues deep visual representations work disentangled representation neural networks factors quantified detected interpretable hidden units reflect special alignment feature space interpretations chimera differences network architectures data sources training conditions lead internal representations greater lesser entanglement zhou contributed equally work zhou bau torralba csail mit bzhou davidbau oliva torralba examine issues propose general analytic framework network dissection interpreting deep visual representations quantifying interpretability using broden broadly densely labeled dataset framework identifies hidden units semantics given cnn aligns concepts building upon preliminary result published begin detailed description methodology network dissection use method interpret variety deep visual representations trained different network architectures alexnet vgg googlenet resnet densenet supervisions supervsied training imagenet object recognition places scene recognition along various supervision tasks show interpretability property representation destroyed rotation without affecting discriminative power examine interpretability affected different training datasets training regularizations dropout batch normalization different data sources experiments reveal units emerge semantic detectors intermediate layers deep visual representations degree interpretability vary widely across changes architecture training conclude representations learned deep networks interpretable previously thought measurements interpretability provide insights structure deep visual representations revealed classification power related work visualizing deep visual representations though cnn models notoriously known black boxes growing number code data dissection results available project page http techniques developed visualize internal representations convolutional neural networks behavior cnn visualized sampling image patches maximize activation hidden units using variants backpropagation identify generate salient image features together natural image prior used invert cnn layer activation image generation network trained invert deep features synthesizing input images synthesizes prototypical images individual units learning feature code image generation network visualizations reveal image patterns learned deep visual representation provide qualitative guide interpretation interpretability units quantitative measure interpretability introduced human evaluation visualizations determine individual units behave object detectors network trained classify scenes however human evaluation scalable increasingly large networks resnet layers therefore aim present work develop scalable method qualitative visualization quantitative interpretation analyzing properties deep visual representations various intrinsic properties deep visual representations explored much research focused studying power cnn layer activations used generic visual features classification transferability activations variety layers analyzed found higher layer units specialized target task susceptibility adversarial input reveals discriminative cnn models fooled particular image patterns analysis correlation different random initialized networks reveal many units converge set representations training question representations generalize investigated showing cnn easily fit random labeling training data even explicit regularization work focuses another less explored property deep visual representations interpretability unsupervised learning deep visual representations recent work unsupervised learning learning exploits correspondence structure comes free unlabeled images train networks scratch example cnn trained predicting image context colorizing gray images solving image puzzle associating images ambient sounds resulting deep visual representations learned different unsupervised learning tasks compared evaluating generic visual features classification datasets pascal voc work provides alternative approach compare deep visual representations terms interpretability beyond discriminative power ramework etwork issection notion disentangled representation rests human perception means concept mixed therefore define interpretability deep visual representation terms degree alignment set concepts quantitative measurement interpretability deep visual representations proceeds three steps identify broad set visual concepts table statistics label type included dataset category scene object part material texture color classes sources ade ade ade opensurfaces dtd generated avg sample gather response hidden variables known concepts quantify alignment hidden pairs process network dissection reminiscent procedures used neuroscientists understand similar representation questions biological neurons since purpose measure level representation disentangled focus quantifying correspondence single latent variable visual concept fully interpretable local coding variable match exactly one concept although expect network learn partially nonlocal representations interior layers past experience shows emergent concept often align combination several hidden units present aim assess well representation disentangled therefore measure alignment single units single interpretable concepts gauge discriminative power representation rather quantifies disentangled interpretability show sec possible two representations perfectly equivalent discriminative power different levels interpretability assess interpretability given cnn draw concepts new broadly densely labeled image dataset unifies labeled visual concepts heterogeneous collection labeled data sources described sec measure alignment hidden unit cnn concept evaluating feature activation individual unit segmentation model concept quantify interpretability layer whole count number distinct visual concepts aligned unit layer detailed sec broden broadly densely labeled dataset able ascertain alignment concepts colors concepts objects assembled new heterogeneous dataset broadly densely labeled dataset broden unifies several densely labeled image datasets ade opensurfaces describable textures dataset datasets contain examples broad range objects scenes object parts textures materials variety contexts examples segmented pixel level except textures scenes given full images addition every image pixel dataset annotated one eleven common color names according human perceptions classified van weijer samples types labels broden dataset shown fig purpose broden provide ground truth set exemplars broad set visual concepts concept labels broden normalized merged original datasets red color yellow color wrinkled texture meshed texture wood material fabric material foot part door part airplane object waterfall object art studio scene beach scene fig samples broden dataset ground truth concept dense annotation top activated images segmented images using binarized unit activation map semantic segmentation annotations segmented annotations fig scoring unit interpretability evaluating unit semantic segmentation every class corresponds english word labels merged based shared synonyms disregarding positional distinctions left top avoiding blacklist overly general synonyms machine car multiple broden labels apply pixel example black pixel label left front cat leg three labels broden unified cat label representing cats across datasets similar unified leg label color label black labels least image samples included table shows number classes per dataset average number image samples per label class totally visual concept classes included scoring unit interpretability proposed network dissection method evaluates every individual convolutional unit cnn solution binary segmentation task every visual concept broden illustrated fig method applied cnn using forward pass without need training backpropagation every input image broden dataset activation map every internal convolutional unit collected distribution individual unit activations computed unit top quantile level determined every spatial location activation map dataset compare unit activation map inputresolution annotation mask concept activation map scaled mask resolution using bilinear interpolation anchoring interpolants center unit receptive field thresholded binary segmentation selecting regions activation exceeds threshold segmentations evaluated every concept dataset computing intersections every pair score unit segmentation concept reported intersection union score across images dataset iouk cardinality set dataset contains types labels present subsets inputs sums computed subset images least one labeled concept category value iouk accuracy unit detecting concept consider one unit detector concept iouk exceeds threshold qualitative results insensitive iou threshold different thresholds denote different numbers units concept detectors across networks relative orderings remain stable comparisons report detector iouk note one unit might detector multiple concepts purpose analysis choose top ranked label quantify interpretability layer count number unique concepts aligned units call number unique detectors figure summarizes whole process scoring unit interpretability segmenting annotation mask using receptive field units top activated images compute iou concept iou evaluating quality segmentation unit objective confidence score interpretability comparable across networks thus score enables compare interpretability different representations lays basis experiments note network dissection works well underlying dataset unit matches concept absent broden score well interpretability future versions broden expanded include kinds visual concepts interpreting deep visual representations testing prepare collection cnn models different network architectures supervision primary tasks listed table network architectures include alexnet googlenet vgg resnet densenet supervised training models trained scratch pretrained imagenet imagenet dataset contains million images classes freeze trained network weights upsample target layer car person grass door uni tion acti fabric segmentation blue conv conv conv conv network probed conv input image evaluate segmentation tasks fig illustration network dissection measuring semantic alignment units given cnn one unit last convolutional layer given cnn probed evaluating performance various segmentation tasks method probe convolutional layer table tested cnn models training none supervised self network alexnet alexnet googlenet alexnet dataset task random imagenet hybrid imagenet imagenet hybrid imagenet imagenet context puzzle egomotion tracking moving videoorder audio crosschannel colorization objectcentric transinv two subsets places database dataset categories kitchen living room coast contains million images scene categories contains million images scene categories hybrid refers combination imagenet training tasks select several recent models trained predicting context context solving puzzles puzzle predicting egomotion learning moving moving predicting video frame order videoorder tracking tracking detecting alignment objectcentric colorizing images colorization inpainting contextencoder predicting crosschannel predicting ambient sound frames audio tracking invariant patterns videos transinv models analyze comparable use alexnet alexnetderived architecture one exception model transinv uses vgg base network following experiments begin validating method using human evaluation use random unitary rotations learned representation test whether interpretability cnns property find conclude interpretability inevitable result discriminative power representation next analyze convolutional layers alexnet trained imagenet trained places confirm method reveals detectors concepts higher layers concepts lower layers detectors concepts emerge scene training show different network architectures alexnet vgg resnet yield different interpretability differently supervised training tasks training tasks also yield variety levels interpretability additionally show impact different training conditions examine relationship discriminative power interpretability investigate possible way improve interpretability cnns increasing width finally utilize interpretable units explanatory factors prediction given cnn human evaluation interpretations using network dissection analyze interpretability units within convolutional layers imagenetalexnet compare human interpretation trained scene classification identical architecture trained object classification imagenet evaluation done raters amazon mechanical turk amt baseline description unit semantics used descriptions unit descriptions collected asking raters write words short phrases describe common meaning pattern selected unit based visualization top image patches three descriptions confidence collected unit canonical description chose common description unit raters agreed description raters agree units may interpretable identify raters shown canonical descriptions visualizations asked whether descriptive units validated descriptions taken set interpretable units compare baseline descriptions labels ran following experiment raters shown visualization top images patches interpretable unit along word short phrase description asked vote whether given phrase descriptive image patches baseline descriptions randomized labels derived using net dissection origin labels revealed raters table summarizes results number interpretable units shown layer average positive votes descriptions interpretable units shown humanwritten labels labels human labels highly consistent units suggesting humans trouble identifying visual concept detectors detectors difficult label fig annotation interface used human raters amazon mechanical turk raters shown descriptive text quotes together fifteen images highlighted patches must evaluate whether quoted text good description highlighted patches table human evaluation network dissection approach interpretable units human consistency network dissection similarly labels given network dissection best found less descriptive lower layers comparison human interpretation labels predicted network dissection plotted fig sample units shown together automatically inferred interpretations manually assigned interpretations taken see predicted labels match human annotation well though sometimes capture different description visual concept crosswalk predicted algorithm compared horizontal lines given human third unit fig confirming intuition color texture concepts dominate lower layers object part detectors emerge measurement interpretability conduct experiment determine whether meaningful assign interpretable concept individual unit two possible hypotheses explain emergence interpretability individual hidden layer units hypothesis interpretability property representation whole individual interpretable units emerge interpretability generic property typical directions representation space hypothesis projecting direction would typically reveal interpretable concept interpretations single units natural basis would meaningful interpretations found direction hypothesis interpretable alignments unusual interpretable units emerge learning converges special basis aligns explanatory factors individual units model natural basis represents meaningful decomposition learned network hypothesis default assumption past found respect interpretability distinction individual high level units random linear combinations high level network dissection allows hypothesis apply random changes basis representation learned alexnet hypothesis overall level interpretability affected change basis even rotations cause specific set represented concepts change hypothesis overall level interpretability expected drop change basis begin representation convolutional units alexnet trained examine effect change basis avoid issues conditioning degeneracy change basis using random orthogonal transformation rotation drawn uniformly applying described interpretability summarized number unique visual concepts aligned units defined sec denoting alexnet find number unique detectors fewer number unique detectors finding inconsistent hypothesis consistent hypothesis also test smaller perturbations basis using fractional powers chosen form minimal geodesic gradually rotating intermediate rotations computed using schur decomposition fig shows interpretability decreases larger rotations applied fig shows examples linearly combined units rotated representation exactly discriminative power original layer writing original network note defines neural network processes rotated representation exactly original operates conclude interpretability neither inevitable result discriminative power prerequisite discriminative power instead find interpretability different quality must measured separately understood repeat complete rotation imagenet times result shown fig observe drop interpretability network drops alexnet originally interpretability alexnet higher alexnet imagenet thus random rotation damages network architectures supervised learning different network architectures affect disentangled interpretability learned representations apply network dissection evaluate range network architectures trained imagenet places simplicity following experiments focus last convolutional layer cnn semantic detectors emerge results showing number unique detectors emerge various network architectures trained imagenet places plotted fig terms network architecture find interpretability resnet densenet vgg green sky object places veined texture orange color color yellow lacelike texture imagenet red color grid pattern perforated texture yellow food material red woven texture yellow banded texture yellow color striped sky object blue grid texture sky object corrugated tree object black white banded texture pink red lined texture red color grass chequered texture sky grass object bed tree car object car pattern crosswalk part horiz lines mountain scene orange muzzle part animal face wheel part blue sky swirly texture round cat object nosed head part face leg part mesh dotted texture windows bed object montain wheels animal faces leg fig comparison interpretability five convolutional layers alexnet trained classification tasks places top imagenet bottom examples units layer shown identified semantics segmentation generated unit shown three broden images highest activation labels shown left labels shown right disagreement seen dominant judgment meaning example human annotators mark first unit places windows detector algorithm matches chequered texture rotation representation rotate rotate rotate rotate fig interpretability changes basis representation alexnet trained places vertical axis shows number unique interpretable concepts match unit representation horizontal axis shows quantifies degree rotation baseline individual units car single unit skyscraper single unit rotate linear combinations iou car combination closest unit iou skyscraper combination closest unit iou iou tree single unit iou tree combination closest unit iou head single unit iou head combination closest unit iou closet single unit iou closet combination closest unit iou random combination alexnet imagenet object scene part material texture color baseline random combination fig complete rotation repeated alexnet trained imagenet respectively rotation reduces interpretability significantly networks number unique detectors rotate baseline object scene part material texture color lac pla ene ces len ybr pla ces ima pla lex ces hyb rid net baseline object scene part material texture color number unique detectors alexnet number unique detectors object scene part material texture color number unique detectors fig interpretability across different architectures trained imagenet places fig visualizations best concept detectors five concepts taken individual units alexnet trained places left compared best detectors concepts taken representation random rotation right concepts iou visualization top activating image patches confirm individual units match concepts better linear combinations cases head detectors visualization linear combination appears highly consistent iou reveals lower consistency evaluated whole dataset googlenet alexnet deeper architectures usually appear allow greater interpretability though individual layer structure different across architecture comparing training datasets find places imagenet discussed one scene composed multiple objects may beneficial object detectors emerge cnns trained recognize scenes alexnet googlenet tab ile ttl ack cas unt ito tra thtu tin ion rta ble ilin ash eld tre units densenet imagenet closet objects ors ike let eep icy tte dpl tor elf ble ing tre ofa alk illo rth ilin sea ben objects units imagenet nta ile ilin tra rse tte tree tra tove raili sur andface lier rta wpa tor sno air illow able cra gnb dle units densenet places objects tra ttl tre sof ild ike tte nce tin oad rta tai lier tra ash rfac rfal rai lin tor irr alm objects units places fig histogram object detectors resnet densenet trained imagenet places respectively unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit dog plant bus airplane fig comparison several visual concept detectors identified network dissection densenet resnet googlenet vgg alexnet network trained two matches among convolutional units network shown segmentation generated unit shown four maximally activating broden images units activate concept generalizations googlenet unit horses dogs white ellipsoids jets textures color textures color colors units airplane train car bus bed house cat dog mountain horse water grass bottle tree boat sofa building toilet plant bird motorbike pool table sky tent sea book pottedplant fence bicycle painting road person curtain sink snow table shelf bathtub windowpane chandelier stairway skyscraper stove rock track washer sidewalk chair court work surface waterfall bridge bench sheep railing tvmonitor mirror palm bookcase highway closet nursery art gallery skyscraper corridor reception airport terminal cockpit shoe shop attic classroom lighthouse playground youth hostel dining room volleyball conference room sauna landing deck mountain snowy auditorium alley castle bowling alley laundromat gas station windmill scenes game room subway interior operating room playroom sandbox staircase pantry escape parking bathroom carrousel parts badlands cemetery butchers shop materials rope bridge beauty salon amusement arcade street waiting room airplane cabin dolmen golf course bus interior corn ice skating beach home theater zen garden art studio park procenium martial arts gym coast living room bullring slum television studio catwalk water tower ice cream parlor bank vault creek textures textures color places object scene parts material creek vineyard supermarket kindergarden classroom locker room building facade bridge aqueduct pasture river objects car orchard boxing ring music studio crevasse ball pit amusement park archive galley fountain lobby topiary garden jewelry shop bar asia earth amphitheater home batters box parlor rubble shopfront clothing store kasbah kitchen junkyard bookstore shopping swimming natural history museum dam barn apartment medina hot spring mountain campsite ballroom arrival driveway excavation window seat construction site clean room utility room artists loft basketball football roundabout screen crosswalk body headboard wheel head hair roof shop window coach back pillow balcony torso monitor food tile carpet paper dotted scenes cracked lined cobwebbed potholed parts chequered material spiralled interlaced zigzagged freckled veined frilly banded striped perforated grooved grid lacelike scaly porous meshed swirly pleated gauzy textures crystalline crosshatched matted paisley studded pitted bubbly sprinkled bumpy marbled braided red continued units objects scene alexnet egomotion textures dog cat grass tree bicycle sea sky water road car painting windowpane mountain motorbike book sidewalk bus mountain snowy wheel hair head ear muzzle arm leg screen food zigzagged dotted chequered banded cobwebbed perforated striped frilly spiralled studded honeycombed grid meshed sprinkled veined porous cracked interlaced crosshatched red yellow objects scene part units sky ceiling grass tree head banded lined dotted perforated grid chequered crosshatched spiralled units bus airplane bed train pool table dog car horse house road boat motorbike tree grass water seat cat shelf ground sea skyscraper stove fence work surface toilet bottle sofa sink building cockpit conference room street closet classroom nursery ball pit highway corridor airport terminal attic bathroom castle art gallery alley dining room kitchen bar mountain snowy living room staircase shoe shop skyscraper lighthouse water tower subway interior bowling alley objects waiting room landing deck laundromat auditorium gas station crevasse ice skating windmill bus interior pantry amphitheater supermarket utility room building facade bookstore reception viaduct car rope bridge airplane cabin bridge park sauna badlands pagoda playroom galley apartment campsite kindergarden classroom artists loft bullring carrousel scenes junkyard bow creek ski slope mountain crosswalk screen head shop window coach food parts potholed paisley materials striped cobwebbed lined sprinkled interlaced studded banded dotted frilly spiralled woven grooved meshed crystalline bumpy scaly braided alexnet imagenet textures alexnet puzzle solving sky ball pit veined chequered meshed striped dotted perforated studded lacelike frilly cobwebbed red units parts units airplane bus car road grass plant windowpane person bed train motorbike horse tree house water pool table table toilet boat mountain dog painting book curtain building grandstand bridge railing swimming pool tent signboard chair work surface skyscraper bottle chandelier seat stove shelf track cabinet cushion objects sofa snow waterfall pillow sink sea rock desk sidewalk plate palm ground sky scenes swivel chair fence ceiling ball pit skyscraper mountain snowy laundromat closet cockpit shoe shop corridor parts conference room material castle windmill staircase classroom playground pantry bowling alley cemetery highway street amusement park lighthouse creek nursery corn textures wheat colorshopping attic youth hostel sandbox bar reception childs room bookstore pagoda slum art gallery butchers shop hair wheel drawer screen head crosswalk roof balcony seat cushion pot shop window hand carpet food chequered striped meshed paisley dotted banded swirly cracked lined perforated grid sprinkled lacelike interlaced frilly spiralled potholed woven freckled studded veined red objects scene units car grass airplane mountain painting tree ceiling dog bus road pool table water horse skyscraper plant motorbike cat bed sea track sink stairway shelf work surface building house waterfall ground sidewalk book chair objects sky windowpane toilet person railing scenes washer signboard table chandelier bird parts ball pit skyscraper materials mountain snowy closet swimming coast corridor auditorium creek highway conference room hair head screen wheel roof headboard crosswalk seat cushion drawer leg arm body shop window food textures chequered color perforated striped grid spiralled dotted swirly lined meshed studded paisley banded cobwebbed grooved porous honeycombed woven interlaced red units car tree grass sea mountain highway head hair banded lined chequered studded zigzagged striped perforated grid cracked meshed gauzy dotted units alexnet ambient sound googlenet places colors places textures places scenes water grass tree car plant windowpane road mountain airplane skyscraper dog sea ceiling building person horse bed track book pool table cabinet chair painting waterfall sidewalk sink shelf sky house stove bus mountain snowy ball pit pantry building facade skyscraper street hair wheel head screen crosswalk shop window food wood lined dotted banded studded grid honeycombed paisley zigzagged meshed cracked chequered perforated sprinkled potholed grooved pleated matted freckled swirly spiralled woven cobwebbed red alexnet places objects grass road sky water tree dog ball pit mountain snowy building facade highway chequered lined banded porous lacelike veined grid frilly perforated cracked potholed freckled studded woven purple orange red yellow blue pink green units fig comparison unique detectors types variety architectures results project page alexnet video tracking objects orm tio tio tio tio tio inc inc inc inc inc object scene part material texture color googlenet object scene part material texture color number unique detectors number unique detectors number unique detectors object scene part material texture color number unique detectors alexnet object scene part material texture color ise elt ise wis elt elt wis elt elt wis object scene part material texture color scene part material texture color colorization tracking dotted texture chequered texture perforated texture head part orm tio tio tio tio tio inc inc inc inc inc head part sky object grass object sky object elt wis elt wis wis wis wis elt elt elt color rac kin tra ovi riz zle der fig top ranked concepts three top categories four selfsupervised networks object part detectors emerge audio detectors person heads also appear puzzle colorization variety texture concepts dominate models training puzzle car object scene part material texture color chequered texture head part fig semantic detectors emerge across different supervision primary training task models use alexnet architecture tested fig shows histogram object detectors identified inside resnet densenet trained imagenet places respectively largest number unique object detectors among networks emergent detectors differ across training data source architecture frequent object detectors two networks trained imagenet dog detectors dog categories classes imagenet training set fig shows examples object detectors grouped object categories object category visual appearance unit detector varies within network also across different networks densenet resnet good detectors bus airplane iou fig showcases unique interpretable units types variety networks fig shows unique interpretable detectors different layers different network architectures trained observe object scene detectors emerge higher layers across architectures alexnet vgg googlenet resnet suggests representation ability increases layer depth compositional structure cnn layers deeper layers higher capacity represent concepts larger visual complexity objects scene parts measurements confirm conclude higher network depth encourages emergence visual concepts higher semantic complexity audio number unique detectors scene part material texture color number unique detectors number unique detectors scene part material texture color number unique detectors number unique detectors alexnet imagenet imagenet googlenet imagenet imagenet fig trained five conv layers comparison interpretability layers alexnet googlenet object selected layers vgg googlenet object object object alexnet resnet included cnns predict neighborhoods two image patches trains networks colorizing images totally investigate networks trained different learning tasks different supervisions affect internal representations compare interpretability deep visual representations resulting learning supervised learning keep network architecture alexnet model one exception recent model transinv uses vgg base network results shown fig observe training creates largest number unique detectors models create many texture detectors relatively object detectors apparently supervision primary task much weaker inferring interpretable concepts supervised training large annotated dataset form makes difference example colorization model trained colorless images almost color detection units emerge hypothesize emergent units represent concepts required solve primary task fig shows typical visual detectors identified cnn models models audio puzzle object part detectors emerge detectors may useful cnns solve primary tasks audio model trained associate objects sound source may useful recognize people cars puzzle model trained align different parts objects scenes image colorization tracking recognizing textures might good enough cnn solve primary tasks colorizing desaturated natural image thus unsurprising texture detectors dominate representations learning recently many work explored novel paradigm unsupervised learning cnns without using millions annotated images namely learning example trains deep representations captioning images compare representations supervised learning learning train cnn scratch using coco captioning dataset alexnet number unique detectors train scratch object scene part material texture color validation accuracy fig example images coco captioning dataset image captioning model network dissection result training scratch using supervision captioning images leads lot emergent object detectors number unique detectors object scene part material texture color number detectors object scene part material texture color training iteration fig evolution interpretability training iterations accuracy validation iteration also plotted baseline model trained iterations marked red line lin lin pou fig effect regularizations interpretability cnns supervision captioning images generates natural language sentence describe image contents specially use image captioning data coco dataset images coco dataset annotated five image captions train cnn plus lstm image captioning model similar features used input lstm generating image captions architecture network dissection results last convolutional layer shown see many object detectors emerge cnn conclude supervision natural language captions contains semantics train cnn training conditions training conditions number training iterations dropout batch normalization random initialization known affect representation learning neural networks analyze effect training conditions interpretability take baseline model prepare several variants using alexnet architecture variants randomly initialize weights train number iterations variant nodropout remove dropout layers baseline model variant batchnorm apply batch normalization convolutional layer baseline model nearly accuracy validation set variant without dropout accuracy variant batch norm accuracy fig shows interpretability units cnns different training conditions find several effects comparing different random initializations models converge similar levels interpretability terms unique detector number total detector number matches observations convergent learning discussed network without dropout texture detectors emerge fewer object detectors batch normalization seems decrease interpretability significantly batch normalization result serves caution discriminative power property representation measured intuition loss interpretability batch normalization batch normalization whitens activation layer smooths scaling issues allows network easily rotate axes intermediate representations training whitening apparently speeds training may also effect similar random rotations analyzed sec destroy interpretability discussed sec however interpretability neither prerequisite obstacle discriminative power finding ways capture benefits batch normalization without destroying interpretability important area future work fig plots interpretability snapshots baseline model different training iterations along accuracy validation set see object detectors part detectors begin emerging iterations iteration processes batch images find evidence transitions across different concept categories training example units turn texture material detectors becoming object part detectors fig keep track six units different training iteration observe units start converging semantic concept early stage example starts detecting mountain snowy early iteration also observe units evolve time detect road first start detecting car airplane respectively fig interpretations units change iterations row shows interpretation one unit network target domain commonly used transfer learning deep features network show good generalization across different domains network also makes training converge faster results better accuracy especially enough training data target domain analyze happens inside representation interpretation internal units evolve transfer learning run two sets experiments imagenet places want see individual units mutate across domains interpretability results model checkpoints different iteration plotted fig see training indeed converges faster compared network trained scratch places fig interpretations units also change finetuning example number unique object detectors first drop keep increasing network trained imagenet slowly dropping network trained places imagenet fig shows examples individual unit evolution happening network trained imagenet network trained imagenet network show six units interpretation beginning finetuning end example network imagenet detects white dogs first mutates detect waterfall detect dogs first mutate detect horse cow respectively lot scene categories places pasture corral contain animals hand network imagenet lot units mutate various kinds dog detectors interestingly though units mutate detect different concepts concepts share similarity colors textures fig zoom two units two finetuning processes plot history concept evolution see units switch top ranked label several times converging concept imagenet flipped white crystalline reaches waterfall concept hand units switch faster example imagenet switches hair dog early stage layer width interpretability alexnet resnet cnns visual recognition grown deeper quest higher classification accuracy depth shown important high discrimination ability seen sec interpretability increase depth well however width layers number units per number unique detectors transfer learning places imagenet validation accuracy imagenet imagenet object scene part material texture color training iteration training iteration fig alexnet imagenet alexnet imagenet imagenet imagenet fig units mutate network imagenet network imagenet six units shown semantics beginning finetuning end layer less explored one reason increasing number convolutional units layer significantly increases computational cost yielding marginal improvements classification accuracy nevertheless recent work shows carefully designed wide residual network achieve classification accuracy superior commonly used thin deep counterparts explore width layers affects interpretability cnns preliminary experiment test width affects emergence interpretable detectors remove layers alexnet triple number units units units triple number units previous conv layers except standard alexnet finally put global average pooling layer fully connect imagenet imagenet fig history one unit mutation imagenet top imagenet low number unique detectors object scene part material texture color alexnet fig comparison standard alexnet widening layer brings emergence detectors networks trained pooled activations final class prediction training obtain similar classification accuracy validation set standard alexnet accuracy lower higher many emergent unique concept detectors conv layers shown fig also increased number units number unique concepts significantly increase may indicate limit capacity alexnet separate explanatory factors may indicate limit number disentangled concepts helpful solve primary task scene classification performance test split compute classification accuracy averaged across classes include classification accuracies six image datasets using deep features plotted fig see deep features supervised trained networks perform much better ones trained networks networks trained places better features datasets networks trained imagenet better features datasets fig plots number unique object detectors representation representation classification accuracy three selected datasets see positive correlation thus supervision tasks encourage emergence concept detectors may also improve discrimination ability deep features interestingly object centric dataset best discriminative representation representation fewer unique object detectors compared hypothesize accuracy representation applied task dependent number concept detectors representation well concept detectors captures characteristics hidden factors transferred dataset discrimination interpretability activations higher layers cnns often used generic visual features noted deep features generalizing well image datasets interesting bridge generalization deep visual representation generic visual features interpretability first benchmark deep features several networks six image classification datasets discriminative power network feed images extract activation last convolutional layer visual feature train linear svm train split evaluate explaining predictions deep features interpret units inside deep visual representation show unit activation along interpreted label used explain prediction given deep features previous work uses weighted sum unit activation maps highlight image regions informative prediction decouple individual unit level segment informative image regions first plot units linear svm trained rank elements feature according svm weights obtain elements deep features contribute class elements units act explanatory factors call top ranked units associated output class units fig shows classspecific units one class respectively example walking dog class top three classspecific units two dog detection unit one person detection unit picnic area class top three units plant detection unit grass detection unit fence detection unit intuitive match visual detectors classes explain suggests visual detectors cnns behave visual features use individual units identified concept detectors build explanation individual image prediction given classifier procedure follows given image let unit activation deep feature resnet gap activation represents value summed activation map unit let top prediction svm response svm learned weight get top ranked units figure ranking unit activations weighted svm weight accuracy accuracy net ima ybr ene rid riz ion ack ing tric zle fra ion bri ale lex lex pla ace riz tra king tce zzl tric oti fra ran vin accuracy net sne accuracy vglac renet lacces rid bri lac oog ces riz tex ion ybr pla puz zle number unique nobject detectors lac tra aud lacack izat ing ion ego ent zle ric tric fra tion ogl accuracy ace pla esn ace pla age bri ces gunique object number detectors rid lac ace ace lac lac lac genrid sch nte riz riz ack ack dio ing ing tce zzl tce ntr ntr fra otio fra tio vin accuracy frameorder objectcentric tracking puzzle moving audio crosschannel frameorder context egomotion colorization audio tracking moving crosschannel colorization frameorder puzzle context egomotion objectcentric accuracy objectcentric tracking puzzle moving audio crosschannel frameorder context egomotion colorization accuracy objectcentric tracking audio puzzle crosschannel context egomotion colorization frameorder moving accuracy number unique object detectors number unique object detectors fig classification accuracy datasets accuracy deep features six image accuracy objectcentric tracking moving puzzle audio crosschannel colorization egomotion context number unique object detectors number unique object detectors objectcentric tracking moving audio crosschannel frameorder context puzzle egomotion colorization accuracy fig number unique object detectors last convolutional layer compared representations classification accuracy three datasets supervised red unsupervised green representations clearly form two clusters top predicted class simply upsample activation map top ranked unit segment image image segmentations using individual unit activation plotted fig unit segmentation explain prediction explicitly example prediction first image gardening explanatory units detect plant grass person flower pot prediction second image riding horse explanatory units detect horse fence dog also plot incorrectly predicted samples figure segmentation gives intuition classifier made mistakes example first image classifier predicts cutting vegetables rather true label gardening second unit incorrectly considers ground table onclusion network dissection translates qualitative visualizations representation units quantitative interpretations measurements interpretability found units deep representation significantly interpretable expected basis representation space investigated interpretability deep visual representations resulting different architectures training supervisions training conditions furthermore shown interpretability deep visual representations relevant power representation generalizable visual feature conclude interpretability important property deep neural networks provides new insights hierarchical structure work motivates future work towards building interpretable explainable systems acknowledgments work partially funded darpa xai program work also partly supported national science foundation grants vannevar bush faculty fellowship program sponsored basic research office assistant secretary defense research engineering funded office naval research grant mit big data initiative csail toyota research institute mit csail joint research center google amazon awards hardware donation nvidia corporation supported facebook fellowship eferences zhou khosla lapedriza oliva torralba object detectors emerge deep scene cnns international conference learning representations modolo ferrari semantic parts emerge convolutional neural networks vondrick pirsiavash torralba generating videos scene dynamics bengio courville vincent representation learning review new perspectives ieee transactions pattern analysis machine intelligence vol bau zhou khosla oliva torralba network dissection quantifying interpretability deep visual representations proc cvpr srivastava hinton krizhevsky sutskever salakhutdinov dropout simple way prevent neural networks journal machine learning research vol ioffe szegedy batch normalization accelerating deep network training reducing internal covariate shift zeiler fergus visualizing understanding convolutional networks proc eccv girshick donahue darrell malik convolutional networks accurate object detection segmentation ieee transactions pattern analysis machine intelligence mahendran vedaldi understanding deep image representations inverting proc cvpr simonyan vedaldi zisserman deep inside convolutional networks visualising image classification models saliency maps international conference learning representations workshop mahendran vedaldi understanding deep image representations inverting proc cvpr dosovitskiy brox generating images perceptual similarity metrics based deep networks advances neural information processing systems nguyen dosovitskiy yosinski brox clune synthesizing preferred inputs neurons neural networks via deep generator networks advances neural information processing systems zhang ren sun deep residual learning image recognition proc cvpr razavian azizpour sullivan carlsson cnn features astounding baseline recognition agrawal girshick malik analyzing performance multilayer neural networks object recognition proc eccv yosinski clune bengio lipson transferable features deep neural networks advances neural information processing systems szegedy zaremba sutskever bruna erhan goodfellow fergus intriguing properties neural networks nguyen yosinski clune deep neural networks easily fooled high confidence predictions unrecognizable images proc cvpr yosinski clune lipson hopcroft convergent learning different neural networks learn representations zhang bengio hardt recht vinyals understanding deep learning requires rethinking generalization international conference learning representations doersch gupta efros unsupervised visual representation learning context prediction proc cvpr noroozi favaro unsupervised learning visual representations solving jigsaw puzzles proc eccv jayaraman grauman learning image representations tied proc iccv agrawal carreira malik learning see moving proc iccv wang gupta unsupervised learning visual representations using videos proc cvpr zhang isola efros colorful image colorization proc eccv springer autoencoders unsupervised learning prediction proc cvpr owens mcdermott freeman torralba ambient sound provides supervision visual learning proc eccv quiroga reddy kreiman koch fried invariant visual representation single neurons human brain nature vol correct label gardening correct label brushing fig segmenting images using top activated units weighted class label deep feature correctly predicted samples incorrectly predicted samples images walking dog images picnic area fig units one class class show three sample images followed top units ranked class weight linear svm predict class svm weight detected concept name iou score shown unit zhou zhao puig fidler barriuso torralba scene parsing dataset proc cvpr bell bala snavely intrinsic images wild acm trans graphics siggraph mottaghi chen liu cho lee fidler urtasun yuille role context object detection semantic segmentation wild proc cvpr chen mottaghi liu fidler urtasun yuille detect detecting representing objects using holistic models body parts proc cvpr cimpoi maji kokkinos mohamed vedaldi describing textures wild proc cvpr van weijer schmid verbeek larlus learning color names applications ieee transactions image processing vol krizhevsky sutskever hinton imagenet classification deep convolutional neural networks advances neural information processing systems szegedy liu jia sermanet reed anguelov erhan vanhoucke rabinovich going deeper convolutions proc cvpr simonyan zisserman deep convolutional networks image recognition huang liu weinberger van der maaten densely connected convolutional networks proc cvpr russakovsky deng krause satheesh huang karpathy khosla bernstein imagenet large scale visual recognition challenge int journal computer vision zhou lapedriza xiao torralba oliva learning deep features scene recognition using places database advances neural information processing systems zhou lapedriza khosla oliva torralba places million image database scene recognition ieee transactions pattern analysis machine intelligence mikjjsra zitnick hebert shuffle learn unsupervised learning using temporal order verification proc eccv gao jayaraman grauman representation learning unlabeled videos pathak krahenbuhl donahue darrell efros context encoders feature learning inpainting proc cvpr wang gupta transitive invariance visual representation learning arxiv preprint diaconis random matrix notices ams vol lin maire belongie hays perona ramanan zitnick microsoft coco common objects context european conference computer vision springer vinyals toshev bengio erhan show tell neural image caption generator proceedings ieee conference computer vision pattern recognition zagoruyko komodakis wide residual networks classifying events scene object recognition proc iccv yao jiang khosla lin guibas human action recognition learning bases action attributes parts proc iccv quattoni torralba recognizing indoor scenes proc cvpr xiao hays ehinger oliva torralba sun database scene recognition abbey zoo proc cvpr fergus perona learning generative visual models training examples incremental bayesian approach tested object categories computer vision image understanding griffin holub perona object category dataset zhou khosla lapedriza oliva torralba learning deep features discriminative localization proc cvpr bolei zhou candidate computer science artificial intelligence lab csail massachusetts institute technology received degree information engineering chinese university hong kong degree biomedical engineering shanghai jiao tong university research interests computer vision machine learning award recipient facebook fellowship microsoft research asia fellowship mit greater china fellowship david bau phd student mit computer science artificial intelligence laboratory csail received mathematics harvard computer science cornell coauthored textbook numerical linear algebra software engineer microsoft google developed ranking algorithms google image search research interest interpretable machine learning aude oliva principal research scientist mit computer science artificial intelligence laboratory csail french baccalaureate physics mathematics received two degrees cognitive science institut national polytechnique grenoble france joined mit faculty department brain cognitive sciences csail research vision memory spanning human perception cognition computer vision human neuroscience received national science foundation nsf career award guggenheim vannevar bush fellowships antonio torralba received degree telecommunications engineering telecom bcn spain degree signal image speech processing institut national polytechnique grenoble france spent postdoctoral training brain cognitive science department computer science artificial intelligence laboratory mit professor electrical engineering computer science massachusetts institute technology mit torralba associate editor international journal computer vision served program chair computer vision pattern recognition conference received national science foundation nsf career award best student paper award ieee conference computer vision pattern recognition cvpr aggarwal prize international association pattern recognition iapr
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semantic photometric bundle adjustment natural sequences rui zhu chaoyang wang lin ziyan wang simon lucey robotics institute carnegie mellon university nov chaoyanw chenhsul slucey abstract problem obtaining dense reconstruction object natural sequence images long studied computer vision classically problem solved application bundle adjustment recently excellent results attained application photometric bundle adjustment pba methods directly minimize photometric error across frames fundamental drawback pba however reliance view points object object surface well textured circumvent limitations propose semantic pba incorporates object prior obtained deep learning within photometric bundle adjustment problem demonstrate state art performance comparison leading methods object reconstruction across numerous natural sequences figure overview proposed method semantic photometric bundle adjustment given sequence images small motion top row recover full dense shape object well camera poses global coordinate system middle row method enables reprojection image plane estimated cameras bottom row optimize photometric consistency well silhouette depth constraints introduction paper primarily concerned goal obtaining dense object reconstructions short natural image sequences one obvious strategy employ classical bundle adjustment across sequence simultaneously recover camera pose points although reliable strategy problematic recover points observed image sequence density reconstruction dependent textured surface object across image sequence recently photometric extensions bundle adjustment proposed directly minimize photometric consistency frames respect pose points borrowing upon terminology shall refer methods collectively herein photometric bundle adjustment pba pba recently proved advantageous classical problems dense reconstructions required due ability directly minimize photometric consistency even recent innovations pba still however fundamentally limited performance numerous efforts within computer vision community bring semantic prior task reconstruction convolutional neural networks cnn proving remarkably useful task one provided scene object category specific labels priors powerful characteristic semantic cnn methods ability circumvent fundamental limitations example offer strategies inferring dense reconstruction object single image even substantial portion projected points recently semantic cnn strategies proposed attempt incorporate multiple frames previous efforts trying attack problem reconstruction supervised learning problem geometry largely treated label predicted although attractive simplicity strategy inherent drawbacks first geometric labels problematic obtain hand labeling error prone rendering lack necessary realism second predicted labels network models adhere geometric constraints photometric consistency making results unreliable recently application geometric constraints within offline cnn learning process entertained including reprojected silhouette matching depth matching even photometric consistency fundamental problem emerging methods however geometric constraints enforced test time dramatically reducing effectiveness given concerns argue instead incorporating geometric constraints semantic cnn strategies offline one instead incorporate object semantics within pba pipeline demonstrate fig results section strategy gives best worlds semantic knowledge object photometric consistency paper propose enhanced semantic pba method works natural sequences classic pba give extensive evaluations synthetic natural sequence domains summarize contributions follows provide first approach kind knowledge semantic pba natural sequences gives global camera poses frame dense shape accuracy denser depth maps systematically evaluate local optimality proposed optimization pipeline well enhanced objective takes use classic pba results initialization regularizer method collect new dataset task shape reconstruction consisting rendered sequences full ground truth cameras depth maps shape canonical pose well natural sequences annotated shapenet models making dataset feasible evaluating pba methods camera estimations methods recover aligned shapes depth maps learning shapenet related work photometric bundle adjustment photometric bundle adjustment pba optimization based method sitting entirely upon visual cue photometric consistency across input frames pba shape recovered jointly optimizing depth maps corresponding visible pixels template frames well camera motion result formulation classic pba solely recover geometry scene completely agnostic semantics works pba aims small motion videos particular shape reconstructing deep learning previously mentioned early deep learning methods solve task object shape reconstruction supervision shape labels recently emerging school thought seeks bring geometry back task including reprojected silhouette matching depth matching even photometric consistency one issue methods assume known cameras global frame fact strong assumption hold natural sequences global camera poses readily availabe others account camera motion creates gap classic pba camera motions instead direct output semantic pba recent work zhu also proposed apply shape priors within pba object reconstruction spite similarity formulation problem zhu approach problematic number ways first performance zhu relies heavily initialization point given cnn predictors trained predominantly rendered images suspicious reliability natural sequences instead utilize reliable source relative camera pose pba initialization second due limitations method weak perspective camera model assumption unreliable initialization source zhu evaluation restricted rendered sequence thus conduct quantitative comparisons actual pba methods camera pose error depth error real world sequences give extensive evaluation method real sequences third zhu give proper analysis characteristics objective function results using inadequate optimization techniques approach paper inspecting property different cost functions propose robust efficient optimization pipeline notation vectors represented bold font matrices bold scalars italicized characters represent sets denote lth sample set images shapes use subscript calligraphic symbols denote functions images defined sampling function pixel coordinates generator offline pba pipeline initializers camera camera pipeline approach camera model figure left pipeline optimization given input sequence first run offline pba pipeline provide rough estimation depth maps future depth map constraint camera motion initialization camera motion parameters style pose initializer also called initialize global pose template frame well style vector starting initialization optimize variables convergence pipeline combined objective photometric consistency lph silhouette matching error lcd inverse depth error linvd right perspective camera model adopt camera model blendertm centering object world frame positioning camera identity pose grey along axis looking origin point show figure camera transformation identity camera parameterized exponential twist solid blue arrow translation red arrows camera local axes approach preliminary camera model assume perspective camera known intrinsics camera extrinsics parameterized concatenation exponential coordinates also known twist rotation translation vector camera projection function written given short window frames pba define first frame target frame frame subsequent frames source frames relative camera pose target frame source frame denoted global pose thus computed composing relative pose source frame global pose target frame define pose composition rule reprojection one point onto frame corresponding global pose framed sampling image reprojected pixel location reprojection reprojection point set given camera pose viewed first reasoning visible part point set masking function mask function implemented projecting points enlarged inverse depth plane factor perform max pooling neighbourhood figure visible point biggest inverse depth mask function gives indices visible points inverse depth map overview method takes rgb sequence taken calibrated camera moving around object category cars airplanes chairs object assumed known rich repository aligned cad models shapenet category define world coordinate system one attached objects chosen cad dataset calibrated perspective camera model parametrized full rotation translation see goal method recover full shape object world frame well parameters camera pose frame adopt shape prior zhu learn shape space repository shapenet cad models use dense point cloud shape parameterization work considering learning shape space point clouds made possible several works shape prior learned categoryspecific point cloud generator written function style vector represents object style output shape prior set ated points defined poses total frames break pose parameters two sets global camera pose target frame relative camera pose source frame corresponding target frame overall pipeline method illustrated fig formulate inference style vector camera pose parameters optimization steps geometric objectives parameters initialized initialization pipeline see optimization steps taken minimize objective parameter space paper propose take advantage cheap rough outputs methods regularize optimization procedure frame get cheap segmentation masks silhouettes recent instance segmentation method fcis considering traditional pba methods gives results inverse depth camera motion also borrow readily available although outputs pba pipelines openmvs add another inverse depth loss estimated depth also take advantage camera motion estimation initialize relative camera pose source frame target frame optimization objective photometric consistency basic objective formulated photometric consistency corresponding pixel pairs target frame source frame classic pba methods usually track set sparse points frames window formulation able get dense correspondence automatically derived reprojection considering visible points may differ frame due camera motion occlusion work formulate photometric consistency lph mpl huber loss silhouette error inverse depth error extra constraints zhu silhouette error utilized extra constraint objective adopt objective estimated silhouette fcis produce instance segmentation show later constraint still effective although masks errorprone write silhouette distance frame chamfer distance set pixel locations side rough silhouette ones projected camera model lcd min min finally apart cheap camera motion able get depth map frame pba pipeline case formulate extra objective term inverse depth error linvd given reprojection module note updated fly considering getting confident camera poses frames optimization steps robustly solved finding scale best aligns estimated camera poses ones offline estimator specifically solve source frame solution average arg min combined objective given lph lcd linvd ablative study weight factors included appendix initialization style pose initialization improve upon existing pipeline provide initialization style template frame camera pose style unlike based regressor adopted use recurrent network architecture leverage sequential information effort alleviate ambiguity style single viewpoint details architecture regressor training process well dataset included appendix generate accurate style available give accurate estimation scale factor instead seek bring estimation loop aligning estimated inverse map reprojected inverse depth arg min camera motion initialized solving solution figure local cost surface yellow dot marks optimum yellow line shows search space initialized offline camera motion vectors exploit cheap silhouette mask background find coarse pose initialization first utilize blendertm render templates varying camera poses retrieve coarse pose finding one template maximum iou target silhouette camera motion initialization one observation photometric consistency error problematic optimization objective one hand lph locally solvers traditional pba template term fixed residual linear problem locally convex problem worsened way variables initialized zhu initialized zeros illustrate show fig plotting cost surface local perturbation show cost surface highly initialization optimum variables initialized far ground truth red arrows however show camera motion parameters initialized correctly search space constrained yellow line better curvature better convergence green arrows inspired observation beginning pipeline run pba pipeline acquire camera poses every frame well point cloud formulation following relation camera model left estimator right unfortunately correspondence use equ initializing camera motion parameters optimization steps optimization pipeline given reasonable initialization steps solve objective based methods particularly found solver gives efficient solution problem summarize optimization pipeline algorithm algorithm optimization objective procedure initializer openmvs equ step maximum iterations update update update update equ return evaluation data preparation rendered data enable evaluation methods zhu feasible rendered domain follow zhu rendering small baseline sequences shapenet cars please refer appendix statistics inherently identical zhu apart perspective camera versus zhu point frame cloud frame frame frame pred seg init figure left examples test sequences show left three examples rendered sequence natural sequence model car real car respectively right visualization style pose initialization also show quantitative results style pose initialization natural sequences three columns right correspond first frame sequence foreground segmentation first frame style pose initialization viewed rendered image respectively natural data view absence natural video dataset collect test set mixture sequences toy model cars real cars carefully choose models ground truth cad available shapenet video shot iphone around rotational motion moderate translation images scaled videos collected toy car models videos real cars evaluation sequence annotate template frame last frame ground truth cad models pose samples test data found fig also visualize qualitative results initialization natural sequences fig although style retrieved regressor precise color details style regressor able yield style prediction close shape evaluation rendered sequences section give qualitative quantitative results methods classic pba methods openmvs ham pba pipeline specifically optimized small motion videos learning based methods gives shapes canonical poses experiments set weights optimization objective use unnormalized rgb values range ablative study different settings included appendix since evaluating synthetic sequences blendertm access full ground truth camera pose every frame world coordinate system well ground truth dense depth map dense shape however given formulation output full shape manner openmvs ham give point cloud entire scene best effort able align three outputs ground truth shape instead choose follow ham measure depth error recovered points reprojecting shape onto image plane estimated cameras compare openmvs ham give relative camera motion every source frames first frame offer ground truth camera pose first frame two methods find transformation align camera pose first frame ground truth scale ambiguity solved shape camera poses methods ideally aligned world coordinate system able measure camera error calculating camera position error distance estimated camera center ground truth camera orientation error acute angle estimated camera orientation ground truth report average error depth maps camera poses table statistics fig results show method achieves comparable camera error openmvs slightly worse ham specifically optimized small motion videos better camera tracking depth map error achieve performance outperforming openmvs ham addition much denser results thanks shape prior produces full shapes moreover thanks shape prior show fig need observations object give confident results even two frames classic methods start considerable amount motion perform camera tracking finally experiment rendered sequences perturbing upon ground truth calculate average convergence show fig method achieves better convergence face large initialization error pose evaluation natural sequences evaluate methods others objectcentric dataset collected dataset includes sequences mixture toy cars real cars sequences annotated aligned cad models retrieved shapenet dataset considering possible get camera poses frames human annotation evaluate depth error annotated first frame sequence well density reprojected points ground truth quantitative frame frame frame frame frame frame frame ham openmvs input frame figure results methods natural sequences comparison openmvs ham deep learning based methods openmvs ham align cameras world coordinate frame note method automatically produces camera poses world frame draw aligned shape estimated camera trajectory blue orientation red annotated camera black two frames marked black dot also project shape estimated camera color reprojected points inverse depths brighter closer last row show sample results left fan right results reported table also show qualitative results natural sequences model cars plus real cars fig sequence consists frames roughly degree camera rotation moderate translation show achieve comparable camera poses denser inverse depths openmvs ham additionally recovers semantic information including full shape detached map global cameras sequence ham gives degraded solution fails camera tracking deification due little motion frame significant lighting change sequence attempt give part results zhu ham openmvs depth error density rendered sequences cam location error cam orientation error natural sequences depth error density table quantitative comparison rendered natural sequences pixel count openmvs ham zhu zhu depth error threshold depth error figure depth error rendered test set left axis shows histogram depth error distribution right axis gives percentage pixels threshold openmvs ham zhu density average depth error number views figure convergence analysis zhu robust initialization error evaluate approach rendered natural settings classic pba methods deep learning based methods prove capable produce dense full shape world coordinates well depth maps quality openmvs ham zhu number views figure depth error rendered test set left axis shows histogram depth error distribution right axis gives percentage pixels threshold figure readers reference also notice learning based methods mostly trained rendered images suffer domain gap test natural sequences conclusion paper propose method semantic photometric bundle adjustment shape reconstruction natural sequence exerts geometric constraints camera pose well full shape generated learned semantic shape prior extensively references alismail browning lucey photometric bundle adjustment slam asian conference computer vision pages springer chang funkhouser guibas hanrahan huang savarese savva song xiao shapenet model repository technical report stanford university princeton university toyota technological institute chicago choy gwak chen savarese unified approach single object reconstruction arxiv preprint engel koltun cremers direct sparse odometry ieee transactions pattern analysis machine intelligence engel cremers direct monocular slam european conference computer vision pages springer fan guibas point set generation network object reconstruction single image arxiv preprint girdhar fouhey rodriguez gupta learning predictable generative vector representation objects european conference computer vision pages springer goldberg hed kaplan tarjan werneck maximum flows incremental search pages gwak choy chandraker garg savarese weakly supervised reconstruction adversarial constraint park jeon kweon highquality depth uncalibrated small motion clip proceedings ieee conference computer vision pattern recognition pages ham chang lucey singh monocular depth small motion video accelerated vision fifth international conference vision ieee gall zheng liu fang surfacenet neural network multiview stereopsis arxiv preprint kar malik learning stereo machine arxiv preprint lin kong lucey learning efficient point cloud generation dense object reconstruction arxiv preprint newcombe lovegrove davison dtam dense tracking mapping computer vision iccv ieee international conference pages ieee guibas deep hierarchical feature learning point sets metric space arxiv preprint tateno tombari laina navab dense monocular slam learned depth prediction arxiv preprint triggs mclauchlan hartley fitzgibbon bundle adjustmenta modern synthesis international workshop vision algorithms pages springer tulsiani zhou efros malik supervision reconstruction via differentiable ray consistency arxiv preprint wang xue sun freeman tenenbaum marrnet shape reconstruction via sketches advances neural information processing systems zhang xue freeman tenenbaum learning probabilistic latent space object shapes via modeling advances neural information processing systems pages yan yang yumer guo lee perspective transformer nets learning object reconstruction without supervision advances neural information processing systems pages haozhi wei fully convolutional semantic segmentation zhou brown snavely lowe unsupervised learning depth video arxiv preprint zhu kiani galoogahi wang lucey rethinking reprojection closing loop shape reconstruction single image ieee international conference computer vision iccv oct zhu wang lin wang lucey objectcentric photometric bundle adjustment deep shape prior arxiv preprint
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jan straggler mitigation distributed optimization data encoding karakus ucla los angeles karakus yifan sun technicolor research los altos suhas diggavi ucla los angeles suhasdiggavi wotao yin ucla los angeles wotaoyin abstract slow running straggler tasks significantly reduce computation speed distributed computation recently approaches applied mitigate effect straggling embedding redundancy certain linear computational steps optimization algorithm thus completing computation without waiting stragglers paper propose alternate approach embed redundancy directly data allow computation proceed completely oblivious encoding propose several encoding schemes demonstrate popular batch algorithms gradient descent applied manner deterministically achieve sample path linear convergence approximate solution original problem using arbitrarily varying subset nodes iteration moreover approximation controlled amount redundancy number nodes used iteration provide experimental results demonstrating advantage approach uncoded data replication strategies introduction solving optimization problems become feasible distributed implementations however efficiency significantly hampered slow processing nodes network delays node failures paper develop optimization framework based encoding dataset mitigates effect straggler nodes distributed computing system approach readily adapted existing distributed computing infrastructure software frameworks since node computations oblivious data encoding paper focus problems form min min kxw represent data matrix vector respectively function mapped onto distributed computing setup depicted figure consisting one central server worker nodes collectively store matrix vector focus batch synchronous optimization methods delayed failed nodes significantly slow overall computation note asynchronous methods inherently robust delays caused conference neural information processing systems nips long beach usa stragglers although convergence rates worse synchronous counterparts approach consists adding redundancy encoding data respectively encoding matrix redundancy factor solving effective problem min minp minp kxw instead proceed computation iteration without waiting stragglers idea inserted redundancy compensate lost data goal design matrix nodes obliviously solve problem without waiting slowest nodes design parameter achieved solution approximates original solution arg minw sufficiently closely since machine learning data analysis tasks one typically interested exact optimum rather sufficiently good solution achieves good generalization error approximation could acceptable many scenarios note also use technique preclude use strategies references therein still implemented top redundancy embedded system potentially improve performance focusing gradient descent algorithms show spectral condition one achieve approximation solution solving without waiting stragglers show sufficient redundancy embedded updates sufficiently large yet strict subset nodes iteration possible deterministically achieve linear convergence neighborhood solution opposed convergence expectation see fig one adjust approximation guarantee increasing redundancy number node updates waited iteration another potential advantage strategy privacy since nodes access raw data still perform optimization task jumbled data achieve approximate solution although paper focus quadratic objectives two specific algorithms principle approach generalized general potentially objectives constrained optimization problems discuss section adding regularization term also simple generalization main contributions follows demonstrate gradient descent constant step size line search applied manner encoded problem achieves universal sample path linear convergence approximate solution original problem using fraction nodes iteration present three classes coding matrices namely equiangular tight frames etf fast transforms random matrices discuss properties iii provide experimental results demonstrating advantage approach uncoded data replication strategies ridge regression using synthetic data aws cluster well matrix factorization movielens recommendation task related work use data replication aid straggler problem proposed studied references therein additionally use coding distributed computing explored however works exclusively focused using coding computation level certain linear computational steps performed coded manner explicit operations performed step specifically used distributed matrix multiplication focused breaking large dot products shorter dot products perform redundant copies short dot products provide resilience stragglers considers gradient descent method architecture data sample replicated across nodes designs code exact gradient recovered long fewer certain number nodes fail however order recover exact gradient potential set stragglers required redundancy factor order number straggling nodes could mean large amount overhead system contrast show one converge approximate solution redundancy factor independent network size problem dimensions section technique also closely related randomized linear algebra sketching techniques used dimensionality reduction large convex optimization problems main difference literature proposed coding technique former focuses reducing kxm ksm figure left uncoded distributed optimization partitioning partitioned right encoded distributed optimization node stores instead uncoded case corresponds problem dimensions lighten computational load whereas coding increases dimensionality problem provide robustness result increased dimensions coding provide much closer approximation original solution compared sketching techniques encoded optimization framework figure shows typical computational model optimization left well proposed network consists machines encoded model right computing machine stores yei optimization process oblivious encoding data stored nodes optimization algorithm proceeds exactly nodes contained uncoded raw data iteration central server broadcasts current estimate worker machine computes sends server yei gradient terms corresponding partition note framework distributed optimization typically communication slow links constitute significant portion overall computation time consider strategy iteration server uses gradient updates first nodes respond iteration thereby preventing slow links straggler nodes stalling overall computation yea get indices first nodes respond iteration similarly given gradient approximation central server computes descent direction history gradients parameter estimates remaining nodes server either send interrupt signal simply drop updates upon arrival depending implementation next central server chooses step size chosen constant decaying needed compute step size exact line search workers compute assume central server hears fastest nodes denoted general compute factor choice goal especially focus case design encoding matrix sequence sets universally converges neighborhood note general scheme guaranteed converge traditionally batch methods like additionally although algorithm works encoded function goal provide convergence guarantee terms original function note exact line search expensive backtracking line search quadratic loss since requires single multiplication algorithms convergence analysis let smallest largest eigenvalues denoted respectively let given order prove convergence consider family matrices aspect ratio redundancy factor sufficiently large submatrix associated subset drop dependence brevity note similar restricted isometry property rip used compressed sensing except required submatrices form although condition needed prove convergence results practice proposed encoding scheme work well even exactly satisfied long bulk eigenvalues lie within small interval discuss several specific constructions relation property section gradient descent consider gradient descent constant step size following theorem characterizes convergence encoded problem algorithm theorem let computed using gradient descent updates set fastest workers constant step size satisfies sequences cardinality initial objective value proof provided appendix relies fact solution effective instantaneous problem corresponding subset lies set therefore gradient descent step attracts estimate towards point set must eventually converge set note order guarantee linear convergence need ensured property theorem shows gradient descent encoded problem based updates nodes results deterministically linear convergence neighborhood true solution sufficiently large opposed convergence expectation note property controlling redundancy factor number nodes waited iteration one control approximation guarantee designed properly see section optimum value original function reached although originally batch method requiring updates nodes stochastic variants also proposed recently key modification ensure convergence hessian estimate must computed via gradient components common two consecutive iterations nodes adapt technique scenario define gradient terms collected descent direction computed get inverse hessian estimate iteration computed min memory length descent direction computed step size determined exact line search using factor convergence result need another assumption matrix addition defining assume note requires one wait sufficiently many nodes finish overlap set fraction nodes thus matrix full rank satisfied assumption node delays satisfied expectation however condition required analysis algorithm may perform well practice even condition satisfied following lemma shows stability hessian estimate lemma satisfied exist constants inverse hessian estimate satisfies proof provided appendix based method using lemma show following result theorem let computed using described gradient updates machines line search updates machines satisfies sequences initial objective value proof provided appendix similar theorem proof based observation solution effective problem time lies bounded set around true solution gradient descent coding enables linear convergence deterministically unlike stochastic variants generalizations although focus quadratic cost functions two specific algorithms approach potentially generalized objectives form kxw simple convex function lasso constrained optimization kxw see well algorithms used problems fista next section demonstrate codes consider desirable properties readily extend scenarios code design consider three classes coding matrices tight frames fast transforms random matrices tight frames frame set vectors exist constants kuk frame tight satisfied case shown constants equal redundancy factor frame form rows tight frame ensures kxw ksxw solution encoded problem figure sample spectrum various constructions low redundancy large normalized figure sample spectrum various constructions high redundancy relatively small normalized therefore solution encoded problem satisfies optimality condition original problem well also strongly convex unique solution note since computation true general arbitrary full rank matrix addition property desired property encoding matrix fact equivalency extends beyond smooth unconstrained optimization convex constraint set well convex objective term subdifferential means tight frames promising encoding matrix candidates constrained optimization shown static equiangular tight frames allow close approximation solution constrained problems tight frame equiangular constant across pairs proposition welch bound let tight frame moreover equality satisfied equiangular tight frame therefore etf minimizes correlation individual elements making submatrix close orthogonal possible promising light property specifically evaluate paley hadamard etfs confused hadamard matrix discussed next experiments also discuss steiner etfs appendix enable efficient implementation fast transforms another computationally efficient method encoding use fast transforms fast fourier transform fft chosen subsampled dft matrix fast walshhadamard transform fwht chosen subsampled real hadamard matrix particular one insert rows zeroes random locations data pair take fft fwht column augmented matrix equivalent randomized fourier hadamard ensemble known satisfy rip high probability random matrices natural choice encoding using random matrices although random matrices computational advantages fast transforms optimalitypreservation property tight frames eigenvalue behavior characterized analytically particular using existing results eigenvalue scaling large gaussian matrices union bound shown max min figure left sample evolution uncoded replication hadamard fwht cases right runtimes schemes different values number iterations scheme note essentially captures delay profile network reflect relative convergence rates different methods denotes ith singular value hence sufficiently large redundancy problem dimension random matrices good candidates encoding well however finite even general encoding scheme optimum original problem recovered exactly property redundancy requirements using analytical bounds sian matrices one see matrices satisfy independent problem dimensions number nodes although tight eigenvalue bounds subsampled etfs numerical evidence figure suggests may satisfy smaller random matrices thus believe required redundancy practice even smaller etfs note theoretical results focus extreme eigenvalues due analysis practice energy gradient associated bulk eigenvalues following proposition suggests mostly also see figure means even satisfied gradient solution approximated closely modest redundancy following result consequence cauchy interlacing theorem definition tight frames proposition rows chosen form etf redundancy eigenvalues equal numerical results ridge regression synthetic data aws cluster generate elements matrix elements dimensions regularization parameter solve problem minw evaluate hadamard matrix redundancy encoded using fwht fast encoding data replication uncoded schemes implement distributed described section amazon cluster using python package worker node instances single central server instance assume central server encodes sends data variables worker nodes see appendix discussion implement efficiently figure shows result experiments aggregated trials baselines consider uncoded scheme well replication scheme uncoded partition replicated times across nodes server uses faster copy iteration seen right figure one speed computation reducing instance resulting reduction runtime note case uncoded fails converge whereas case stably converges also observe data replication scheme converges average worst case convergence much less smooth since performance may deteriorate copies partition delayed figure test rmse left right nodes server waits top bottom responses perfect refers case figure total runtime nodes different values fixed iterations scheme matrix factorization movielens dataset next apply matrix factorization dataset movie recommendation task given sparse matrix movie ratings dimension users movies rij specified user rated movie withhold randomly ratings form split goal recover user vectors movie vectors embedding dimension rij xti user movie global biases respectively optimization problem given min rij xti kxi kyj observed choose achieves test rmse close current best test rmse dataset using matrix problem often solved using alternating minimization minimizing first repetition step decomposes row column made smaller sparsity solve first extract rij observed solve resulting sequence regularized least squares problems variables distributedly using coded repeat first experiment distributed coded solved master node encoding data locally distributing encoded data worker nodes appendix discusses implement step efficiently overhead associated initial step included overall runtime figure movielens experiment run single machine ram order simulate network latency artificial delay exp imposed time worker completes task small problem instances solved locally central server using function additionally parallelization done ridge regression instances order isolate speedup gains distribution reduce overhead create bank encoding matrices paley etf hadamard etf given problem instance subsample columns appropriate matrix match dimensions overall observe encoding overhead amortized distributed optimization figure gives final performance distributed various encoding schemes epochs shows coded schemes robust small full table results given appendix http acknowledgments work supported part nsf grants references ananthanarayanan ghodsi shenker stoica effective straggler mitigation attack clones nsdi volume pages beck teboulle fast iterative algorithm linear inverse problems siam journal imaging sciences berahas nocedal method machine learning advances neural information processing systems pages candes tao decoding linear programming ieee transactions information theory candes tao signal recovery random projections universal encoding strategies ieee transactions information theory drineas mahoney muthukrishnan faster least squares approximation numerische mathematik dutta cadambe grover computing large linear transforms distributedly using coded short dot products advances neural information processing systems pages fickus mixon tremain steiner equiangular tight frames linear algebra applications geman limit theorem norm random matrices annals probability pages goethals seidel orthogonal matrices zero diagonal canad math karakus sun diggavi encoded distributed optimization ieee international symposium information theory isit pages ieee lee lam pedarsani papailiopoulos ramchandran speeding distributed machine learning using codes information theory isit ieee international symposium pages ieee mahoney randomized algorithms matrices data foundations trends machine learning mokhtari ribeiro global convergence online limited memory bfgs journal machine learning research paley orthogonal matrices studies applied mathematics pilanci wainwright randomized sketches convex programs sharp guarantees ieee transactions information theory riedl konstan movielens dataset silverstein smallest eigenvalue large dimensional wishart matrix annals probability pages complex hadamard matrices equiangular tight frames linear algebra applications tandon lei dimakis karampatziakis gradient coding systems workshop mlsys nips wang joshi wornell using straggler replication reduce latency parallel computing acm sigmetrics performance evaluation review welch lower bounds maximum cross correlation signals ieee transactions information theory yadwadkar hariharan gonzalez katz learning straggler avoiding predictive job scheduling journal machine learning research lemmas proofs ignore normalization constants objective functions brevity let fta yea set let yea denote solution effective yea instantaneous problem iteration arg min stronger versions following lemma proved include weakened version result completeness lemma proof define note xek kxek triangle inequality implies kxek kxw kxek consider ksa ykkxek follows expanding yea yea since minimizer function follows fact optimality follows inequality follows definition matrix norm since true choose gives kxek plugging back get completes proof lemma fea feta fea proof since kxw fea similarly fea linear recursive inequality get considering inequalities multiplying summing lemma fea convex proof sufficient show minimum eigenvalue bounded away zero easily shown fact unit vector lemma let positive definite matrix condition number ratio maximum eigenvalue minimum eigenvalue given unit vector proof let subspace spanned let matrix whose columns form orthonormal basis represented implies since still positive definite matrix defining qri note quantity interested equivalently represented note unit vector dqv since condition number ratio two elements larger since otherwise could find unit vectors contradiction representing cos sin angle written cos sin note minimizing inner product equivalent maximizing function tan tan tan tan setting derivative zero find maximizing given therefore cos sin desired result proof lemma first note xwt also consider krt implies krt setting consider trace implies also shown similar det det kuj det kuj det implies det det since trace bounded determinant bounded away zero must exist proofs theorem theorem throughout section consider particular iteration denote min max denote minimum maximum eigenvalues matrix also denote solution effective instantaneous problem iteration yea ignore normalization constants objective functions arg min yea feta yea finally define fea proof theorem using convexity choices fea fea get xdt get get fea follows fact follows since follows strong convexity inequality using definition get fea feta fea lemma implies finally lemma implies concludes proof proof theorem using convexity expression step size fea fea get xdt get get xdt xdt xdt xdt xdt get get kxd xdt get get kxdt kdt get get kbt get fea fea xdt follows fact term parenthesis increasing kxdt assumption follows assumption follows defining function quadratic form follows definition lemmas follows strong convexity min lemma implies follows choosing follows using definition inequality obtain fea feta fea hence applying first lemma lemma get desired result full results movielens experiment tables give test train rmse movielens recommendation task random split uncoded replication gaussian paley hadamard train rmse test rmse runtime train rmse test rmse runtime train rmse test rmse runtime table full results movielens distributed nodes total runtime hours uncoded scheme running full batch rmse runtime hours uncoded replication gaussian paley hadamard train rmse test rmse runtime train rmse test rmse runtime table full results movielens distributed nodes total runtime hours uncoded scheme running full batch rmse runtime hours efficient encoding using steiner etf first describe steiner etf based construction proposed steiner equiangular tight frames let power let real hadamard matrix let ith column consider matrix column incidence vector distinct subset instance note rows exactly elements construct steiner etf matrix replacing row distinct column normalizing instance example general procedure results matrix redundancy factor full generality steiner etfs constructed larger redundancy levels refer reader full discussion constructions efficient distributed encoding steiner etf allows distributed efficient implementation encoding given matrix note encoding matrix consists blocks corresponding row consider following partition horizontal block note multiplication block computed directly finding column indices block given multiplication implemented simply taking hadamard transform corresponding rows whose indices given case dimension mismatch one append zero rows remove rows make multiplication therefore one partition blocks sis across worker nodes worker node read corresponding rows pool data given iinik blocks assigned worker apply fast hadamard transform block note processing blocks parallelized within node using multiple cores practice observed performance steiner etf significantly improves rows shuffled encoding implemented nodes exchange rows encoding however incurs significant communication cost practical approach could one block assigned multiple nodes random block encoded multiple nodes worker nodes drop subset encoded rows row retained exactly one node based rule note effect nodes randomly exchange encoded rows space complexity one might raise point many datasets sparse sparsity lost encoding significantly increasing memory usage address issue first consider case node access entire dataset sake argument worker node would compute gradient corresponding block using order operations represented following parenthesization since would require multiplications note given block rows therefore order compute order operations one would need store rows corresponding rows apply transformation corresponding whenever needed node assigned blocks would require storing sparse rows means memory usage would increase constant factor order redundancy factor
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effective ahp based metaheuristic approach solve supplier selection problem tamal tanmoy chakraborty pranab dan department industrial engineering management west bengal university technology sector salt lake city kolkata india email abstract supplier selection problem based electing best supplier group prespecified candidates identified multi criteria decision making mcdm proportionately significant terms qualitative quantitative attributes fundamental issue achieve quantifiable unquantifiable attributes aim accomplish best solution abovementioned problem article portrays metaheuristic based optimization model solve problem initially analytic hierarchy process ahp implemented generate initial feasible solution problem thereafter simulated annealing algorithm exploited improve quality obtained solution taguchi robust design method exploited solve critical issues subject parameter selection technique order verify proposed methodology numerical results demonstrated based tangible industry data keywords supplier rating vendor section ahp simulated annealing metaheuristic taguchi method introduction decades supplier selection problem dragging attention researchers practitioners vicinity supply chain management study therefore several techniques proposed academicians professionals solve multi criteria decision making mcdm problem dickson stated contemporary firms alternatives select suitable technique decipher complicated job selecting suppliers techniques varies adoption uncomplicated methods selection vendor offering lowermost tender effective complicated methods exploit unquantifiable attributes supplier selection problem although many empirical studies already contributed proposed area research yet researchers practitioners involved improving solution methodologies due involvement growing complexities technology innovation manufacturing industry procurement cost raw materials outside suppliers unquantifiable attributes quality materials delivered time delivery substantially crucial nearly total cost involved whole product development phase weber corresponding author email phone fax current benton since early set hard estimate unquantifiable attributes turned key components supplier selection problem helped developing novel complicated methodologies ellram selecting optimal set suppliers basis certainly increase sustainability firms present global competition thompson traditionally supplier selection problem based purchasing cost quality delivery functionality raw materials consideration aspects enhances complexities problem polynomial time selection decision becomes extensively significant procurement department due several different levels success rival suppliers stated circumstances aissaoui haouari hassini vendor lowermost tender might best delivering material proper quality case government bodies practice optimization model vendor selection processes would extremely helpful private firms government framework legal need obey formal rules procedures control vendor selection processes therefore decision models could immensely supportive maintaining transparency fair business schooner evident past literature precise sophisticated methodologies urgently required solve supplier selection problems scott matter kanagaraj jawahar implemented simulated annealing algorithm saa tool supplier selection problems obtain optimal solutions quickly saen introduced technique based data envelopment analysis dea rank suppliers presence nondiscretionary factors integrated model decision support system proposed gnanasekaran velappan manimaran supplier selection reduce cost enhance product quality help reliabilitybased total cost ownership rbtco model integrates purchasing maintenance stoppage costs along realistic constraints based product reliability weight restraint nonlinear integer programming nlip used develop mathematical formulation rbtco model present article multiobjective supplier selection problem introduced incorporated faulty materials delay delivery cost decisive factors thereafter simulated annealing based optimization model implemented solve problem initial solution procedure generated using analytic hierarchy process ahp order select optimal set parameters taguchi design experiments doe technique exploited rest article structured following manner section presents brief literature survey proposed area study section demonstrates formulation proposed multiobjective problem model section elaborates proposed solution methodology numerical results depicted section followed managerial implications conclusion research literature survey fun hung reported supplier selection methodologies practiced facilitate selection process substantial impact selection outcomes numerous supplier selection methods established categorized decades petroni braglia proposed linear weighting method using supplier rating based different attributes faster inexpensive method instigate although various drawbacks limitations also indicated study cost proportion timmerman aggregate cost proprietorship ellram based aggregate cost approaches assemble cost components supplier selection process fiscal units flexible methods exact methods due complexities time involvement methods moderately expensive implement mathematical programming approaches primarily exploit quantifiable factors approaches comprise principal component analysis pca petroni braglia artificial neural network ann choy lee low bello stated research pca approach advantageous terms competency managing various differing aspects ann approaches also useful cost minimization time reduction multiple attribute utility theory method practiced global vendor selection problems surroundings complex uncertain zhao bross chen lin huang demonstrated fuzzy set theory technique controls situations supplier performance evaluation approach helpful managers purchase suppliers according choice fuzzy weighted additive mixed integer linear programming developed model aggregates weighted membership functions objectives construct relevant decision functions objectives different relative importance proposed weighted linear program supplier selection problem paper demonstrates transformation technique mathematical model solve problem without optimizer amid ghodsypour brien demonstrated fuzzy multiobjective additive model consider imprecision information along order quantities supplier using price breaks objective functions used minimizing net cost rejected items late deliveries satisfying constraints capacity demand requirement authors stated weighted model similar problem environment amid ghodsypour brien help analytic hierarchy process ahp proposed model could utilized find appropriate order supplier saen zohrehbandian proposed dea approach quantity discount policy select best supplier stated hybrid model using data envelopment analysis dea decision trees neural networks nns evaluate supplier performance model comprised two parts first part utilized dea categorizes suppliers several clusters thereafter second part used data company train decision trees intelligent neuro model precised results obtained however saen stated approach based analysis based dea rank suppliers presence volume discounts author proposed another dea based method rank suppliers presence weight limitations factors saen analytical hierarchical process ahp exploited method supplier selection problems multi criteria decision making technique implemented rank alternatives several criteria believed considered permits managers formulate complicated problems form hierarchical relationship saaty ahp comparatively straightforward method practice technique integrates tangible intangible attribute problems detail survey tahriri supplier selection methods portrays ahp frequently employed method supplier selection ahp hierarchy typically contains three distinct levels objectives factors alternatives ahp suggests way rank alternative choices based manager decisions relating significance criteria due fact ahp preferably appropriate abovementioned problem problem hierarchy provides analysis based impact given level next higher level saaty managerial judgments stated terms comparisons entries specified level hierarchy based influences next higher level comparisons signifies approximation proportion weights two criteria compared since ahp exploits proportionate scale personal decisions relative weights reflect relative importance norms attaining objective hierarchy tam tummala yahya kingsman used saaty ahp method determine primacy selecting suppliers authors employed vendor rating determining allot business inadequate progress work utilized akarte handfield walton sroufe jing liu hai rajkumar kannan jayabalan also utilized ahp technique study integral part supplier selection literature generally used traditional methods traditional techniques efficient solution state space large various constraints cause vendor selection problem complicated articles utilized methods metaheuristics recently arunkumar karunamoorthy makeshwaraa rezaei davoodi kubat yuce ding benyoucef xie proposed genetic algorithm based metaheuristic approach solve supplier selection problems multiobjective environment present research introduces efficient metaheuristic approach based simulated annealing solve supplier selection problem problem formulation supplier selection processes primarily reliant specific objectives solved problem relevant constraints related objectives article case leading construction firm india considered derive multiobjective optimization model firm constructs commercial buildings well residential units large scale sez shopping malls hospitality retail units logistics industrial squares etc due heavy construction approach company requires various raw materials steel beams cement light weight bricks cast iron etc proposed model articulated considering constrictions managed rationally firm select suppliers raw material supplied supplier would different constraints characteristics percentage faulty materials supplied percentage delay delivery unit purchasing cost materials supplier certainly capacity supply firm specific requirements material certain period raw material supplied supplier initial procurement cost defined unit cost jth material supplied kth supplier qjk amount material type procured ith supplier total substandard delivery defined percentage faulty items jth material supplied kth supplier qjk amount material type procured ith supplier total delay delivery defined delay percentage delivery jth material supplied kth supplier qjk amount material type procured ith supplier supplier selection problem three objectives simultaneously considered minimize total procurement cost minimize total number faulty items supplied iii minimize total number delay days delivery procuring various raw materials various suppliers quality function demonstrates number faulty items supplied suppliers faulty items generally detected receiving firm relocating raw materials inventory firm strategy return back substandard items suppliers request replace items within stipulated time substantially one week therefore project fixed schedule supply substandard material could incur total cost terms one week delay time thus quality function equation total delay delivery equation could transformed cost component using hence proposed multiobjective model aims minimizing total cost function subject xjk capacity supplier supplying material yjk demand material firm certain period constraint confirms supplier supplies according capacity constraint confirms total raw material procured harmonize firm demand proposed multiobjective model validated using simulated annealing approach considering abovementioned objectives constraints gives optimal selection results suppliers order define approach quick solution generation method believed identified generated feasible solution assumed used initial solution algorithm therefore research analytic hierarchy process ahp utilized serve purpose next section would demonstrate details solution methodologies involved approach research methodology facilitate present research work authors visited eminent construction firm operating kolkata india currently firm involved big project based development special economic zone sez sector rajarhat newtown salt lake city kolkata authors prepared questionnaires based information gathered professionals abovementioned firm basis experts opinion ahp analysis carried study proposed optimization methodology developed using simulated annealing algorithm achieve optimal solutions algorithm simulates natural annealing process particles solid organize thermal equilibrium introduction obtained book aarts korst general applications concerns combinatorial optimization problems following form predetermined set feasible solutions algorithm exploits neighbourhood structure control parameter called temperature resemblance natural annealing process governs search behaviour iteration neighbour solution current solution figured improved objective function value solution accepted current solution swapped alternatively attain better objective function value solution recognized specific probability depending difference objective function values temperature parameter pseudocode exhibits general method pseudocode initialize repeat generate candidate solution evaluate candidate determine current solution reduce temperature termination condition met article factors affect decision makers strength analysed process follows steps using ahp method weight factors obtained qualitative expressions supplier weight also achieved final composite criteria weights determined ahp rank suppliers according composite scores therefore according ranks suppliers could selected simulated annealing takes ranks obtained previous step attempts search optimal set suppliers based objectives defined section initial solution generation article ahp exploited select initial set suppliers solution initial feasible solution method based optimization model analytical hierarchy process ahp ahp decision technique exploits hierarchical relationships represent problem primacies substitutes acquired based opinion experts saaty method consists several important steps outlining shapeless problem shape obtaining ahp hierarchical relationships forming pairwise comparison matrices approximating relative weights examining consistency finally attaining overall ranking lee chen chang ahp empower managers represent interface several factors complicated shapeless circumstances technique based pairwise comparison decision variables respect factors substitutes pairwise comparison matrix obtained size number criteria compared ahp method adopted article stated step hierarchical relationship problem obtained presented figure proposed ahp method decomposes problem three levels saaty first level demonstrates main objective selection suppliers second level depicts criteria last level reports six suppliers compared figure hierarchical relationship supplier selection problem step calculation pairwise comparison matrix level required pairwise comparison ranking scale used criteria evaluation saaty scale crisp scale ranging presented table scale values assigned criteria based experts opinion ahp questionnaire sheet prepared authors pairwise comparison matrix criteria presented table last column table depicts overall importance criteria ahp procedure presented consider comparison matrix size criteria eigenvector size eigenvalue find ranking priorities namely eigen vector initialization take squared power matrix find row sums normalize array find set main take squared power matrix find row sums normalize array find find elements close zero else set set step stop table saaty point scale importance value definition equal strong moderate strong fairly strong strong absolute strong intermediate values description two factors equally contributing objective one factor marginally superior one factor strongly superior one factor stongly superior highest level superiority one factor according negotiation required table comparison matrix main attributes cost quality delivery cost quality delivery relative weight table table present pairwise comparison matrices suppliers criteria selected research last columns pairwise matrices present calculated relative weights suppliers table pairwise comparison matrix suppliers respect cost relative weight table pairwise comparison matrix suppliers respect quality relative weight table pairwise comparison matrix suppliers respect delivery relative weight overall rating supplier computed adding product relative weight criterion relative weights suppliers considering corresponding criteria table table demonstrates supplier overall ranking composite score best supplier followed supplier computation consistency index consistency ratio ahp procedure requires computation consistency ratio ensure precision obtained solution consistency index pairwise comparison matrix calculated using random consistency index computed maximum eigenvalue size pairwise comparison matrix thus consistency ratio obtained using computed understand consistency solution obtained general believed table provides computed values pairwise comparison matrices computed values therefore computed results acceptable table composite score matrix four matrices cost quality delivery composite score rank table values pairwise comparison matrices cost quality delivery criteria fitness function procedure procedure examines fitness score solution generated solution neighbourhood fitness calculation one significant steps metaheuristic method decides solution stored one eliminated multiobjective function required compute fitness score order facilitate computation transformed expressed equation two weights also assigned equation considered total cost computed summing total procurement cost cost since firm gives importance delivery issues rather quality cost issues thus assigned prefixed values respectively proposed simulated annealing algorithm subsection describes proposed algorithm depth initial input solution string generated ahp technique therefore initial input string obtained table using ranks suppliers size solution string total number suppliers evaluated bit string represents rank corresponding supplier string index therefore states supplier retain ranks respectively thereafter multiobjective procedure set maximize equation symbolizations used algorithm introduced scur current solution neighbourhood solution best solution found far tinit initial temperature tfinal freezing temperature current temperature temperature reducing factor markov chain length iter iteration number current fitness value fbest best fitness value steps proposed algorithm summarized follows step obtain initial solution using analytic hierarchy process method step evaluate calculate corresponding fitness value step set set sbest scur step initialize heuristic parameters tinit tfinal iter step count repeat steps step generate new supplier rank configuration neighbourhood searching performing randomly selecting two suppliers interchanging ranks step read suppliers rank configuration steps generate corresponding neighbourhood solution step fbest sbest scur count count step step fbest count count step step compute scur obtain random variable range step step set scur step step else step iter iter step freezing temperature tfinal reached step reduce temperature using function procedure repetitively employed solution achieved attains highest fitness score parameters counters initialized step special move namely utilized proposed algorithm guide solution searching procedure spotted ordinarily leads improved solutions effortlessly competently practiced principle component finding better neighbourhood solution step algorithm also verifies number instances neighbourhood solutions become static number attains constant value fitness value current configuration compared optimal solution obtained thus far conclude whether prolong iterations stop best solution achieved experiments verifications order apply proposed algorithm solution methodology effects changing values various parameters studied determining optimal set parameters crucial respect therefore article taguchi robust design method taguchi employed determine optimal parameters set taguchi method parameters selection parameters initial temperature tinit temperature reducing factor markov chain length final temperature tfinal taken constant value termed factors factor three discrete levels table hence orthogonal array used recommends sets taguchi experiments prerequisite results evaluated using analysis variance anova technique parameter settings experiment shown table table levels parameters tested levels parameters tinit table presents results corresponding anova analysis ratio table variance ratios ratios factors determined test significance confidence level employed spot significance factors values factors tinit investigated values parameters seen less critical level degrees freedom suggests parameters significant factors proposed approach response table table depicts average response characteristic level factors table include ranks based delta statistic compares relative magnitude effects ranks assigned based values using level averages response table optimal set levels factors could determined yields best result table experimental settings taguchi experiments experiments tinit responses table anova table factors tinit residual error total degrees freedom factor sum squares mean square variance ratio value ranks indicate initial temperature tinit greatest influence followed temperature reducing factor markov chain length factor level fixed way highest response could achieved table main effects plot figure show optimal solution obtained tinit set respectively table response table levels rank tinit convergence analysis convergence analysis procedure quite simple supplier selection problem convergence curve iterations proposed metaheuristic technique presented figure first iteration fitness score attained value since procedure designed maximize fitness function iteration counts therefore iteration attained value increase final optimal solution based experimentation reported article observed fitness score increased iteration counts till reaches best fitness score iteration thereafter fitness score continues remain constant even number iterations increased therefore convergence property established test problem hand proposed approach executed iterations took cpu seconds attain best solution proves computational efficiency figure main effects plot figure convergence curve algorithm computational results data six vendors collected construction firm periodic demand metric ton tmt steel bar depicted table strategy procurement department distribute order among best three vendors avoid biasness supplier certain capacity supply materials given table supplier supply faulty materials according percentage defective items calculated supply according pace percentages delays delivery also provided table supplier firm project schedule moderately rigid therefore delay delivery incurs overall cost project delay cost calculated experts firm closely lacs inr per day cost incurs due substandard supply generally converted delay cost stated firm manager calculated using first component equation ahp method depicts supplier best three suppliers total procurement cost obtained three vendors outlined table thereafter procedure executed different result obtained states supplier best total procurement cost computed method found less ahp result shown table although total faulty materials supplied total delay days almost identical methods method attains closely better solution ahp method monetary term recovers lacs inr firm observation indicates technique efficient less complex simplicity simulation solution obtained negligible computational time seconds thus proposed method shown outperform ahp technique table collected vendor details data firm vendors max order quantity metric ton unit cost inr percent ton percent delay delivery table comparison result ahp result ahp result result vendor order defective late delivery quality cost delay cost total procurement cost cost inr vendor order defective late delivery quality cost delay cost total procurement cost cost inr managerial implications study accomplished research significant managerial implications soft computing approach proposed article exploited critical managerial decision making tool beneficial optimizing vendor network successful resource allocation vendor improvement curriculums optimizing vendor network managers employ method choose vendors without biasness particular vendor reduces chance failure supplier network would help every supplier grow evenly management firm deliver suppliers possible standards enhancement target time could anticipated complementing another managerial insight study affirms already recognized supplier selection methodology ahp may best methodology techniques metaheuristics substantially attain better solution maximize profit firm firms main objective conclusions article portrays novel based metaheuristic algorithm select supplier particular indian firm problem nature problem formulated using multiobjective mathematical model reflects essential optimization criteria research initial feasible solution proposed based technique obtained using ahp technique order quicken computation work exploits taguchi robust design approach select optimal set parameters algorithm crucial influencing performance technique uniqueness work lies practicing two different decision making techniques solving mcdm problem model past literature metaheuristic approach evaluate enhance ahp ranking vendors never carried perform said analysis authors collected industrial data national construction firm computational results presented section demonstrate method outperforms ahp technique performing better ahp method supplier selection problem proposed procedure produces nearly improved solution work experimental study considered main criteria problem cost delivery delay however many intricate could also considered make work realistic future work accomplished utilizing technique complex supplier selection problems incorporating conflicting criteria considering risk factors suppliers profiles related issues possible extension research acknowledgement authors grateful anonymous reviewers valuable comments suggestions improving quality paper references aarts korst simulated annealing boltzmann machine john wiley sons new york usa aissaoui haouari hassini supplier selection order lot sizing modeling review computers operations research vol akarte web based casting supplier evaluation using analytic hierarchy process journal operational research society vol amid ghodsypour brien weighted additive fuzzy multiobjective model supplier selection problem price breaks supply chain international journal production economics vol amid ghodsypour brien weighted model fuzzy multiobjective supplier selection supply chain international journal production economics article press arunkumar karunamoorthy makeshwaraa optimization technique vendor selection quantity discounts using genetic algorithm journal industrial engineering international vol bello case study approach supplier selection process thesis submitted university puerto rico mayaguez zhao bross supplier selection process emerging markets case study volvo bus corporation china thesis submitted university sweden chen lin huang fuzzy approach supplier evaluation selection supply chain management international journal production economics vol choy lee intelligent supplier management tool benchmarking suppliers outsource manufacturing expert systems applications vol dickson analysis vendor selection systems decisions journal purchasing vol ding benyoucef xie approach using genetic search supplier selection proceedings winter simulation conference ellram supplier selection decision strategic partnerships journal purchasing materials management vol gnanasekaran velappan manimaran integrated model supplier selection using fuzzy analytical hierarchy process steel plant case study international journal procurement management vol handfield walton sroufe applying environmental criteria supplier assessment study application analytical hierarchy process european journal operational research vol kanagaraj jawahar simulated annealing algorithm optimal supplier selection using total cost ownership model international journal procurement management vol kubat yuce supplier selection genetic algorithm fuzzy ahp proceedings international symposium intelligent manufacturing systems lee chen chang fuzzy ahp bsc approach evaluating performance department manufacturing industry taiwan expert systems applications vol fun hung new measure supplier performance evaluation iie transactions vol liu hai voting analytic hierarchy process method selecting supplier international journal production economics vol efficient simple model multiple criteria supplier selection problem european journal operational research vol petroni braglia vendor selection using principal component analysis journal supply chain management vol rajkumar kannan jayabalan vendor selection analytic hierarchy process international journal procurement management vol rezaei davoodi deterministic inventory model supplier selection imperfect quality applied mathematical modelling vol saaty analytic hierarchy process planning priority setting resource allocation new york saen using analysis ranking suppliers presence volume discount offers international journal physical distribution logistics management vol saen zohrehbandian data envelopment analysis approach supplier selection volume discount environments international journal procurement management vol saen using data envelopment analysis ranking suppliers presence nondiscretionary factors international journal procurement management vol saen restricting weights supplier selection decisions presence factors applied mathematical modelling vol schooner commercial purchasing chasm united state government evolving policy practice arrowsmith trybus eds public procurement continuing revolution kluwer law international hague scott best value contracts lessons learned paving road quality national contract management journal vol taguchi taguchi robust technology development bringing quality engineering upstream journal electronic packaging vol tahriri osman ali yusuff review supplier selection methods manufacturing industries suranaree journal science technology vol tam tummala application ahp vendor selection telecommunications system omega vol thompson vendor profile analysis journal purchasing materials management vol timmerman approach vendor performance evaluation engineering management review ieee vol weber current benton vendor selection criteria methods european journal operational research vol supplier selection hybrid model using dea decision tree neural network expert systems applications vol yahya kingsman vendor rating entrepreneur development programme case study using analytic hierarchy process method journal operational research society vol jing decision model supplier selection considering trust chinese business review vol
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triple massey products vanish fields aug duy bstract show absolute galois group field vanishing triple massey product property several corollaries structure maximal absolute galois groups deduced furthermore vanishing higher massey products proved ntroduction let field separable closure galois group gal called absolute galois group every galois group profinite group one may ask special properties absolute galois groups among profinite groups difficult problem moment properties found however discovered properties great interest considerable depth classical papers published artin schreier developed theory real fields showed particular finite subgroups absolute galois groups groups order recently remarkable work rost voevodsky proved conjecture thereby establishing special property galois cohomology absolute galois groups relatively recently two new conjectures vanishing conjecture kernel conjecture proposed see conjectures based number previous considerations one motivation coming topological considerations see dgms another motivation program describe various series absolute galois groups kernels simple galois representations see msp paper shall consider special case vanishing conjecture exception section consider well papers vanishing conjecture formulated case base field contains primitive root unity paper consider stronger version conjecture condition field sake simplicity shall recall definition products case refer reader sections general case see definition definition vanishing massey product partially supported natural sciences engineering research council canada nserc grant ndt partially supported national foundation science technology development nafosted grant duy property also see section reviews basic definitions facts related massey products conjecture vanishing conjecture let prime number integer let field absolute galois group vanishing massey product property respect paper show conjecture true use word triple instead theorem theorem let field prime number absolute galois group vanishing triple massey product property respect briefly recall definition triple massey products corresponding vanishing property let profinite group prime number consider finite field trivial discrete let differential graded algebra inhomogeneous continuous cochains coefficients see nsw section write corresponding cohomology groups denote subgroup consisting use trivial action coefficients hom let elements assume case say triple massey product defined exist cochains say sometimes simplicity defining system triple massey product observe hence define value triple massey product respect defining system cohomology class set values runs set defining systems called triple massey product say triple massey product vanishes definition say vanishing triple massey product property respect every triple massey product vanishes whenever defined vanishing conjecture claims field prime absolute galois group vanishing triple massey product property proved hopkins wickelgren global field characteristic vanishing triple massey product property triple massey products vanish fields proved result valid field proved vanishing triple massey product property respect global field containing primitive root unity efrat matzri provided alternative proofs mentioned results matzri proved prime field containing primitive root unity vanishing triple massey product property paper shall provide cohomological proof main result see theorem also remove assumption contains primitive root unity see theorem thus every absolute galois group vanishing triple massey product property fundamental new restriction absolute galois groups see subsection significant consequences structure quotients absolute galois groups indeed subsection able describe strong restrictions shape relations zassenhaus filtration modulo maximal absolute galois group shape relations excludes possibility certain triple commutators occurring relations new significant restrictions shape relations canonical quotients absolute galois groups description relations modulo appears close optimal description structure paper follows section recall basic material cohomology bicyclic groups section discuss heisenberg extensions section using material developed sections provide alternative cohomological proof main result vanishing triple massey products respect contains primitive root unity received nice preprint matzri planned make proof completed proof received preprint also want notice referring directly results see remark brief explanation one avoid using material section however think section might independent interest used work tools theory central simple algebras paper use cohomological techniques instead remark provide yet another short direct variant key part proof theorem considering details proof makes possible prove vanishing triple massey products general setting consider formation profinite group collection open subgroups indexed set discrete see chapter chapter section definition formation shall call formation field formation satisfies two axioms open normal subgroup set elements fixed action every element short exact sequence duy acts trivially written multiplicative way recall axiom guarantees field formation also field formation see chapter section main interest absolute galois group set finite separable extensions approach valid general setting may applications anabelian geometry also approach clarifies key properties sufficient proofs nearly simultaneously arxiv posting first version paper efrat matzri posted arxiv paper replacement efrat matzri also provide cohomological approach theorem approach similar flavor proofs paper still different feel papers taken together provide definite complementary insight new fundamental property absolute galois groups mentioned remark provide second alternative proof vanishing triple massey products proof able show specific element triple massey product vanishes results paper results heisenberg extensions subsection lemma already used construction important galois groups namely succeeded extending crucial ideas paper together ideas galois theory find explicit constructions galois extensions gal fields primes example theorem proof play portant role finding crucial submodule needed construction required galois extension containing galois group isomorphic section prove vanishing massey products form copies vanishing massey products form copies see theorem first vanishing deduced also results second vanishing appears new see also wic corollary another result vanishing massey products acknowledgements grateful jochen adam topaz initially began correspondence vanishing triple massey products pursuing different strategy also grateful eliyahu matzri sending beautiful preprint shortly arxiv posting although yet discuss details paper ido efrat eliyahu matzri danny neftin kirsten wickelgren thank interest great encouragement grateful anonymous referee careful reading paper providing insightful comments valuable suggestions used improve exposition triple massey products vanish fields ohomology bicyclic groups section study cohomology cyclic bicyclic groups number basic results need subsequently paper recalled main references ckm pages pages cohomology cylic groups abelian group element order denote let cyclic group order choose generator recall augmentation homomorphism following resolution trivial multiplication resp even resp odd resolution determines complex hom homg homg make natural identification homg complex becomes implies particular ker explained chapter viii isomorphism depend choice generator described choice defines homomorphism coboundary associated short exact sequence trivial sends element isomorphism map sends cohomology bicyclic groups let bicyclic group choose two generators order order duy define chain complex follows defined following conditions convenience put canonical basis obtain free resolution trivial define ker augmentation ideal ker obtain following exact sequence also consider following exact sequence zng let resolution yields following complex homg hom homg homg make natural identifications hom complex becomes explicit descriptions maps matrix form given particular ker triple massey products vanish fields exact sequences yield following commutative diagram homg homg hom homg diagram implies natural injection coker coker note isomorphism since natural identifications homg homg shall describe explicitly objects maps diagram first homg map becomes sends follows observation map obtained applying functor homg composite maps mod zng hand surjection yields injection hom homg identify hom image homg ker ker map becomes described explicitly follows consider composite map homg homg hom duy given map given class let denote class modulo hti order natural identification identifying cup product see subsection let homomorphism set follows observation ckm page inflation map given note also similarly homomorphism therefore observe following commutative diagram induces following commutative diagram homg homg homg homg triple massey products vanish fields summary identifications homg homg diagram becomes natural injection coker coker given eisenberg extensions norm residue symbols let field containing primitive root unity element shall write character corresponding via kummer map hom assume galois extension galois group satisfies character defines homomorphism hom hom formula let element norm residue symbol defined cup product interpreted norm residue symbol precisely consider exact sequence identified group roots unity via choice obtain duy one see chapter xiv proposition following fact chapter xiv proposition also used frequently sequel proposition res ker ker heisenberg extensions subsection provide short alternative version material section see also chapter section assume elements linearly independent modulo galois extension whose galois group generated let let group unipotent tries consider map sends following embedding problem map gal last isomorphism gal one sends assume norm residue symbol trivial hence exists see chapter xiv proposition iii set lemma let element let proof observe lemma follows identity representation let composition projection triple massey products vanish fields proposition assume let element let defined homomorphism lifts heisenberg extension resker proof see every gal implies extension galois let gal resp resp since gal extension hence implies order hand hence implies also order set implies order gal generated also define isomorphism gal letting composition gal desired lifting corollary let notation proposition let map res ker proof since homomorphism obtain therefore desired duy riple assey products triple massey products fields containing primitive roots unity subsection assume field containing primitive root unity let elements assume triple massey product defined theorem always assume linearly independent modulo linearly independent modulo also fix two elements let element defined right lemma let corollary map res ker since exists map element triple massey product consider following commutative diagram res ker res ker lemma resker res ker resker proof resker res ker res ker res ker res ker res ker res ker res ker follows commutativity diagram since thus let triple massey products vanish fields let galois group gal respectively extension gal respectively gal define define lemma proof lemma lemma consider galois action diagram subsection becomes natural isomorphism coker coker given duy following results subsection except theorem use presentation cohomology classes section corollary triple element proof follows immediately previous lemma lemma element image proof since isomorphism suffices show image hence hilbert theorem exists therefore imv desired corollary exists inf proof since image exist implies inf inf inf desired remark observe coker identify naturally group coker group notation see remark corollary composition map coker coker exactly map denoted page proof lemma shows element kernel following exact sequence inf res make natural identification one check map inf ker triple massey products vanish fields natural isomorphism mentioned page corollary follows proposition consider following commutative diagram res inf res res ker inf res lemma res res ker inf proof ckm page identification via identifying cup product res commutativity diagram obtain res ker inf inf res inf inf desired corollary exists inf proof lemma lemma res ker res ker inf statement follows proposition corollary exist particular triple massey product contains proof corollaries inf duy let since injective obtain desired last statement replace consider elements also defining system triple massey product hence contains theorem let field containing primitive root unity let elements triple massey product contains whenever defined proof assume defined also assume case assume linearly dependent modulo let defining system resker res ker res ker res ker res ker res ker res ker res ker proposition replace consider element also defining system triple massey product hence contains case assume linearly dependent modulo linearity massey products see lemma enough show contains contains theorem may shall assume since coefficients characteristic divide define straightforward check triple massey products vanish fields pick map defining system resker res ker res ker res ker res ker res ker res ker res ker proposition use argument case implies contains also follows general result proposition see also theorem case assume linearly independent linearly independent corollary says triple massey product contains remark case proof show specific element massey triple product vanishes leads another proof case hence theorem avoids using corollary theorem exists coboundary proof proof lemma exists lemma implies hence therefore exists similarly lemma implies hence duy therefore exists let implies consequence theorem obtain another proof case proof theorem another proof case proof theorem let use notation replace also defining system triple massey product corollary implies contains inf inf hence contains inf embedding problems triple massey products arbitrary field weak embedding problem profinite group diagram consists profinite groups homomorphisms surjective homomorphisms profinite groups considered paper assumed continuous addition also surjective call embedding problem weak solution homomorphism call finite weak embedding problem group finite kernel defined ker let group unipotent entries let subgroup matrices entries except position may identify group unipotent entries omitted representation let composition projection use similar notation representations note group homomorphism triple massey products vanish fields recall following result lemma direct consequence dwy theorem lemma let profinite group prime number following statements equivalent vanishing triple massey product property respect every homomorphism finite weak embedding problem weak solution lifted homomorphism proposition let profinite group prime number let open subgroup whose index coprime assume vanishing triple massey product property respect also vanishing triple massey product property respect proof shall prove condition lemma group let homomorphism consider weak embedding problem assumption lemma weak embedding problem induced weak solution let cohomology class corresponds extension following commutative diagram res duy particular res since weak embedding problem weak solution see hoechsmann lemma nsw chapter proposition note statement hoechsmann lemma nsw deals embedding problems proof goes well weak embedding problems since coprime order see restriction map res injective chapter corollary proposition hence hoechsmann lemma implies weak embedding problem weak solution done theorem let field prime number absolute galois group vanishing triple massey product property respect proof charf maximal free therefore vanishing triple massey product property respect assume charf let primitive root unity let finite extension degree divides implies coprime since vanishing triple massey product property theorem follows also vanishing triple massey product property proposition consequences subsection assume odd prime number case treated theorem theorem recall profinite group prime number zassenhaus filtration defined inductively least integer greater equal two closed subgroups means smallest closed subgroup containing commutators similarly means smallest closed subgroup containing powers let set let free set generators see nsw definition let filtration element may written uniquely bij cijk cijk bij cijk convenience call canonical decomposition modulo respect basis also set uij bij uij triple massey products vanish fields denote maximal absolute galois group given field theorem let set elements assume exists element distinct indices canonical decomposition modulo uij ukj uki ukl cijk every different factors occur canonical decomposition modulo realizable field proof follows immediately theorem theorem corollary theorem let set elements assume exists element distinct indices canonical decomposition modulo uij uil ciji respectively cijj every different factor occur canonical decomposition modulo realizable field proof follows immediately theorem theorem corollary vanishing higher assey products massey products let profinite group prime number consider finite field trivial discrete let differential graded algebra inhomogeneous continuous cochains coefficients nsw write corresponding cohomology groups denote subgroup consisting use trivial action coefficients hom see references therein general setups let integer let elements definition collection aij elements aij called defining system massey product following conditions fulfilled ail duy cohomology class called value product relative defining system denoted nfold massey product subset consisting elements written form defining system speak triple massey product note case triple massey product defined convenience introduce following definition definition let integer let elements collection aij elements aij called complete defining system following conditions fulfilled ail note complete defining system complete defining system complete defining system massey product defined contains vanishing higher massey products let unital commutative ring let positive integer assume every integer invertible following binomial polynomials ring polynomials one variable coefficients following elementary lemma lemma let notation one ring polynomials coefficients two variables remark let notation one also results presented stated proved using obk vious modifications instead using triple massey products vanish fields element hom define corollary let element hom let integer proof lemma desired corollary let element hom let positive integer system complete defining system copies proof follows immediately corollary proposition let field containing primitive root unity let elements linearly independent modulo let positive integer assume massey product copies defined complete defining system form aij aij duy proof corollary system aij complete defining system set shall prove induction exist alr imply immediately system aij complete defining sytem since exists set assume induction hypothesis exist system alr defining system massey product value massey product respect ease notation denote res ker res res res res res res res res res res res proposition exists hence exists triple massey products vanish fields set alr desired proposition let field containing primitive root unity let elements linearly independent modulo let positive integer assume massey product copies defined complete defining system form aij aij proof proposition exists system aij complete defining system aij set shall prove induction exist alr imply immediately system aij complete defining system since exists set assume induction hypothesis exist system alr duy defining system massey product value massey product respect ease notation denote res ker res res res res res res res res res res res proposition exists hence exists set alr desired theorem let field containing primitive root unity let elements linearly independent modulo let positive integer assume massey products copies defined contain massey products copies defined contain proof recall following formal property massey products defined han defined triple massey products vanish fields see kra theorem observe also every statement follows two previous propositions eferences ckm dwy dgms kra msp nsw artin schreier algebraische konstruktion reeller abh math sem univ hamburg reprinted artin collected papers eds lang tate new york artin schreier eine kennzeichnung der reell abgeschlossenen abh math sem univ hamburg reprinted artin collected papers eds lang tate new york kanevsky sansuc des surfaces cubiques diagonales diophantine approximation transcendence theory bonn lecture notes springer berlin dwyer homology massey products maps groups pure appl algebra efrat zassenhaus filtration massey products representations profinite groups adv math efrat matzri vanishing massey products brauer groups math bull efrat matzri triple massey products absolute galois groups appear eur math efrat descending central sequence absolute galois groups amer math efrat galois groups cohomological functors appear trans amer math fenn techniques geometric topology london math soc lect notes cambridge deligne griffiths morgan sullivan real homotopy theory manifolds invent math hopkins wickelgren splitting varieties triple massey products pure appl algebra kraines massey higher products trans amer math soc matzri triple massey products galois cohomology preprint merkurjev essential pgl amer math soc spira witt rings galois groups ann math triple massey products galois theory appear eur math kernel unipotent conjecture massey products odd rigid field appendix efrat adv math triple massey products global fields doc math construction unipotent galois extensions massey products preprint neukirch schmidt wingberg cohomology number fields grundlehren der mathematischen wissenschaften fundamental principles mathematical sciences berlin serre local fields translated french greenberg graduate texts mathematics new serre galois cohomology translated french ion revised author corrected reprint english edition springer monographs mathematics berlin duy wic sharifi twisted heisenberg representations local conductors thesis university chicago sharifi massey products ideal class groups reine angew math tignol produits algebra villegas relations quadratic forms certain galois extensions manuscript ohio state university http weiss cohomology groups pure applied mathematics vol academic press new wickelgren obstruction sections massey products galois theory arithmetic geometry proceedings conferences kyoto october nakamura pop schneps tamagawa eds advanced studies pure mathematics vol mathematical society japan epartment athematics estern niversity ondon ntario anada address minac epartment athematics estern niversity ondon ntario anada athematics ietnam cademy cience echnology oang uoc iet anoi ietnam address duytan nstitute
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energy storage arbitrage markets via reinforcement learning hao wang baosen zhang department electrical engineering university washington seattle feb email zhangbao abstract paper derive temporal arbitrage policy storage via reinforcement learning price arbitrage important source revenue storage units designing good strategies proven difficult highly uncertain nature prices instead current model predictive dynamic programming approaches use reinforcement learning design optimal arbitrage policy policy learned repeated charge discharge actions performed storage unit updating value matrix design reward function reflect instant profit decisions also incorporate history information simulation results demonstrate designed reward function leads significant performance improvement compared existing algorithms ntroduction energy storage provide various services load shifting energy management frequency regulation grid stabilization power grid economic viability receiving increasing attention one discussed revenue sources energy storage temporal arbitrage charging low prices discharging higher prices storage units take advantage price spreads electricity market prices make profits time application received significant attention research community especially since growing penetration intermittent renewable generations resulting volatile electricity market prices however even increase price spread remains nontrivial design arbitrage policies make significant even positive profit difficulties come fact future prices unknown difficult forecast may even paper aim develop arbitrage policy energy storage framework using reinforcement learning example arbitrage using energy storage studied see references within authors studied using batteries flywheels arbitrage nyiso found batteries potentially profitable using data authors analyzed generic storage system pjm market discovered arbitrage value nearly doubled due higher price variations authors formulated linear optimization problem compare arbitrage profits energy storage technologies several major electric markets similar studies also carried different markets australian national electricity market european electricity markets february draft crucially studies assumed perfect knowledge electricity prices therefore implemented arbitrage strategies recent works started explicitly take electricity price uncertainty account designing arbitrage strategies authors proposed stochastic dynamic program optimally operate energy storage system using available forecast authors formulated stochastic optimization problem storage owner maximize arbitrage profit uncertainty market prices studies need forecast electricity prices performances heavily rely quality forecast however market prices highly stochastic notoriously difficult forecast well overcome reliance price predictions authors employed approximate dynamic programming derive biding strategy energy storage nyiso market without requiring prior knowledge price distribution however strategy often highly computationally expensive authors proposed online modified greedy algorithm arbitrage computationally straightforward implement require full knowledge price distributions needs estimate bounds prices assume storages big enough always true practice aforementioned challenges motivate develop easily implementable arbitrage policy using reinforcement learning policy outperforms existing ones without explicitly assuming distribution policy able operate constantly changing prices may time repeatedly performing charge discharge actions different prices proposed policy learns best strategy maximizes cumulative reward key technical challenge turns design good reward function guide storage make correct decisions specifically make following two contributions paper formulate energy storage operation markov decision process mdp derive policy optimally control energy storage temporal arbitrage market design reward function reflect instant profit decisions also incorporate historical information simulation results demonstrate designed reward function leads significant performance improvements compared natural instant reward function addition using real historical data show proposed algorithm also leads much higher profits existing algorithms remainder paper ordered follows section present optimization problem energy storage arbitrage section iii provide reinforcement learning approach obtain arbitrage policy numerical simulations using real data discussed section section concludes paper rbitrage odel ptimization roblem consider energy storage battery operating electricity market finite operational horizon objective energy storage maximize arbitrage profit charging low prices discharging prices high assume energy storage price taker operation affect market prices denote discharged power storage time charged february draft power storage time let prices denoted formulate arbitrage maximization problem amp follows max subject amp emin emax cmax dmax variables denote efficiencies constraint specifies dynamics energy level time constraints amount energy storage emin emax bounds maximum charge discharge rates denoted cmax dmax respectively storage optimization problem amp linear program characterize optimal solution next lemma lemma optimal charge discharge profiles satisfy least one time min cmax emax min dmax emin lemma states energy storage charge discharge time also optimal charge discharge power hit boundary per operational constraints specifically storage decides charge charge either maximum charge rate cmax reaching maximum energy level emax similarly discharge power either maximum discharge rate dmax amount reach minimum energy level emin binary structure important design reinforcement learning algorithm next section future prices known optimization problem amp easily solved provide offline optimal strategy decisions however offline solution practical good price forecast available reality future prices known advance energy storage needs make decisions based current historical data words decisions functions price information current time slot denoted arbitrage policy maximizing profit therefore amp constrained sequential decision problem solved dynamic programming potentially high dimensionality state space february draft makes dynamic programming computationally expensive potentially unsuitable applications like price arbitrage moreover price forecast markets extremely challenging mismatch power supply demand attributed many different causes iii einforcement earning lgorithm solve online version amp use reinforcement learning reinforcement learning general framework solve problems actions taken depend system states cumulative reward optimized iii current state past actions known system might energy storage arbitrage problem four properties different electricity prices lead different actions future energy storage level depends past actions energy storage aims maximizing total arbitrage profit iii energy storage priori knowledge prices knows past history actual price profiles following describe setup amp detail state space define state space energy arbitrage problem finite number states specific system state fully described current price previous energy level discretize price intervals energy level intervals represents even price intervals lowest highest denotes energy level intervals ranging emin emax action space per lemma energy storage charge discharge time moreover optimal charge discharge power always reach maximum allowable rates denote maximum allowable rates min dmax emin min cmax emax therefore action space energy storage consists three actions charge full rate hold discharge full rate action denotes discharge either maximum rate dmax unitl storage hits minimum level emin action denotes charge maximum rate cmax storage reaches maximum level emax reward time taking action state energy storage receive reward energy storage knows good action according objective function amp energy storage aims february draft maximize arbitrage profit charging low prices discharge high prices therefore define reward max charge reward hold discharge instant reward charge discharge energy storage charges rate time pay spot price reward negative contrast energy storage discharges rate earn revenue reward straightforward natural design actually effective reason negative reward charge makes energy storage perform conservatively learning process thus arbitrage opportunity explored motivates develop effective reward avoid conservative actions introduce average price reward idea comes basic principle arbitrage charge low prices discharge high prices average price works simple indicator determine whether current price low high compared historical values max specifically new reward defined charge hold reward discharge average price calculated smoothing parameter note simple average weighs past prices equally instead use moving average leverage past price information also adapt current price change see reward energy storage charges price lower average price get positive reward otherwise receive loss spot price greater similarly reward encourages energy storage discharge high price giving positive reward reward outperforms reward exploring arbitrage opportunities achieving higher profits also mitigates prices since weights current price much heavier prices distant past show numerical comparisons section algorithm state action reward defined obtain charge discharge policy using popular subclass algorithms energy storage maintains value matrix entry defined pair state action energy storage takes action spot price value matrix updated follows max february draft parameter learning rate weighting past value new reward discount rate determining importance future rewards taking action state transits energy storage updates value matrix incorporating instant reward reward future value state time energy storage learn value action states converges optimal values obtain optimal arbitrage policy specifically algorithm derive arbitrage policy arg max optimal arbitrage policy guranteed finite mdp state energy storage always chooses best action maximizes value matrix algorithm energy storage arbitrage initialization time slot set iteration count initialize repeat observe state based price energy level decide best action using method based calculate reward using reward update according energy level end operation end algorithm energy arbitrage presented algorithm avoid learning algorithm getting stuck solutions employ algorithm exploits best action following also explores actions could potentially better specifically using algorithm randomly choose actions probability choose best action probability umerical esults section evaluate two reward functions also compare algorithm baseline synthetic prices realistic prices synthetic prices generate independent identically distributed prices realistic price use hourly prices iso new england market january december realistic price depicted figure see averaged price flat instantaneous prices fluctuate significantly periodic spikes february draft price average price price time hour fig pjm price synthetic price first evaluate two reward functions synthetic price uniformly distributed hours set cmax dmax emin emax cumulative profits rewards depicted figure profits stay flat first hours algorithm exploring environment different prices afterwards algorithm using reward starts make profit achieves reward end understand reward reward affect storage operation plot evolution energy level hour horizon fig see algorithm using reward performs conservatively reward makes algorithm actively charge discharge take advantage price spread therefore reward leads profitable arbitrage strategy real historical price evaluate two reward functions using realistic prices iso new england market plot cummulative profits two rewards training figure see reward fails make profit using reward produces high profit demonstrates effectiveness designed reward able adapt price changes makes profit continuously cumulative profit reward reward time hour fig cumulative profits synthetic prices february draft price time hour energy level mwh synthetic price time hour energy level mwh energy level algorithm using reward time hour energy level algorithm using reward fig price energy levels hour period using reward reward synthetic prices also plot evolution energy levels operational horizon figure see algorithm using reward capture price differences makes price flat contrast algorithm using reward able charge low prices hours hold energy prices low discharge hours respectively price reaches relatively high point cumulative profit reward reward time hour fig cumulative profits prices february draft price time hour energy level mwh price time hour energy level mwh energy level algorithm using reward time hour energy level algorithm using reward fig price energy levels hour horizon reward reward historical data comparison baseline algorithm discussion demonstrates reward performs much better reward thus stick reward compare algorithm baseline algorithm called online modified greedy algorithm algorithm uses thresholding strategy control charge discharge online fashion configure parameters baseline according arbitrage profits two algorithms simulated battery rate depicted figure baseline algorithm get algorithm earns times baseline profit profit baseline decreases rate increases algorithm achieves even higher profit times baseline profit reason baseline algorithm relies estimate price information lacks adaptability prices algorithm updates average price adapt price changes thus performs better february draft baseline algorithm algorithm cumulative profit cumulative profit baseline algorithm algorithm time hour time hour battery battery fig cumulative profits baseline algorithm algorithm onclusion paper derive arbitrage policy energy storage operation markets via reinforcement learning specifically model energy storage arbitrage problem mdp derive policy control energy storage design reward function reflect instant profit decisions also incorporate history information simulation results demonstrate designed reward function leads significant performance improvement algorithm achieves much profit compared existing baseline method consider degradation battery future work acknowledgment work partially supported university washington clean energy institute eferences eyer corey energy storage electricity grid benefits market potential assessment guide sandia national laboratories vol sioshansi denholm jenkin weiss estimating value electricity storage pjm arbitrage welfare effects energy economics vol woo horowitz moore pacheco impact wind generation electricity price level variance texas experience energy policy vol byrne estimating maximum potential revenue grid connected electricity storage arbitrage regulation sandia national laboratories kim poor scheduling power consumption price uncertainty ieee transactions smart grid vol borenstein efficiency electricity pricing energy journal sutton barto reinforcement learning introduction february draft walawalkar apt mancini economics electric energy storage energy arbitrage regulation new york energy policy vol bradbury pratson economic viability energy storage systems based price arbitrage potential electricity markets applied energy vol mcconnell forcey sandiford estimating value electricity storage wholesale market applied energy vol zafirakis chalvatzis baiocchi daskalakis value arbitrage energy storage evidence european electricity markets applied energy vol abdulla hoog muenzel suits steer wirth halgamuge optimal operation energy storage systems considering forecasts battery degradation ieee transactions smart grid krishnamurthy uckun zhou thimmapuram botterud energy storage arbitrage price uncertainty ieee transactions power systems jiang powell optimal bidding electricity market battery storage using approximate dynamic programming informs journal computing vol qin chow yang rajagopal online modified greedy algorithm storage control uncertainty ieee transactions power systems vol weron electricity price forecasting review look future international journal forecasting vol watkins dayan machine learning vol howard dynamic programming markov processes oxford england john wiley hourly online available https february draft
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interdependent security games networks behavioral probability aug ashish hota shreyas abstract consider class interdependent security games networks node chooses personal level security investment attack probability experienced node function investment investment neighbors network existing work settings considers players contrast studies behavioral decision theory shown individuals often deviate behavior making decisions uncertainty particular true probabilities associated uncertain outcomes often transformed perceived probabilities highly nonlinear fashion users influence decisions paper investigate effects behavioral probability weightings nodes optimal investment strategies resulting security risk profiles arise nash equilibria interdependent network security games characterize graph topologies achieve largest smallest worst case average attack probabilities nash equilibria total effort games equilibrium investments weakest link best shot games introduction interdependent security games class strategic games security risk faced player often manifested probability successful attack depends personal investment security investments interacting players laszka kunreuther heal broad framework capture security interdependencies independent stakeholders networked systems large literature class problems laszka kunreuther heal varian manshaei motivated applications cybersecurity airline security epidemic risks much work interdependent security games considers players risk averse sense classical expected utility theory laszka hand rich literature decision theory behavioral economics showing human behavior consistently significantly deviates predictions classical expected utility theory camerer studies highlighting significance biases irrationalities human information security domains christin schneier garg camp theoretical analyses deviations classical notions rational behavior scarce literature interdependent security games empirical investigations also limited christin goal paper initiate rigorous investigation impact behavioral decisiontheoretic models interdependent security games context security games one preliminary version work appeared proceedings gamesec hota sundaram authors school electrical computer engineering purdue university ahota work supported grant purdue research foundation important behavioral deviations classical expected utility framework way individuals perceive probability uncertain outcome cyber attack particular empirical studies show individuals tend overweight small probabilities underweight large probabilities thus true probabilities typically transformed highly nonlinear fashion perceived probabilities used tversky kahneman gonzalez transformations captured form probability weighting functions paper analyze effects behavioral probability weighting players equilibrium strategies interdependent security games networks consider three canonical manifestations security risk forms total effort weakest link best shot games models first introduced varian studied extensively literature model several scenarios cybersecurity domain described total effort games networks probability successful attack node affine decreasing function average security investments node neighbors total effort externality studied abstraction several cybersecurity problems grossklags grossklags johnson instance attacker tries slow file transfers networks success attack depends aggregate effort participating agents grossklags similar externalities arise underinvestment security user potentially causes increasing spam activity others communicate laszka nguyen authors consider similar formulation security risk faced node weighted linear combination investment investments neighbors authors discuss multiple settings externalities arise web authentication spam verification amin amin study related setting set independent control systems interact shared communication network failure probability function number controllers invested weakest link game node secure least secure node neighborhood best shot game player maximum investment neighborhood must successfully attacked attack node successful weakest link externalities prevalent cybersecurity domains successful breach one subsystem often increases vulnerability connected subsystems giving attacker increased access otherwise restricted parts best shot externalities arise systems redundancies attacker must breach secure subsystem attack successful best shot externalities also arise censorship resilient communication information available node long one neighbors possesses information johnson summarize main findings effects network structure probability weighting players strategies equilibrium attack probabilities three games table security investments often exhibit characteristics public good varian different risk externalities described indeed studied context public good games hirshleifer also growing interest networked public goods recent years also various behavioral characteristics affect perceived values gains losses tversky kahneman hota security externalities similar total effort formulation also studied broader interdependent security game literature context inefficiency equilibria jiang incomplete information network topology pal hui cyber insurance schwartz settings security risk faced node determined actions node immediate neighbors different line work models epidemic risks cascading failures spreading network nowzari externality weakest link impact network structure expected fraction nodes successfully attacked pne highest graphs among graphs given average degree connected graph nodes identical investments pne best shot nodes nonzero investments pne form maximal independent set total effort impact weighting function graphs interior pne secure probability weighting graph sufficiently dense attack probabilities arise pne values close depending game parameters probability weighting pne never fully insecure always exist node nonzero investment table summary main results pure nash equilibrium pne characteristics different attack probability functions galeotti analysis behavioral probability weighting complements line research probability weighting discussed previous section focus paper understanding effects nonlinear weighting true probabilities individuals making decisions risk weightings comprehensively studied behavioral economics psychology literature camerer recently wireless communications mandayam smart grid saad behavioral perception probabilities human decision makers certain fundamental characteristics including possibility effect overweighting probabilities close certainty effect underweighting probabilities close iii diminishing sensitivity end points characteristics usually captured inverse weighting function prominent parametric forms weighting functions proposed kahneman tversky tversky kahneman gonzalez gonzalez prelec prelec illustrated figure general analysis nash equilibrium total effort games section weakest link best shot games section assume derivative weighting function satisfies following properties assumption probability weighting function following properties unique minimum denoted xmin xmin strictly concave xmin strictly convex xmin impose following additional requirements weighting functions order obtain certain results effects network structure equilibria section perceived probability true probability figure shape probability weighting function different parametric forms proposed gonzalez gonzalez prelec prelec kahneman tversky tversky kahneman assumption probability weighting function following properties xmin strictly convex xmin remark assumption hold true parametric forms weighting functions proposed kahneman tversky tversky kahneman gonzalez gonzalez prelec prelec ranges parameter values functions shape figure particular assumptions hold ranges parameter values estimated empirical studies human subjects booij contains review several studies results effect intensity overweighting underweighting weighting functions equilibrium attack probabilities section need consider specific parametric form weighting function purpose use prelec weighting function prelec due analytical tractability particular true probability outcome prelec weighting function given exp exp exponential function parameter controls curvature weighting function weighting function linear smaller function sharper overweighting low probabilities underweighting high probabilities useful property function regardless value xmin words minimum value interdependent security games paper consider interdependent security games networks let denote undirected network graph set nodes node independent player game representing instance entity system security investment node denoted security risk attack probability experienced node function investment investment direct neighbors denote set neighbors node investment profile nodes vector true probability successful attack node given function node incurs security investment attack successful incurs loss expected utility true probability successful attack eui ease notation define extended neighborhood node denoted set nodes including neighboring nodes denote size extended neighborhood node work consider three canonical models interdependent security games initially presented grossklags models differ attack probability function described total effort weakest link best shot note prior works grossklags games defined complete graphs consider general graph topologies analysis focus total effort attack probability function since results also potential implications classes security games considered literature nguyen since focus present work understand effects behavioral probability weighting functions node degrees nash equilibrium security levels focus case security risk node influenced identical way neighbors investments formally define notion equilibrium best response interdependent security games follows definition strategy profile pure nash equilibrium pne every player eui eui definition best response player given investment profile neighbors set argmaxsi eui strategy profile pure nash equilibrium every player words pne exists vector best response mappings possesses fixed point osborne rubinstein equilibria without probability weighting establish baseline present following proposition describes main results grossklags regarding properties best response player total effort game paper considered complete graphs result extends directly general graphs refer player true expectation maximizer maximizes expected utility without behavioral probability weighing proposition consider player total effort game extended neighborhood size best response dli cii dli cii special case dli cii investment optimal strategy note except pathological case dli cii best response player either fully protect remain completely unprotected size neighborhood increases best response jumps investing investing behavior arises since marginal utility true expectation maximizer independent strategy strategies players neighborhood cost parameters homogeneous across players set nodes small enough degrees make investment high degree nodes invest zero nash equilibrium total effort game furthermore degree regular graphs equilibrium arises players investing investing players either fully secure fully unprotected analysis show behavioral probability weighting best responses equilibria much richer structural properties vary smoothly weighting parameters network structure pure nash equilibria total effort games behavioral probability weighting section consider total effort games networks probability weighting functions satisfying assumption cost parameters first prove existence pne class games establishing existence fixed point best response mapping see definition probability weighting expected utility player investment given eui total investment security neighbors quantity size extended neighborhood player defined marginal utility given solutions satisfy first order necessary condition optimality true attack therefore candidate solutions player best response note probability faced player without probability weighting illustrate nature solutions first order condition figure prelec weighting function parameter one see figure dli cii xmin investing best response player irrespective strategies suppose dli cii xmin case first order condition dli cii two distinct interior solutions corresponding true attack probabilities xmin xmin illustrated figure note degree node increases increases well eui eui dli cii xmin unique solution min player prefers invest case proof similar case proof lemma figure interior solutions denoted example shown horizontal line prelec weighting parameter total investment neighbors player player strategy change true attack probability interval words extended neighborhood size player directly change probability successful attack make following assumptions assumption player dli cii xmin let size extended neighborhood cost parameters lcii first condition implies given player contain required maintaining continuity best response varies discuss implications last two assumptions later analysis unless otherwise stated results section hold assumption probability weighting functions assumption neighborhood sizes cost parameters start following characterization best response player particular show assumptions best response player unique continuous monotonically decreasing aggregate investment nodes neighborhood proof presented appendix lemma suppose dli cii player given aggregate investment neighboring nodes best response player given otherwise solution xmin use properties best response proven lemma establish existence pne total effort games networks behavioral probability weighting theorem consider total effort game graph weighting functions players satisfy assumption cost parameters neighborhood sizes satisfy assumption game admits pure nash equilibrium proof player dli cii xmin best response invest regardless investments neighbors otherwise best response unique continuous strategies neighbors lemma addition strategy space player compact convex thus according brouwer fixed point theorem exists fixed point best response mapping corresponds pne consequence lemma pne investments extended neighborhood player dli cii xmin expressed min max particular xmin converse second identity holds remark equilibrium computation recently authors gharesifard showed best responses players given lemma continuous best response dynamics converge pne strategy profile furthermore strategy profile satisfies computed solving linear complementarity program similar result obtained present expanded discussion appendix general strategy profiles satisfy equation need unique therefore need unique pne however special case complete graphs classical setting total effort games grossklags show strategy profiles nash equilibria unique true equilibrium attack probability experienced players proposition consider total effort game complete graph player playerspecific weighting function satisfying assumption cost ratio lcii satisfying assumption nash equilibria true probability successful attack nodes proof presented appendix strategy profile neighborhood satisfies second identity refer pne interior equilibrium words interior equilibrium strategy profile satisfies adjacency matrix graph vector neighborhood sizes allones vector vector players denotes hadamard product thus every interior equilibrium true attack probability faced node existence interior equilibrium always guaranteed except certain special cases players homogeneous nodes graph identical degrees graph proposition consider graph degree homogeneous players xmin symmetric strategy profile node invests constitutes interior pne xmin proof straightforward substituting every player equation existence secure equilibrium assumption ensures best response player remains continuous strategies players helps establish existence pne theorem possible also show existence pne players invest equilibrium second third conditions assumption hold players proposition suppose every player total effort game either exists pne players invest proof consider player assume neighboring nodes investing player nonnegative interval marginal utility best response invest similarly lcii player optimal investment follows case proof lemma substituting yields eui eui lcii lcii occur weighting function sufficiently overweights small attack probabilities overweighting small attack probabilities discourages player reducing investment even players fully secure would lead large perceived increase attack probabilities relatively secure state results fully secure equilibrium fully secure pne possibly coexists equilibria identifying conditions pne exists potential implications designing incentive mechanisms encourage users achieve secure pne effects network structure section focus understanding effect network structure degrees nodes security investments attack probabilities pne total effort games consider players homogeneous weighting functions cost parameters order isolate effects degrees investments use characterization pne strategy profile given equation analysis section assume weighting functions satisfy assumptions weighting functions cost parameters homogeneous across players quantity function size extended neighborhood recall dli xmin properties function basis analytical results section remark note increasing function strictly increasing xmin illustrated figure investments nodes overlapping neighborhoods without probability weighting proposition indicated lower degree node always invests least much higher degree node monotonicity hold general behavioral probability weighting nonetheless prove certain monotonicity properties monotonically decreasing function holds assumption shown lemma let quantity defined ity weighting function satisfies decreasing xmin monotonically proof lemma presented appendix prove following result using lemma recall denotes extended neighborhood node proposition consider total effort game network homogeneous players whose weighting functions satisfy assumptions let cost parameters satisfy assumption let pne proof consider equilibrium strategy profile player invests otherwise result holds trivially equation fact obtain follows monotonicity shown lemma result must lemma know decreasing function pne characterization node invests aggregate investment extended neighborhood subject investment within node neighbors desired investment larger therefore node increase personal investment hand node large degree smaller large number neighbors rely meet target investment upper bound average probability successful attack equilibrium pne strategy profile denote average true probability successful attack note also equal expected fraction nodes successfully attacked pne strategy profile obtain upper bound every player proof presented appendix proposition consider total effort game graph homogeneous players satisfy assumptions addition suppose dli every player pne strategy profile attack probability node furthermore exists interior pne strategy profile remark leaf node satisfies assumption must otherwise violates first condition assumption since decreasing according lemma assumption implies holds every node graph state main result section show graphs achieve highest graph topologies average degree theorem consider total effort game graph assumptions let dli xmin every player let davg average extended neighborhood sizes xavg pne strategy profile xavg xmin xavg davg proof let function xmin defined inverse dli assumptions know strictly increasing strictly convex xmin therefore inverse function strictly increasing concave davg yields strict concavity xavg equality holds davg every node result states graphs identical node degrees larger worst case furthermore graph becomes dense bound grows average degree nodes graphs smallest average attack probability bound subsection answer complementary question regarding graph topologies smallest upper bound order highlight dependence use slightly modified notation proof following result particular denote xdi player proposition consider total effort games homogeneous players satisfy among connected graphs nodes tions let xmin star graph achieves smallest xdi proof recall remark xdi increasing function graph tree remove setp edges resulting subgraph tree reduces neighborhood sizes decreases xdi remains show among trees star graph minimizes xdi consider tree star graph consider node highest extended neighborhood size since graph star must exist leaf node connected node let neighbor denoted neighborhood size argue xdi decreases remove edge add edge operation xdu increases xdu node xdv decreases xdv node nodes xdi remains unchanged compute change xdi xdu xdu xdv xdv xdu xdu xdv xdv xdv xdv xdv xdv first inequality follows theorem two players extended neighborhood sizes xdu xdu second inequality holds therefore following strict concavity shown theorem tree star graph construct another tree reduces xdi figure prelec weighting functions parameters proof proposition relies properties quantities arise due nature behavioral weighting functions addition result star graphs achieve smallest security risk upper bound artifact total effort risk externalities assume attack node depends average security investment neighborhood words attacks targeted sense trying disconnect network interesting avenue future work characterize network topologies maximize complex network value functions considered gueye schwartz cerdeiro behavioral probability weighting targeted attacks comparative statics weighting functions section compare effect intensity probability weighting average equilibrium probability successful attack graphs analyzing graphs isolates effects heterogeneity degrees effects weighting function showed previous section graphs possess interior equilibrium upper bounds arise broader class graphs order compare two different weighting functions use parametric form proposed prelec parameter weighting function linear decreases magnitudes overweighting underweighting increase consider two total effort games graph size extended neighborhood nodes let cost parameters among players across two games first game homogeneous players weighting parameter second game homogeneous players weighting parameter words players first game significant overweighting underweighting true probabilities compared players second game let solutions equations xmin prelec weighting function weighting parameter proposition attack probability node equal respective interior pnes illustrate figure initially smaller starts increase quantity depends values state formally following lemma proof presented appendix lemma consider two prelec weighting functions parameters respectively let exists unique present main result section follows lemma theorem let true probability successful attack node interior pne graph nodes prelec weighting function parameter let intersection point defined lemma otherwise result shows attack probability node pne high enough greater players substantial underweighting probabilities smaller view increased security investments highly beneficial terms reducing perceived attack probabilities result attack probability node pne smaller game weighting parameter hand attack probability less players smaller find perceived reduction attack probabilities sufficient make high investment however players weighting functions closer linear observe greater perceived reduction probability due increased investment result players smaller average attack probability interior equilibria keep fixed increase neighborhood size eventually end regime equilibrium attack probability greater thus expected fraction nodes successfully attacked equilibrium smaller behavioral probability weighting graph sufficiently dense large number edges given number nodes vice versa weakest link best shot games nash equilibrium strategies weakest link best shot games special properties security level neighborhood determined investment single player one smallest largest investment respectively therefore first state following results characterize security investment isolated player proofs following results appendix lemma consider weighting function satisfies assumption let xmin exists unique proposition let defined lemma optimal investment single player otherwise optimal investment xmin analyze pne weakest link games proposition consider weakest link game connected graph homogeneous players pne nodes make identical investments defined lemma continuum pure nash equilibria successful attack probabilities nodes greater equal xmin additional equilibria including ones previous case attack probabilities close proof consider node security investment let smallest investment neighborhood attack probability node node reduces investment cost investment decreases attack probability remains unchanged therefore pne investment node must equal minimum investment extended neighborhood result connected graphs nodes must make identical security investments pne proposition states single node investing isolation would prefer invest suppose nodes identical security investment true attack probability node since player marginal utility positive player would unilaterally deviate make smaller investment therefore investment nodes would result pne xmin optimal investment single player invest therefore strategy profile node invests sufficiently small pne attack probability every node strategy profile players positive marginal utility prefer invest due continuity utility functions equilibria exist addition set equilibria attack probabilities least note investment nodes pne since note long graph remains connected structure plays role equilibrium investments attack probabilities first part result identical investments players holds true expectation maximizers players well main differences weighting functions twofold first large enough possible equilibrium true expectation maximizers players invest proposition probability weighting range possible equilibrium investments resulting attack probabilities greater second xmin investment players give rise pne true expectation maximizers probability weighting exist attack probabilities supported pne either close least finally xmin defined though optimal investment single player still case investment players give rise pne finally discuss pnes arise best shot games proposition consider best shot network security game homogeneous players strategy profile pne set nodes form maximal independent set invest according proposition nodes invest proof best shot game attack probability node function highest investment extended neighborhood suppose neighborhood player making investment accordance proposition since optimal investment single node make neighbors find investing level thereby reducing attack probabilities profitable therefore optimal strategy invest eliminates security investment cost csi attack probability unchanged therefore two nodes make nonzero investment must adjacent furthermore every node making zero investment must neighbor invests nonzero amount therefore set nodes making nonzero investment must belong maximal independent set converse also true maximal independent set investments determined proposition nodes investing constitute pne independent set characterization holds true expectation maximizers well nonlinear weighting functions change level investment true expectation maximizers nonzero figure graph topology analyzed example investment level one boundary points either proposition probability weighting equilibria entirely unprotected nonzero equilibrium investment one interior solutions numerical examples illustrate theoretical characterizations impacts network structure probability weighting equilibrium investments two numerical examples presented example consider graph shown figure nodes node decision maker prelec weighting function parameter extended neighborhood sizes range parameters satisfy assumptions total effort externalities sequential best response dynamics converged pne strategy profile example pne leaf nodes investment node invests nodes invest investments satisfy note node larger neighborhood size compared node yet invests node pne contrast equilibria arise without probability weighting proposition equilibrium investments either investment node node smaller degree furthermore investments leaf nodes larger neighbors shown proposition best shot externality set nodes form maximal independent set nonzero investment proposition graph shown figure nodes form maximal independent set another set consists nodes prelec weighting function value defined lemma accordingly equilibrium investment profile nodes maximal independent set invest nodes invest hand equilibrium investment nodes example discussed effects network structure pne investments total effort best shot games impact intensity overweighting underweighting probabilities graphs illustrated following example corroborates theoretical findings section example consider two graphs nodes shown figure nodes graph figure respectively figure extended neighborhood sizes respectively consider two types players prelec weighting functions parameters respectively derivative weighting functions players shown figure let players first consider graph shown figure players weighting functions parameter attack probability every node interior graph nodes graph nodes figure graph topologies analyzed example equilibrium equal players parameter corresponding attack probability case equilibrium secure players whose weighting functions closer linear contrast graph shown figure attack probabilities every node interior pnes equal players weighting function parameters respectively noted theorem succeeding discussion example illustrates equilibrium secure players substantial degree overweighting underweighting captured smaller parameter value graph sufficiently dense discussion conclusion studied class interdependent security games networks players exhibit certain behavioral attributes perception attack probabilities making security investment decisions analyzed three canonical interdependent security game models total effort weakest link iii best shot games nash equilibria total effort games much richer structural properties behavioral probability weighting corresponding equilibria players true expectation maximizers sharper overweighting small probabilities may disincentivize node reducing investment neighbors make high investments underweighting large probabilities leads equilibria nodes completely unprotected opposed equilibria without probability weighting effect behavioral probability weighting beneficial terms reducing security risk probability successful attack sufficiently high pne hand attack probability moderately high players weighting functions closer linear secure equilibrium obtained upper bound expected fraction nodes successfully attacked pne terms average degree nodes furthermore among class graphs given average degree davg davg graph pne achieves upper bound conversely star graphs achieve smallest average security risk upper bound among connected graphs graph connected nodes make identical investments weakest link game continuum attack probabilities independent graph structure arise pne hand strategy profile pne best shot game nodes making nonzero security investments form maximal independent set cases equilibria never completely unprotected behavioral probability weighting references saurabh amin galina schwartz shankar sastry security interdependent identical networked control systems automatica adam booij bernard van praag gijs van kuilen parametric analysis prospect theory functionals general population theory decision yann rachel kranton martin amours strategic interaction networks american economic review colin camerer george loewenstein matthew rabin advances behavioral economics princeton university press diego cerdeiro marcin dziubinski sanjeev goyal contagion risk network design working paper nicolas christin network security games combining game theory behavioral economics network measurements decision game theory security pages springer nicolas christin serge egelman timothy vidas jens grossklags benjamins empirical study incentivizing users ignore security advice financial cryptography data security pages springer richard cottle pang richard stone linear complementarity problem volume siam andrea galeotti sanjeev goyal matthew jackson fernando leeat yariv network games review economic studies vaibhav garg joseph camp heuristics biases implications security design ieee technology society magazine bahman gharesifard behrouz touri tamer jeff shamma convergence piecewise linear strategic interaction dynamics networks ieee transactions automatic control richard gonzalez george shape probability weighting function cognitive psychology jens grossklags benjamin johnson uncertainty security game game theory networks pages jens grossklags nicolas christin john chuang security insurance management networks heterogeneous agents acm conference electronic commerce pages assane gueye jean walrand venkat anantharam design network topology adversarial environment decision game theory security pages springer jack hirshleifer voluntary provision public goods public choice ashish hota shreyas sundaram interdependent security games behavioral probability weighting decision game theory security pages springer ashish hota siddharth garg shreyas sundaram fragility commons risk attitudes games economic behavior libin jiang venkat anantharam jean walrand bad selfish investments network security transactions networking benjamin johnson jens grossklags nicolas christin john chuang security experts useful bayesian nash equilibria network security games limited information esorics pages springer howard kunreuther geoffrey heal interdependent security journal risk uncertainty aron laszka mark felegyhazi levente buttyan survey interdependent information security games acm computing surveys csur tianming narayan mandayam users interfere protocols prospect theory wireless networks using random access data pricing example ieee transactions wireless communications mohammad hossein manshaei quanyan zhu tansu alpcan tamer hubaux game theory meets network security privacy acm computing surveys csur benjamin yolken john mitchell nicholas bambos security among interdependent organizations computer security foundations symposium pages ieee kien nguyen tansu alpcan tamer stochastic games security networks interdependent nodes international conference game theory networks pages ieee cameron nowzari victor preciado george pappas analysis control epidemics survey spreading processes complex networks ieee control systems magazine efe real analysis economic applications volume princeton university press martin osborne ariel rubinstein course game theory mit press ranjan pal pan hui modeling internet security investments tackling topological information uncertainty decision game theory security pages springer drazen prelec probability weighting function econometrica pages walid saad arnold glass narayan mandayam vincent poor toward grid behavioral perspective proceedings ieee bruce schneier psychology security progress pages springer galina schwartz saurabh amin assane gueye jean walrand network design game reliability security failures allerton conference communication control computing pages galina schwartz nikhil shetty jean walrand contracts fail reflect allerton conference communication control computing pages ieee amos tversky daniel kahneman advances prospect theory cumulative representation uncertainty journal risk uncertainty hal varian system reliability free riding economics information security pages springer proofs pertaining equilibrium characterization proof lemma proof ease notation drop subscript since proof holds every player let total security investment neighbors size extended neighborhood player assumption interval falls one four different cases case case interval lies right therefore attack probability figure thus consequently case case therefore player feasible investment strategy first order condition satisfied equality investment resulting attack probability result value would satisfy first order necessary condition optimality hand however assumption would therefore result candidate optimal investment also satisfies second order sufficient condition since xmin since optimal solution must property continuous linearly decreasing boundary values respectively remaining two cases case case interval lies region therefore true attack probability result case case three candidate solutions utility maximization true attack probability resulting strategies players first show player would always prefer invest investing denote resulting attack probability investing note using compute inequality due fact strictly concave xmin xmin therefore player always prefer boundary solution potential interior solution leads possibility best response might discontinuous jump value region however show second third conditions assumption player would always prefer invest investing compute last inequality follows function strictly decreasing function indeed therefore player would always prefer invest investing regardless value including proof proposition proof without loss generality let players ordered nci largest solution solution player define player players investing pne true attack probability pne must least objective attack probability would pne since would always exist player positive investment would prefer invest suppose two pnes different probabilities successful attack consider strategy profile smaller attack probability note ruled possibility two exhaustive cases either player player let player definition quantities therefore similarly case second pne true attack probability players would continue invest possibility players investing nonzero amounts true attack probability would decrease contradicting assumption proof case player follows identical arguments linear complementarity problem formulation exploit structure best response presented represent nash equilibrium strategy profile solution linear complementarity problem lcp proposition consider total effort game graph vertices adjacency matrix let dli cii xmin every player assumptions strategy profile nash equilibrium game solution lcp given jth entry proof solution lcp vector write vector investments players strategy profile set variables assume nonzero values corresponding investment definition lcp solution secondly player second inequality ensures therefore investment player solution lcp feasible formally given vector matrix lcp problem finding solution vector iii finally gives every node result therefore investments satisfy second part equation also resulting investments satisfy third part equation finally first part equation holds concludes proof comprehensive discussion lcps different solutions algorithms found cottle structure lcp often determines performance different algorithms show lcp defined proposition satisfies certain properties proven proposition guarantee convergence lemke pivotal method converge solution pne strategy profile problem non degenerate convergence result due cottle proposition lcp defined proposition matrix copositive sol proof proof first statement consider vector second part consider solution lcp must result must vector thus sol proofs pertaining effects network structure proof lemma proof denote inverse function xmin consider function second inequality follows differentiating inverse function xmin weighting function satisfies since strictly decreasing strictly decreasing interpreted size extended neighborhood node proof proposition proof first show attack probability node pne recall lemma best response player aggregate investment neighbors condition satisfied since result investment player pne lies investment node resulting attack probability otherwise sum investments node neighbors satisfy first order condition equality case resulting attack probability exactly upper bound average attack probability follows averaging nodes interior pne player experiences attack probability exactly equal therefore bound holds equality proofs pertaining impact weighting function first state prove following lemma whose result useful proof lemma lemma function exp strictly decreasing proof compute exp exp thus result since implies therefore proof lemma proof first derivative prelec weighting function given therefore given exp definition furthermore result becomes smaller thus exists uniqueness follows strict monotonicity proved lemma order prove second third parts lemma suffices show therefore compute previous discussion lemma therefore concludes proof proofs pertaining weakest link best shot games proof lemma proof consider function xmin xmin xmin xmin furthermore therefore must root xmin suppose exist convex xmin since strictly contradiction thus unique second part suppose exists convexity contradicts definition proof proposition proof single player investing isolation three candidate solutions utility maximization solutions defined section note since finite investing security utility maximizer analysis case lemma therefore potential interior solution satisfies first order condition boundary solution player always prefers boundary solution compare utilities solutions compute thus lemma player prefers invest otherwise optimal investment
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link selection hybrid systems statistical queueing constraints marwan hammouda sami anna maria vegni harald haas nov peissig abstract radio frequency visible light communications vlc technologies investigated indoor environments enhance network performances address specific qos constraints paper explore benefits employing technologies qos requirements imposed limits buffer overflow delay violation probabilities important metrics designing low latency wireless networks particularly consider scenario utilizes vlc links data transmission indoor environment propose link selection process transmitter sends data link sustains desired qos guarantees considering data source employ maximum average data arrival rate transmitter buffer bounds data buffering delay main performance measures formulate performance measures assumption links subject average peak power constraints furthermore investigate performance levels either one two links used data transmission used simultaneously finally show impacts different physical layer parameters system performance numerical analysis index terms visible light communications constraints buffering delay bound link selection hammouda peissig institute communications technology leibniz hannover hanover germany peissig vegni comlab laboratory department engineering roma tre university roma italy haas institute digital communications research development centre university edinburgh edinburgh work supported european research council starting ntroduction demand mobile communications triggered quest technical solutions support stringent qos constraints thanks significant advances white light emitting diodes leds research availability extensive unregulated spectrum visible light communication vlc emerged promising technology utilize leds simultaneously data transmission illumination since many unique aspects compared communication technologies moreover improve data security light penetrate surrounding walls also sustain green agenda minimize installation costs require extensive infrastructure nevertheless attention must paid certain limitations challenges vlc systems smaller coverage strong dependence components achievable rates vary spatial fluctuations order overcome constraints researchers proposed hybrid systems end users benefit wide coverage area systems support stable rates vlc systems provide networks practically feasible vlc systems coexist without causing interference operate environment offices rooms comparing hybrid systems systems employ either vlc authors demonstrated remarkable increase data transmission throughput energy efficiency delay performance hybrid systems moreover authors projected hybrid system use vlc links communication links communication system authors ones investigated handover load balancing mechanisms respectively alternatively considering outdoor environment authors studied transmission scenario system switch links vlc links comparing ratio levels link regarding system setting authors assumed vlc links transmission rates proposed transmission scheme transmitter sends data employing links simultaneously aforementioned studies analyzed hybrid systems mostly physical layer perspective concentrate data link layer metrics limits buffer overflow buffering delay probabilities much needed noting dramatic increase demand reliable services recent years mobile video traffic making total global data traffic end addition physical layer performance metrics need qos metrics analysis tool physical layer features performance levels layer context authors performed analysis physical layers many different scenarios regarding markovian data arrival process transmitter buffer statistically varying data service process wireless channel authors characterized maximum average data arrival rate transmitter buffer presence statistical constraints buffer overflow probability hand best knowledge studies investigated performance levels vlc systems example authors employed effective capacity performance measure resource allocation schemes vlc systems heterogeneous networks composed vlc links note authors concentrated case constant data arrival rates transmitter buffer realistic certain practical settings details effective capacity refer paper assuming modeled data arrival process transmitter buffer transmitter use vlc channels data transmission investigate performance gains achieved hybrid system operates statistical qos constraints inflicted limits buffer overflow delay violation probabilities perform analysis hybrid systems physical layers employ first maximum average data arrival rate transmitter buffer considering asymptotic buffer overflow probability approximation buffering delay violation probability main performance measures propose mathematical toolbox system designers performance analysis hybrid systems work low latency conditions summarize main contributions paper follows assuming vlc links subject average peak power constraints express maximum average data arrival rate transmitter data service process transmitter receiver support qos constraints either vlc link used links simultaneously used data transmission propose three different link usage strategies base two proposed strategies assumption receiver multihoming capability thus data transmission possible one link either vlc link third strategy assume link aggregation possible data transmitted links simultaneously following power sharing policy obtain data backlog buffering delay violation probability bounds considering proposed link usage strategies particularly provide rudimentary model communication systems operate qos constraints employ vlc links two different mechanisms model easily invoked settings two different mechanisms well reason behind communications boost performance levels increased degree freedom therefore order introduce model smoothly make easier readers understand objective also benefit existing literature vlc studies however best knowledge analytical framework provided paper investigate qos performance addressed studies one aspect hybrid system vlc links provide transmission rates links provide rates vary time communication setting depends solely link may suffer low transmission rates longer data backlogs transmitter buffer however communication setting utilize vlc links instance take advantage constant transmission rate vlc link transmission rate link falls certain level rest paper organized follows introduce hybrid system section provide detailed descriptions vlc channels section iii provide performance analysis link selection process section substantiate results numerical demonstrations conclude paper section relegate proofs appendix ystem odel consider network access controller provides connection either access point vlc access point positioned different locations indoor environment seen figure herein assume scenario network access controller acts transmitter user acts receiver sequel use transmitter receiver instead network access controller user respectively initially analytical framework provided paper easily extended scenario details refer remark section iii fig hybrid system transmitter receives data source sources stores packets buffer subsequently sends data packets frames seconds receiver following given transmission strategy receiver considered equipped also note vlc coverage area generally smaller coverage area finally consider system assume network controller constrained fixed average power budget denoted pavg data transmission sequel initially introduce vlc channels describe data source model channel model data transmission channel relation time instant expressed complex channel input output access point transmitter receiver respectively complex channel input subject average power constraint pavg pavg peak power constraint ppeak ppeak complex channel fading gain arbitrary distribution finite average power furthermore consider channel assume fading gain stays constant one transmission frame seconds available bandwidth channel channel fading gain lth time frame note symbols transmitted one time frame moreover fading gain changes independently one frame another meanwhile additive noise model also considered receiver circularly symmetric complex gaussian random variable variance noise samples assumed independent identically distributed assume reliable data transmission exists long transmission rate channel lower equal instantaneous mutual information channel input particular transmission rate lth time frame bits per frame lower equal instantaneous mutual information bits per frame channel input channel output reliable data transmission occurs bits decoded correctly receiver assume large enough decoding error probability negligible lower bound maximum mutual information channel capacity provided input circularly truncated gaussian distribution therefore set instantaneous data transmission rate lower bound assume input circularly truncated gaussian distribution specifically reliable transmission lth time frame bpavg exp bits per frame solutions following equations bppeak exp bppeak bppeak pavg bppeak exp ppeak changes one time frame another maximum instantaneous mutual information frame function channel fading gain vlc channel model assume transmitter employs intensity detection principally vlc access point transmitter equipped led data modulated intensity emitted light receiver collects light using generates assumption based known result literature transmitting data rates less equal instantaneous mutual information high reliability thus decoding error negligible electrical current voltage proportional intensity received light besides know vlc channels typically composed well components however majority collected energy comes components typical indoor scenarios therefore assume vlc channel flat dominant component channel gain vary data transmission long receiver accordingly relation vlc channel vlc access point transmitter receiver time instant given follows channel input output respectively conversion efficiency detector responsivity amperes per watt additive noise receiver real gaussian random variable variance noise samples independent identically distributed moreover optical channel gain recall data transmitted vlc link modulated light illuminates environment hence assuming operation range radiated optical power limited pmin pmax light modulate data power levels pmin pmax pmin pmax result data bearing symbol pmin limited follows pmax pmin ppeak hence relation assuming expected value bounded pavg pavg available bandwidth optical channel set transmission rate channel bits per frame lower bound channel capacity defined follows pavg exp ppeak ppeak variations vlc channels fading mitigated since area much larger light wavelength sec thus vlc channels known fact almost true regardless frame duration user stationary mobile indoor environments users either stationary move slowly fig state transition model data arrival process pavg ppeak ppeak pavg ppeak unique solution pavg ppeak constant values mutual information vlc link change time due strong channel component channel gain change input distribution refer source model regarding data arrival process transmitter buffer consider discretetime markov states time frame source state one time frame data source sources arrives transmitter buffer state consider constant data arrival rate bits per frame number bits arriving transmitter buffer zero state shown fig transition probability state state denoted transition state state denoted probability staying state probability staying state hence state transition matrix becomes let pon poff probabilities data arrival process states respectively pon poff following equality project certain data arrival models markov processes instance voice sources generally modeled states pon poff pon poff subsequently pon average data arrival rate transmitter buffer poff hence bits per frame finally note following analysis easily extended source models iii erformance nalysis section investigate performance levels aforementioned system achieves opportunistically exploiting vlc channels data transmission herein data initially stored transmitter buffer transmission assume certain constraints applied amount data buffer buffering delays therefore examine system qos constraints associated buffer overflow data backlog buffering delay express decay rate tail distribution queue length loge lim stationary queue length see fig buffer overflow threshold denotes decay rate tail distribution data backlog accordingly approximate buffer overflow probability large threshold qmax qmax notice buffer overflow probability decays exponentially rate controlled also defined qos exponent basically larger implies stricter qos constraints whereas smaller corresponds looser constraints recall outgoing service transmitter queue given data sent channel given data sent vlc channel data arrival markov process bits per frame state zero bits state hence assuming buffer size infinite considering independent data arrival data service processes exist unique respectively asymptotic generating functions total amount bits arriving transmitter buffer total service transmitter channel total service transmitter vlc channel respectively theorem particular asymptotic generating functions loge loge refer example obtaining generating functions using express maximum average data arrival rate transmitter buffer service process channel sustain loge loge given derivation refer appendix likewise using following steps appendix also express maximum average data arrival rate transmitter buffer service process vlc channel sustain loge loge given accordingly relation pavg ppeak moreover special case expressions provide effective capacity maximum sustainable constant data arrival rate channel process given qos constraints another special case independent past current states loge loge link selection policy section focus channel selection process transmitter employs set maximum average data arrival rate qos constraints objective channel selection process particular transmitter chooses channel service process maximizes average data arrival rate transmitter buffer notice transmission rate vlc channel constant whereas transmission rate channel varies due changes channel fading gain due fact channel fading gains known receiver well transmitter provide following proposition proposition aforementioned system transmitter sends data receiver vlc link following condition given qos exponent holds loge proof see appendix proposition states maximum attainable transmission rate vlc channel greater side transmitter perform transmission vlc link statistical variations channel deteriorates buffer stability meanwhile special case loge specifically constant rate vlc channel greater effective capacity channel vlc channel chosen data transmission following present two transmission strategies data transmitted link highest instantaneous transmission rate one time frame data transmitted links simultaneously transmission strategy analysis obtain performance levels transmitter chooses either two channels data transmission following link selection process based maximum average data arrival rates service processes channel support hand exists fast stable handover mechanism transmitter receiver transmitter forward data receiver link provides maximum lower bound instantaneous mutual information corresponding channel instance lower bound instantaneous mutual information channel lth time frame greater lower bound instantaneous mutual information vlc channel transmitter sends data link corresponding time frame otherwise prefers sending data vlc link respectively establish channel selection criterion follows transmitter sends data link bpavg tvb exp otherwise sends data vlc link aforementioned selection test also considered outage condition channel link outage noting channel fading gain changes independently one time frame another channel generating function service process becomes loge conditional expectation given probability density function hence maximum average data arrival rate transmitter buffer hybrid service process sustain qos constraints specified becomes loge transmission strategy different aforementioned protocols consider transmitter sends data links simultaneously frame assume receiver multihoming capability assume transmission scheme data streams transmitted vlc links different independent indeed scenario feasible since light waves cause interference assume power allocation policy two links average power constraint link set pavg pavg one vlc link set pavg pavg pavg total average power constraint instantaneous transmission rate channel lth time frame becomes pavg exp bits per frame solutions similarly following transmission rate vlc channel becomes avg ppeak exp ppeak pavg ppeak ppeak pavg ppeak unique solution follows total transmission rate frame sum transmission rates links generating function readily expressed loge loge maximum average data arrival rate equal loge finally remark optimal value maximizes sum transmission rate obtained numerically remark employing aforementioned strategies requires perfect knowledge vlc channels transmitter side frame assume channel estimation performed receiver forwarded transmitter feedback channel increases signaling overhead moreover applying transmission strategy increases implementation complexity power sharing performed transmission frame implementation perspective exploiting transmission strategy limited receiver multihoming capability enables perform link aggregation receive data different transmission technologies simultaneously hand maximum average data arrival rate performance measure thus selection process explained proposition considered operation performed periods multiple transmission frames therefore implementation complexity decreases operation also proposed remark selection process also employed cases access point controller choose two transmission links example let transmission links let maximum average data arrival rates transmitter buffer sustained links proposition paper updated solution following maximization problem transmission link max similarly let instantaneous transmission rates provided link lth time frame link selection criterion transmission strategy updated solution following maximization problem transmission link max subsequently generating function service process loge pri pri probability transmitter chooses link data transmission finally receiver multihoming capability transmission strategy applied power sharing maximize total transmission rate frame impacts handover delay handover delay occurs transmitter moves one link case transmission strategy strategy data transmission one time frame performed link provides maximum instantaneous transmission rate frame particularly transmitter switches one link stay link end time frame comparing instantaneous transmission rates links given denotes duration one single handover phase let initially assume frame duration larger handover phase sake simplicity divide frame duration equal particularly series data transmission phase followed one handover process transmitter changes transmission link another series data transmission phase transmitter stays transmission link analytical representation model buffer activity end markov process shown fig states first states state state represent data transmission vlc link state represents handover process vlc link link similarly subsequent states state state represent data transmission link state represents handover process link vlc link notice state transition probability state state data transmission link completed link change may occur end nth hand state either transmitter changes link system enters state probability transmitter stays link system enters state probability similarly state either transmitter changes link system enters state probability transmitter stays link system enters state probability finally system moves state state state state probability end one handover phase transmitter starts data transmission seen transmitter sends data link otherwise sends data vlc link therefore specific case express pji pji transition matrix otherwise pji state transition probability state state generating function hybrid system provided follows chap example loge spectral radius matrix diagonal matrix whose components moment generating functions processes states notice transmitted bits removed transmitter buffer ends nth frames therefore moment generating functions nth frames respectively however bits removed states service rates states effectively zero hence moment generating functions fig state transition model hybrid scenario handover states moreover unique qos exponent obtained bounds aforementioned results provide performance analysis particularly analysis obtained number time frames large hand nonasymptotic bounds regarding statistical queueing delay characterizations transmitter buffer interest system designers well therefore address framework network calculus consider theorem states minimal bound queue length found given buffer overflow probability particularly given data service process section markov modeled data arrival process section buffer threshold expressed inf given buffer overflow probability loge max sup sup loge supt supt loge pon poff buffer violation probability free parameters notice also generating function service process whereas generating function arrival process depends current state moreover remark goes infinity expressed herein refer calculation details likewise data service process section employed inf sup loge loge sup notice minimized moreover smaller zero therefore service channel chosen service process sup loge supt assuming fast stable handover mechanism transmitter receiver service channel selection process described transmission strategy characterize delay bound follows inf qrv loge qrv sup max given remark let assume protocol exists transmitter buffer minimal bound buffering delay expressed follows theorem qrv inf inf inf service process service process transmission strategy employed remark consider performance measure maximum average data arrival rate data buffer formulate link selection employing transmission rates provided vlc links analytical framework easily extended general scenario regarding rate allocations user transmission links data arrival processes transmitter buffer regard refer fig section employ multiple access fdma multiple access tdma protocols numerical illustrations moreover fig vlc channel model paper different system sum throughput maximized system average power consumption minimized studies framework joint resource allocation link assignment process employed provided optimization problems principle mixed integer programming problems mathematically intractable therefore main optimization problems decomposed solvable iterative algorithms provided remark link selection process easily adopted scenarios receiver mobile one needs consider generating function vlc link changing transmission rates loge base channel selection process link increases maximum average data arrival rate transmitter buffer transmission strategy transmission strategy channel fading gains vlc links instantaneously known transmitter time frame aforementioned analysis different currently paper even transmission rates vary due mobility also note user mobility normally low indoor scenarios regard refer umerical esults section present numerical results substantiate theoretical findings unless otherwise specified set transmission time frame milliseconds assume led vlc access point transmitter lambertian radiation pattern transmitter receiver planes parallel assume transmitter directed downwards receiver directed upwards depicted fig however theoretical results easily adopted different positional settings table imulation parameters vlc system led half intensity viewing angle field view fov physical area channel bandwidth mhz conversion efficiency optical concentrator gain vertical distance noise power spectral density system channel bandwidth mhz path loss exponent rician factor standard deviation ambient temperature herein channel gain given follows rect surface area distance led transmitter receiver normal distance transmitting receiving planes moreover optical concentrator gain photodiode angle incidence field view maximum angle light emitted led detected addition cos lambertian index led half intensity viewing angle rect rect otherwise finally thermal noise power noise power spectral density regarding channel consider rician fading distribution rician factor channel realizations independent identically distributed circularly symmetric complex gaussian random variables mean variance setting reasonable value reflect channel characteristics millimeter wave range communications well path loss decibels function distance access point transmitter receiver given dref loge dref dref path loss reference distance dref operating frequency ghz addition path loss exponent represents shadowing fig maximum average arrival rates vlc links function average power limit pavg different values power ratio qos exponent bpf bits per frame effect assumed gaussian random variable standard deviation expressed finally thermal noise power receiver boltzmann constant ambient temperature table summarizes simulation parameters unless otherwise stated finally setting average transmission power constraint pavg transmission strategies define power ratio vlc links pavg ppeak pavg ppeak transmission strategies consider scenario transmitter vlc access point access point shown fig receiver located distance vlc access point user located horizontal distance cell center distance access point fig plot maximum average data arrival rates transmitter buffer functions average power constraint pavg different power ratios vlc links employed respectively results fig fig observe maximum average data arrival rates increase faster increasing average power constraint vlc empirical values different indoor scenarios provided instance value ranges varies inside buildings pavg dbm pavg dbm fig maximum average arrival rates vlc links function qos exponent different values average power limit pavg source statistics bpf bits per frame link link instance maximum average data arrival rate increases bits per frame pavg increasing dbm dbm vlc link whereas increases bits per frame link observe behavior peak power constraint increases explain result stochastic nature transmission rates channel particularly instantaneous transmission rate channel becomes low data packets accumulated transmitter buffer therefore order sustain qos constraints transmitter buffer accept data lower arrival rates hand transmission rate vlc channel constant maximum average data arrival rate increases almost linearly increasing transmission rate vlc channel moreover performance link better vlc link average power constraint lower performance vlc link better link average power constraint higher fig plot functions qos exponent different values pavg dbm fig pavg dbm fig performance link better vlc link low whereas performance link decreases faster increasing becomes less performance vlc link words stochastic nature channel prevents link supporting data arrival rates transmitter buffer qos constraints stringent indeed maximum average data arrival rate link supports approaches zero exponentially increasing regardless average power constraint source statistics however link support higher data arrival rates qos fdma power bandwidth tdma time fig maximum average arrival rates vlc links function number served receivers equivalently receiver allocated resources different values led viewing angle pavg dbm bpf bits per frame constraints looser observe increasing decreasing results better performance values vlc links probability state pon average arrival rate transmitter buffer increase however effect source statistics performance values much less qos constraints especially link words randomness service process higher impact system performance randomness arrival process typical indoor scenarios vlc access points serve multiple receivers applied sharing available resources power time bandwidth among served receivers herein assume transmitter employs commonly known fdma tdma schemes links fig plot maximum average data arrival rate per user given users uniformly positioned within coverage area vlc access point observe performance per user link generally much higher performance per user vlc link number receivers basically system serve users link vlc link qos constraints interest addition results fig agree results fig performance vlc link highly affected decreasing average power constraint finally see decreasing led viewing angle performance vlc link becomes better transmission power concentrated smaller areas notice also vlc channel gain affected lambertian index herein show performance sensitivity vlc links allocated transmission resources power time bandwidth given available resources equally shared among users also show framework easily invoked scenario explore system performance respect receiver location set cartesian coordinates vlc access point coordinates access point coordinates receiver particularly consider following strategies strategy transmitter sends data link maximum average arrival rate expressed strategy transmitter sends data vlc link maximum average arrival rate expressed transmission strategy transmitter sends data link provides highest transmission rate maximum average arrival rate expressed transmission strategy transmitter sends data links simultaneously following power allocation policy maximize transmission rate maximum rate expressed fig plot maximum average arrival rate function coordinates receiver location different qos constraints seen fig fig position receiver impact performance levels link necessarily performance level stays almost constant receiver stays defined range however performance levels strategies affected position receiver maximum average data arrival rate increases receiver gets closer point constant transmission rate vlc access point receiver increases stochastic nature link mitigated increasing rate vlc link seen fig fig maximum average data arrival rate goes zero strategy receiver goes coverage area vlc access point similarly seen fig fig fig fig maximum average data arrival rate becomes equal one strategy receiver goes coverage area vlc access point furthermore transmission strategy provides higher performance levels strategy vlconly strategy transmitter employing transmission strategy sends data link instantaneous transmission rate link higher rate vlc link mitigates lower transmission rates link fig maximum average arrival rates different selection strategies function receiver position terms different values pavg dbm bpf bits per frame cell center cell edge fig maximum average arrival rate function different values qos exponent user position terms pavg dbm sending data vlc link finally transmission strategy outperforms strategies however performance gap transmission strategy transmission strategy necessarily large hence advantageous employ transmission strategy order avoid hardware complexity follows addition multihoming capability transmission strategy fig assuming handover process causes transmission delay handover process takes seconds plot maximum average data arrival rate transmission strategy function considering different user locations noting smaller means longer handover period observe transmission performance highly affected handover process increasing performance levels approach values obtained handover delay moreover maximum average data arrival rates higher fig fig constant transmission rate vlc link higher user center subsequently fig plot maximum average data arrival rates functions vertical distance vlc access point receiver set position receiver horizontal distance vlc cell center performance level vlc link zero cell area small cover point user stands tan cell radius performance levels strategies except strategy increase value decrease increasing increase led viewing angle relatively less increase beginning therefore pavg dbm pavg dbm fig maximum average arrival rate different transmission strategies function vertical distance different average power limit pavg bpf bits per frame user effectively getting closer cell center rate vlc link words gain achieved getting closer cell center higher expected degradation due increasing cell radius however increasing beyond certain value user gets far away vlc access point hence radiated power spreads area eventually leads decreased transmission rates vlc link therefore gain vlc link vanishes distance vlc access point becomes larger delay bounds aforementioned results analyze system performance following provide results regarding bounds bounds buffering delay experienced data transmitter buffer particularly plot delay bound function state transition probability state state data arrival process different values state transition probability state state data arrival process set average data arrival rate value close average data service transmission rate transmission channel note average data service rate transmission channel depends chosen transmission strategy delay bound highest strategy seen fig whereas lowest strategy seen fig however arrival rate strategy supports higher rate strategy supports interestingly hybrid strategies support higher arrival rates less delay bounds transmission strategy outperforms others herein system takes transmission kbpf transmission kbpf transmission kbpf transmission kbpf fig delay bounds different transmission strategies function transition probability different values pavg dbm bpf bits per frame advantage occasional higher rates links mitigates lower rates link constant transmission rate vlc link moreover increasing decreasing cause delay bound increase finally explore effects data arrival rate delay bound performance fig set consider different average power constraints pavg dbm delay bounds increase asymptotically average arrival rate approaches average data service rates channels average data arrival rate greater average data service rate one channel system becomes unstable long buffering periods expected moreover seen fig delay bounds minimum strategy ranges average data arrival rates smaller strategies seen fig delay bounds maximum strategy however hybrid strategies outperform others hybrid strategies take advantage vlc link rate link goes drastically utilize link channel conditions better vlc link provides stability transmission transmission transmission transmission fig delay bounds different transmission strategies function arrival rate different values pavg decreases delay bounds link increases range average data arrival rate supported finally fig see increasing average power potentially improve system performance terms buffering delay onclusions paper analyzed performance hybrid system statistical qos constraints inflicted limits buffer overflow delay violation probabilities provided analysis regarding physical layers employing maximum average arrival rate transmitter buffer delay bounds main performance measures proposed analyzed three strategies vlc links utilized data transmission formulated performance levels achieved proposed strategies numerical results shown technology beneficial lower average power constraints looser qos requirements moreover shown utilizing vlc technology data transmission either alone hybrid transmission strategy potentially enhance system performance terms delay performance lowers buffering delay bounds compared technology particularly data arrival rates transmitter buffer low vlc links provide lower queueing delays links links support higher data arrival rates transmitter buffer ppendix derivation recall data arrival rate buffer data source state pon probability state thus average data arrival moreover rate loge loge solving aforementioned equation given obtain loge result formulate maximum average data arrival rate set follow steps proof proposition based link selection process vlc link selected maximum average arrival rate supported vlc link given maximum average arrival rate supported link given since logarithm monotonic increasing function condition satisfied let expressed following quadratic inequality solving equation results two solutions two ranges loge loge setting note loge hence implies loge therefore loge solution region completes proof eferences rahaim vegni little hybrid radio frequency broadcast visible light communication system proc ieee global telecommun globecom workshops basnayaka haas design analysis hybrid radio frequency visible light communication system ieee trans chowdhury katz cooperative data download move indoor hybrid hotspot coverage trans emerging telecommun vol basnayaka haas hybrid vlc systems improving user data rate performance vlc systems proc ieee veh technol conf spring bao zhu song protocol design capacity analysis hybrid network visible light communication ofdma systems ieee trans veh vol shao khreishah rahaim elgala ayyash little indoor hybrid internet access system proc ieee int conf mobile hoc sensor kashef ismail abdallah qaraqe serpedin energy efficient resource allocation mixed heterogeneous wireless networks ieee sel areas commun vol vegni little handover vlc systems cooperating mobile devices proc int conf computing netw commun icnc wang wang qian dai yang efficient vertical handover scheme heterogeneous systems opt commun vol liang tian fan bai novel vertical handover algorithm hybrid visible light communication lte system proc ieee veh technol conf fall shao khreishah delay analysis unsaturated heterogeneous small cell wireless networks case coexistence ieee trans wireless vol zhang hanzo cooperative load balancing hybrid visible light communications wifi ieee trans vol wang haas dynamic load balancing handover hybrid networks ieee lightwave vol stefan burchardt haas area spectral efficiency performance comparison vlc femtocell networks proc ieee int conf commun icc wang basnayaka haas optimization load balancing hybrid networks ieee trans vol kazemi uysal touati outage analysis hybrid systems based markov chain modeling proc ieee int workshop opt wireless commun iwow chatzidiamantis karagiannidis kriezis matthaiou diversity combining hybrid systems psk modulation proc ieee int conf commun icc cisco visual networking index global mobile data traffic forecast update white paper http paper chang stability queue length delay deterministic stochastic queueing networks ieee trans autom control vol negi effective capacity wireless link model support quality service ieee trans wireless vol choudhury lucantoni whitt squeezing atm ieee trans vol kelly notes ective bandwidths stochastic networks theory applications akin gursoy effective capacity analysis cognitive radio channels quality service provisioning ieee trans wireless vol hammouda akin peissig effective capacity cognitive radio broadcast channels proc ieee global telecommun conf globecom ozmen gursoy wireless throughput energy efficiency random arrivals statistical queuing constraints ieee trans inf theory vol jin zhang dong hanzo resource allocation constraints communication ieee access vol jin zhang hanzo resource allocation constraints heterogeneous femtocell ieee transactions wireless communications vol letzepis fabregas outage probability gaussian mimo optical channel ppm ieee trans vol ozarow shamai wyner information theoretic considerations cellular mobile radio ieee trans veh vol shamai capacity average quadrature gaussian channels ieee trans inf theory vol komine nakagawa fundamental analysis communication system using led lights ieee trans consumer electronics vol kahn barry wireless infrared communications proceedings ieee vol pohl jungnickel von helmolt channel model wireless infrared pimrc jungnickel european view next generation optical wireless communication standard proc ieee conf standards commun networking cscn oct dimitrov haas principles led light communications towards networked cambridge university press lapidoth moser wigger capacity optical intensity channels ieee trans inf theory vol heffes lucantoni markov modulated characterization packetized voice data traffic related statistical multiplexer performance ieee sel areas vol chang zajic effective bandwidths departure processes queues time varying capacities proc ieee infocom vol chang performance guarantees communication networks springer science business media cheung huang huang tradeoff predictive scheduling integrated cellular networks ieee sel areas vol chang performance guarantees communication networks springer science business media fidler rizk guide stochastic network calculus ieee commun surveys tutorials vol ciucu burchard liebeherr scaling properties statistical bounds network calculus ieee trans inf theory vol fidler becker boundaries composable model sources systems ieee trans wireless vol zhou yang dynamic network resource optimization hybrid vlc radio frequency networks selected topics mobile wireless networking mownet international conference dehghani soltani safari haas limited feedback resource allocation visible light communication networks proceedings international workshop visible light communications systems acm barry kahn krause lee messerschmitt simulation multipath impulse response indoor wireless optical channels ieee sel areas vol akl tummala indoor propagation modeling ghz ieee networks international association science technology development rappaport mcgillem uhf fading factories ieee sel areas vol neskovic neskovic paunovic modern approaches modeling mobile radio systems propagation ieee communications surveys tutorials vol
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developing machine learning systems formal mathematics daniel selsam percy liang david dill jun abstract noisy data objectives model misspecification numerical instability cause undesired behaviors machine learning systems result detecting actual implementation errors extremely difficult demonstrate methodology developers use interactive proof assistant implement system state formal theorem defining means system correct process proving theorem interactively proof assistant exposes implementation errors since error program would cause proof fail case study implement new system certigrad optimizing stochastic computation graphs generate formal proof gradients sampled system unbiased estimates true mathematical gradients train variational autoencoder using certigrad find performance comparable training model tensorflow introduction machine learning systems difficult engineer many fundamental reasons first foremost implementation errors extremely difficult alone localize many potential causes undesired behavior machine learning system example implementation error may lead incorrect gradients cause learning algorithm stall symptom may also caused noise training data poor choice model unfavorable optimization landscape inadequate search strategy numerical instability issues common often assumed undesired behavior caused one result actual implementation errors persist stanford university stanford correspondence daniel selsam dselsam proceedings international conference machine learning sydney australia pmlr copyright author standard methodology test empirically debug program test methodology verify mathematically debug specify program prove figure comparison methodology standard methodology developing machine learning systems instead relying empirical testing expose implementation errors first formally specify system required terms underlying mathematics try formally prove system satisfies specification process proving exposes implementation errors systematically program prove debug loop eventually terminates system proof correctness indefinitely without errors even difficult detect stochastic programs since errors may distort distributions random variables may require writing custom statistical tests detect machine learning systems also difficult engineer require substantial expertise mathematics linear algebra statistics multivariate analysis measure theory differential geometry topology even understand machine learning algorithm supposed thought correctly even simple algorithms gradient descent intricate justifications large gap mechanics intended mathematical semantics theano bergstra development almost decade yet recent github issue https reporting model loss continually diverges middle training various experiments comparing behavior systems team agree likely implementation error writing neither cause error set models affects determined developing machine learning systems formal mathematics paper demonstrate practical methodology building machine learning systems addresses challenges enabling developers find eliminate implementation errors systematically without recourse empirical testing approach makes use tool called interactive proof assistant gordon gordon melham harrison nipkow owre coq development team moura consists programming language language state mathematical theorems set tools constructing formal proofs theorems note use term formal proof mean proof formal system checked machine approach developers use theorem language state formal mathematical theorem defines means implementation terms underlying mathematics multivariate analysis upon implementing system using programming language developers use proof tools construct formal proof theorem stating implementation correct first draft implementation often errors process interactive proving expose errors systematically yielding impossible proof obligations implementation errors fixed developers able complete formal proof certain implementation errors respect specification moreover proof assistant check formal proof automatically human needs understand proof correct order trust figure illustrates process proving correctness machine learning systems requires building tools insights two distinct fields program verification leroy klein chlipala chen aimed prove properties computer programs formal mathematics rudnicki gonthier gonthier hales aimed formally represent generate proofs mathematical theorems fields make use interactive proof assistants tools libraries design patterns developed two fields focus different problems remained largely incompatible methodology outlined familiar program verification community reasoning formally mathematics underlies machine learning familiar formal mathematics community proving sophisticated mathematical properties large stochastic software systems new goal poses many new challenges explore challenges demonstrate softplus sigmoid cost softplus cost figure example stochastic computation graph representing simple variational autoencoder stochastic nodes indicated rounded rectangles loss function graph expected value cost node stochastic choices case single sample gaussian distribution ticality approach implemented new machine learning system certigrad optimizing stochastic computation graphs schulman stochastic computation graphs extend computation graphs underly systems like tensorflow abadi theano bergstra allowing nodes represent random variables defining loss function graph expected value sum leaf nodes stochastic choices see figure example stochastic computation graph implement system lean theorem prover moura new interactive proof assistant still active development integration programming mathematical reasoning ongoing design goal formally state prove functional correctness stochastic backpropagation algorithm sampled gradients indeed unbiased estimates gradients loss function respect parameters note provable correctness need come expense computational efficiency proofs need checked development introduce runtime overhead although algorithms verify work lack many optimizations running time training machine learning systems spent multiplying matrices able achieve competitive performance simply linking optimized library matrix operations used eigen guennebaud demonstrate practical feasibility empirically trained variational bayes aevb model kingma welling mnist using adam kingma found performance comparable training model tensorflow note validity theorem becomes contingent eigen matrix operations functionally equivalent versions formally proved correct developing machine learning systems formal mathematics informal summarize contributions present first application formal proof techniques developing machine learning systems describe methodology detect implementation errors systematically machine learning systems demonstrate approach practical developing performant implementation sophisticated machine learning system along machinecheckable proof correctness motivation developing machine learning systems many program optimizations involve extensive algebraic derivations put mathematical expressions example suppose want compute following quantity efficiently diag log expand density functions grind algebra hand eventually derive following closed form expression log formal figure translating informal usages integral gradient formal representation note whereas informal examples ambiguous interpret without additional information lean representation always unambiguous language three capabilities provided new interactive proof assistant lean moura lean implementation logical system known calculus inductive constructions coquand huet design inspired coq proof assistant coq development team development makes use certain features unique lean present equally applicable coq lesser extent interactive theorem provers nipkow explain motivate relevant features lean walk applying methodology toy problem writing program compute gradient softplus function write standard functional programs lean softplus def splus log exp also represent abstract operations integrals gradients implement procedure compute quantity include part larger program run first experiment plots encouraging hoped ruling many possible explanations eventually decide scrutinize procedure closely implement monte carlo estimator quantity compare procedure random inputs find estimates systematically biased algebra carefully might notice sign wrong easier compiler checked algebra found erroneous step better yet algebra first place could guarantee result background lean theorem prover develop software systems implementation errors need way write computer programs mathematical theorems mathematical proofs intended meaning integral function intended meaning gradient derivative function point figure shows represent common idioms informal mathematics formal representation note whereas informal examples ambiguous interpret without additional information lean representation always unambiguous represent mathematical theorems lean well example use following predicate state particular function differentiable point prop fact return type prop indicates computer program executed rather represents mathematical theorem also state assume basic properties gradient linearity developing machine learning systems formal mathematics returning running example state theorem particular function computes gradient softplus function def prop splus suppose try write program compute gradient softplus function follows proof correct order trust although execute functions gsplus directly core logic lean since real number infinite object stored computer execute approximation inside lean virtual machine gsplus answer def gsplus exp application gsplus represents proposition implementation gsplus correct indeed computes gradient softplus function inputs try formally prove theorems lean interactively theorem gsplus lean gsplus user lean gsplus splus user introduce lean gsplus splus user gsplus splus lean exp log exp user lean exp exp exp lines beginning lean show current state proof displayed lean time consists collection goals form assumptions conclusion every line beginning user invokes tactic command modifies proof state way lean automatically construct proofs original goals given proofs new ones tactic rewrites exhaustively known gradient case uses rules log exp addition constants identity function final goal clearly provable means found implementation error gsplus luckily goal tells exactly gsplus needs return gsplus exp exp fix implementation gsplus proof script failed succeeds generates proof revised gsplus note need even attempted implement gsplus starting proof since process revealed program needs compute revisit phenomenon process proving theorem lean constructs formal proof certificate automatically verified small executable whose soundness based argument embedding core logic lean set theory whose implementation heavily scrutinized many developers thus human needs able understand case study certified stochastic computation graphs stochastic computation graphs directed acyclic graphs node represents specific computational operation may deterministic stochastic schulman loss function graph defined expected value sum leaf nodes stochastic choices figure shows stochastic computation graph simple variational autoencoder using methodology developed system certigrad allows users construct arbitrary stochastic computation graphs primitives provide main purpose system take program describing stochastic computation graph run randomized algorithm stochastic backpropagation expectation provably generates unbiased samples gradients loss function respect parameters overview certigrad briefly describe components certigrad analogues traditional software mathematics libraries type represents tensors particular shape along basic functions exp log operations gradient integral assumptions tensors gradient rules exp log facts proved terms assumptions gradient rule softplus also type represents probability distributions vectors tensors reasoned mathematically also executed procedurally using number generator implementation data structure represents stochastic computation graphs well implementation stochastic backpropagation also functions optimize stochastic computation graphs various ways integrating parts objective appealing property lost axiom assumed true discuss issue complete development found developing machine learning systems formal mathematics function well basic utilities training models stochastic gradient descent specification collection theorem statements collectively define means implementation correct certigrad one main theorem states stochastic backpropagation procedure yields unbiased estimates true mathematical gradients also theorems state individual graph optimizations sound proof many helper lemmas decompose proofs manageable chunks tactic scripts generate proofs lemmas theorems appearing system also tactic programs automate certain types reasoning computing gradients proving functions continuous optimized libraries stochastic backpropagation function written lean proved correct execute primitive tensor operations eigen library linear algebra small amount code wrap eigen operations use inside lean virtual machine rest section describes steps took develop certigrad include sketching architecture designing mathematics libraries stating main correctness theorem constructing formal proof though many details specific certigrad case study designed illustrate methodology expect projects follow similar process note certigrad supports tensors introduces notational complexity conceptual difficulty simplify presentation follows assuming values scalars informal specification first step applying methodology write informally system required suppose stochastic computation graph nodes simplify notation takes single parameter together define distribution values nodes let cost function sums values leaf nodes primary goal write stochastic backpropagation algorithm bprop graph bprop cost equation may seem sufficient communicate specification human mathematical background precision needed communicate computer next step formalize background mathematics real numbers tensors probability distributions state formal analogue equation computer understand although believe possible develop standard libraries mathematics future developers use needed develop mathematics libraries certigrad scratch designing mathematics libraries whereas traditional formal mathematics goal construct mathematics first principles gonthier hales need concern foundational issues simply assume standard mathematical properties hold example assume type real numbers without needing construct cauchy sequences likewise assume integration operator reals satisfies properties without needing construct either riemann sums note axioms must chosen great care since even single false axiom perhaps caused single missing precondition principle allow proving false theorem would invalidate property formal proofs trusted without however many preconditions appear mathematical theorems integrability almost always satisfied machine learning contexts developers ignore using axioms omit preconditions necessarily lead proving theorems missing corresponding preconditions practice developer extremely unlikely accidentally construct vacuous proofs exploiting axioms first draft system purposely omitted integrability preconditions axioms simplify development later make axioms sound propagate additional preconditions throughout system could fully trust formal proofs despite convenience axiomatizing mathematics designing libraries still challenging two reasons first many different ways formally represent mathematical objects question needed experiment understand tradeoffs different representations second needed extend several traditional mathematical concepts support reasoning executable computer programs rest example seemingly harmless axiom without precondition used prove absurdity system assumes axiom formal proof correctness could trusted without inspection since proof may exploit contradiction developing machine learning systems formal mathematics section illustrates challenges considering problem faced designing representation probability distributions certigrad representing probability distributions challenge devise sufficiently abstract representation probability distributions satisfies following desiderata reason probability density functions continuous random variables way reason arbitrary deterministic functions applied random variables execute distribution procedurally using pseudorandom number generator rng mathematical procedural representations distribution guaranteed correspond mathematics recognizable somebody familiar informal math behind stochastic computation graphs first define types represent mathematical procedural notions probability distribution mathematics define func functional takes realvalued function scalar def func type intended semantics func represents distribution expected value sampling define prog procedure takes rng returns vector along updated rng def prog type rng rng also assume primitive continuous distributions primdist func prog consist probability density function corresponding sampling procedure principle could construct distributions uniform variates expediency treat distributions primitive gaussian gauss primdist finally define type distributions dist abstractly represents programs may mix sampling primitive distributions arbitrary deterministic computations dist denoted func function reason mathematically prog function run execute rng readers familiar functional programming construction similar monad allow three ways constructing dist corresponding sampling primitive distribution sample returning value deterministically det composing two distributions compose sample pdf prog primdist dist det dist compose dist dist dist dist type dist scg type scg dist cost scg figure basic types functions need formally state specification dist represents distribution expected value function scg represents computation graph nodes dist function samples scg yields distribution values nodes cost sums values leaf nodes graph curly braces around argument indicates inferred context need passed explicitly mathematical semantics three constructors straightforward sample pdf prog pdf det compose procedural semantics run sample pdf prog rng prog rng run det rng rng run compose rng let rng run rng run rng defined run correspond consider stochastic program correct prove relevant theorems func denotation sample passing rng prog denotation formal specification background mathematics place next step write formal specification first design types every object function appearing informal description start need type scg represent stochastic computation graphs nodes function takes scg scalar parameter distribution real numbers dist also need function cost takes graph values nodes sums values leaf nodes figure provides full types objects appear specification write analogue informal specification presented equation def bprop scg prop scg bprop cost given mathematics libraries implementing developing machine learning systems formal mathematics objects functions appearing specification scg straightforward functional programming interactive proof conventional wisdom one would write program trying prove correct interactive proof process provides much helpful information system needs began working proof immediately drafting specification split proof two steps first implemented simplest possible function satisfied specification computed gradient single parameter time memoize proved correct second implemented performant version computed gradient multiple parameters simultaneously using memoization proved equivalent first one first step started placeholder implementation immediately returned zero let interactive proof process guide implementation whenever proof seemed require induction particular data structure extended program recurse data structure whenever proof showed branch program needed return value given expectation worked backwards determine value return proving first step also exposed errors specification form missing preconditions specification hold needed make additional assumptions graph identifier node graph unique leaf node scalar wellformed also needed assume generalization differentiability requirement mentioned schulman subset nodes determined structure graph must differentiable matter result stochastic choices gradsexist second step wrote memoizing implementation starting proof used process proving test debug although code memoizing simple short still managed make two implementation errors one conceptual one syntactic luckily process proving necessarily exposes implementation errors case made clear fix completed main proof correctness proving lemmas proof depends lemmas turned true except missing preconditions proving expose additional implementation errors also completed main proof axioms still unsound see made axioms sound propagated changes def bprop scg prop scg wellformed gradsexist integralsexist candiffunderints bprop cost figure final specification simplified problem scalars opposed tensors single parameter actual system supports tensors differentiating respect multiple parameters found specification required two additional preconditions functions integrated theorem statement indeed integrable integralsexist many preconditions needed pushing gradient integral expected loss satisfied candiffunderints however tracking additional preconditions lead changes actual implementation figure shows final specification optimizations also use methodology verify optimizations involve mathematical reasoning developing machine learning models one often starts model induces gradient estimator unacceptably high variance informal mathematics hand derive new model objective function induces better gradient estimator approach user write models use process interactive proving confirm induce objective function common transformations written proved correct users need write first model second derived proved equivalent automatically part certigrad wrote program optimization integrates multivariate isotropic gaussian distribution proved optimization sound also verified optimization reparameterizes model random variables depend parameters need backpropagated specifically optimization replaces node samples diag graph three nodes first samples scales shifts result according respectively applied two transformations sequence yield variational bayes aevb estimator kingma welling developing machine learning systems formal mathematics epoch model optimization procedure tensorflow running cpu cores found expected losses decrease rate certigrad takes longer per epoch figure expected loss epoch running time epoch figure results running certified procedure aevb model compared tensorflow system trains well takes longer per epoch verifying backpropagation specific models even though proved bprop satisfies formal specification sure compute correct gradients particular model unless prove model satisfies preconditions specification although preconditions technically undecidable practice machine learning models satisfy simple reasons wrote heuristic tactic program prove specific models satisfy preconditions used verify bprop computes correct gradients aevb model derived running system proved system correct idealized mathematical context real numbers actually execute system need replace real numbers program numbers although technically invalidates specification introduce numerical instability cases class errors well understood higham could ruled well principle harrison boldo ramananandro conceptually distinct algorithmic mathematical errors methodology designed eliminate improve performance also replace tensors optimized tensor library eigen approximation could introduce errors system whatever reason eigen methods use functionally equivalent ones formally reason course developers could achieve even higher assurance verifying optimized tensor code well experiments certigrad efficient experiment trained aevb model encoding network decoding network mnist using optimization procedure adam kingma compared expected loss running time system discussion primary motivation develop machine learning systems approach may provide significant benefits even building systems need perfect perhaps greatest burden software developers must bear needing fully understand system works found formally specifying system requirements able relegate much burden computer able synthesize fragments system able achieve extremely high confidence system without needing think pieces system fit together approach responsible ensuring local properties developer establishes imply overall system correct although using methodology develop certigrad imposed many new requirements increased overall workload substantially found whole made development process less cognitively demanding many ways methodology adopted incrementally example specifications need cover functional correctness theorems need proved unsound axioms used omit certain preconditions traditional code wrapped axiomatized eigen developing certigrad pursued ideal complete machinecheckable proof functional correctness achieved extremely high level confidence system correct however realized many benefits partial synthesis reduced cognitive process proving lemmas although could certain found bugs made axioms sound filled gaps formal proofs hindsight eliminated bugs early process well pure version methodology may already applications expect pragmatic use methodology could yield many benefits relatively little cost could useful developing wide range machine learning systems varying standards correctness developing machine learning systems formal mathematics acknowledgments thank jacob steinhardt alexander ratner cristina white william hamilton nathaniel thomas vatsal sharan providing valuable feedback early drafts also thank leonardo moura tatsu hashimoto joseph helfer helpful discussions work supported future life institute grant references abadi agarwal ashish barham paul brevdo eugene chen zhifeng citro craig corrado greg davis andy dean jeffrey devin matthieu ghemawat sanjay goodfellow ian harp andrew irving geoffrey isard michael jia yangqing jozefowicz rafal kaiser lukasz kudlur manjunath levenberg josh dan monga rajat moore sherry murray derek olah chris schuster mike shlens jonathon steiner benoit sutskever ilya talwar kunal tucker paul vanhoucke vincent vasudevan vijay fernanda vinyals oriol warden pete wattenberg martin wicke martin yuan zheng xiaoqiang tensorflow machine learning heterogeneous systems url http software available bergstra breuleux bastien lamblin pascanu desjardins turian bengio theano cpu gpu math expression compiler python scientific computing conference moura leonardo kong soonho avigad jeremy van doorn floris von raumer jakob lean theorem prover system description automated springer gonthier georges formal theorem notices ams gonthier georges asperti andrea avigad jeremy bertot yves cohen cyril garillot roux mahboubi assia connor russell biha sidi ould proof odd order theorem interactive theorem proving springer gordon michael edinburgh lcf mechanised logic computation gordon michael melham tom introduction hol theorem proving environment higher order logic guennebaud jacob http eigen hales thomas adams mark bauer gertrud dang dat tat harrison john hoang truong kaliszyk cezary magron victor mclaughlin sean nguyen thang tat formal proof kepler conjecture arxiv preprint harrison john hol light tutorial introduction international conference formal methods computeraided design springer boldo sylvie jourdan leroy xavier melquiond guillaume verified compilation floatingpoint computations journal automated reasoning harrison john verification using theorem proving international school formal methods design computer communication software systems springer chen haogang ziegler daniel chajed tej chlipala adam kaashoek frans zeldovich nickolai using crash hoare logic certifying fscq file system proceedings symposium operating systems principles acm higham nicholas accuracy stability numerical algorithms siam chlipala adam bedrock structured programming system combining generative metaprogramming hoare logic extensible program verifier acm sigplan notices volume acm coq development team coq proof assistant reference manual version inria coquand thierry huet calculus constructions information computation kingma welling variational bayes arxiv kingma diederik jimmy method stochastic optimization adam arxiv preprint klein gerwin elphinstone kevin heiser gernot andronick june cock david derrin philip elkaduwe dhammika engelhardt kai kolanski rafal norrish michael formal verification kernel proceedings acm sigops symposium operating systems principles acm leroy xavier formal verification realistic compiler communications acm developing machine learning systems formal mathematics nipkow tobias paulson lawrence wenzel markus proof assistant higherorder logic volume springer owre sam rushby john shankar natarajan pvs prototype verification system automated springer ramananandro tahina mountcastle paul meister lethin richard unified coq framework verifying programs computations proceedings acm sigplan conference certified programs proofs acm rudnicki piotr overview mizar project proceedings workshop types proofs programs schulman john heess nicolas weber theophane abbeel pieter gradient estimation using stochastic computation graphs advances neural information processing systems
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dec ding injective ding projective ding flat modules complexes james gillespie abstract characterize ding modules complexes rings show ring ding projective resp ding injective resp ding flat coincide gorenstein projective resp gorenstein injective resp gorenstein flat modules turn nothing modules appearing cycle exact complex projective resp injective resp flat modules prove similar characterization chain complexes complex ding projective resp ding injective resp ding flat component ding projective resp ding injective resp ding flat along way generalize results stovicek obtain interesting corollaries example show noetherian ring exact chain complex gorenstein injective components must cotorsion cycle modules complex flat module hand coherent ring cycles exact complex projective components must satisfy absolutely pure module introduction let gorenstein ring sense left right noetherian ring finite injective dimension left right module rings exact chain complexes projective injective flat nice homological properties particular exact complex projectives complex homr also exact projective module general complexes called totally acyclic complexes projectives modules appearing cycle complex called gorenstein projective gorenstein rings exact complex projectives totally acyclic cycles gorenstein projective similar statements hold exact complexes injectives flats corresponding gorenstein injective gorenstein flat modules gorenstein homological algebra study modules complexes theory particularly satisfying gorenstein rings case results traditional homological algebra analog gorenstein homological algebra book standard reference many authors studied subject stovicek recently proved coherent analog result concerning totally acyclic complexes injectives prop raises question projective flat analogs answered paper explaining first recall left coherent ring one finitely date december james gillespie generated left ideals finitely presented ring left right coherent called coherent include noetherian von neumann regular rings lesson learned many results homological algebra extend noetherian coherent rings replacing finitely generated modules finitely presented modules process injective modules replaced absolutely pure modules way ring coherent ring finite absolutely pure dimension left right module introduced since coherent ring noetherian modules coincide injective modules rings nothing gorenstein rings whenever noetherian author noted work ding mao see especially provides natural way extend notions gorenstein homological algebra noetherian coherent rings process gorenstein modules replaced ding modules example say module ding projective cycle module exact complex projectives homr remains exact flat modules see definitions admittedly seems strange require homr remain exact flat modules rather projectives feels like requiring much however shown ding projectives cofibrant objects especially nice quillen model structure category left whenever coherent implies every module approximated ding projective full subcategory ding projective modules naturally form frobenius category think associated stable category projective stable module category analog usual gorenstein projectives noetherian rings simply seem true general first result paper extension rings well known result concerning gorenstein rings described first paragraph part result stovicek motivated analogous question ding projectives ding flats theorem let ring left right coherent ring finite absolutely pure dimension let exact chain complex component projective cycle ding projective indeed homr remains exact flat modules component injective cycle ding injective indeed homr remains exact absolutely pure component flat cycle ding flat indeed remains exact absolutely pure consequently gorenstein modules coincide ding modules whenever ring discussing proof methods note gorenstein chain complexes also studied quite bit example known time gorenstein rings chain complex gorenstein injective resp gorenstein projective component gorenstein injective resp gorenstein projective see yang liu ding modules complexes liang characterize ding complexes ring next result surprising refinement characterization theorem let ring chain complex ding projective component ding projective ding injective component ding injective ding flat component ding flat consequently gorenstein complexes coincide ding complexes whenever ring briefly describe methods used highlight results paper first injective cases proved completely different fashion projective analogs flat analogs relatively easy injective case generalize approach stovicek record useful result concerning cotorsion pairs direct limits cotorsion pair closed direct limits whenever satisfies two three property short exact sequences see proposition injective cases theorems follow corollaries approach leads string interesting corollaries appearing sections perhaps interesting corollary implies noetherian ring exact chain complex gorenstein injective components must cotorsion cycle modules complex flat module projective case requires completely different approach injective one proof projective part theorem relies result proved theorem obtain projective statement theorem fact need generalize result category chain complexes section devoted theorem main result finally following interesting result proved theorem theorem let coherent ring exact complex projectives absolutely pure module regarding structure paper preliminaries section first address injective cases sections sections devoted projective case section flat case preliminaries throughout paper denotes general ring identity mean left unless stated otherwise category denoted abelian category chain complexes abelian categories let abelian category denote corresponding category chain complexes case denote convention differentials chain complexes lower degree chain complex also chain homotopy category denoted objects also chain complexes morphisms chain homotopy classes james gillespie chain maps given chain complex nth suspension denoted complex given given object denote complex consisting concentrated degrees elsewhere denote complex consisting degree elsewhere given two chain complexes define hom comq plex abelian groups hom hom get functor hom note functor takes exact sequences left exact sequences exact projective similarly contravariant functor hom sends exact sequences left exact sequences exact injective exercise check homology satisfies hom abelian category comes yoneda ext groups particular denote group equivalences classes short exact sequences baer sum operation subgroup consisting degreewise split short exact sequences split exact following lemma gives connection hom lemma chain complexes isomorphisms hom particular chain complexes hom exact iff chain map homotopic iff chain map homotopic case recall usual tensor product chain complexes given resp complex right resp left tensor product defined degree boundary map defined generators degree element modified hom tensor complexes denotes category chain complexes hom often referred internal hom case commutative hom object note cycles degree internal hom coincide external hom functor hom homch idea fact useful define alternate internal hom follows given define hom complex hom hom differential hom hom defined notice degree component hom exactly homch way get internal hom hom useful categorical considerations example hom left exact functor exact projective category ding modules complexes hand hom exact injective corresponding derived functors denote ext satisfy ext complex whose degree extich similarly usual tensor product chain complexes characterize categorical flatness need modified tensor product left derived torsion functor denote defined terms usual tensor product follows given complex right complex left define complex whose nth entry boundary map given defines complex get bifunctor right exact variable denote corresponding left derived functors tori refer reader details finitely generated projective complexes standard characterization projective objects following complex projective exact complex cycle projective also recall definition chain complex finitely generated whenever collection subcomplexes exists finite subset standard fact finitely generated bounded finitely generated say chain complex type projective resolution finitely generated projective complexes certainly finitely presented hence finitely generated absolutely clean level complexes character duality level absolutely clean modules introduced generalizations flat modules coherent rings injective modules noetherian rings notions category also studied recall definitions results definition call chain complex absolutely clean chain complexes type equivalently ext complexes right type hand call chain complex level chain complexes right type refer prop proof following proposition chain complex absolutely clean exact absolutely clean chain complex level exact level recall character module defined homz right resp left whenever left resp right rmodule construction extends chain complexes given chain complex homz since injective cogenerator category abelian groups functor homz preserves reflects exactness james gillespie proposition immediately gives following corollary due perfect character module duality absolutely clean level modules theorem corollary chain complex left resp right modules level homz absolutely clean complex right resp left modules chain complex left resp right modules absolutely clean homz level complex right resp left modules cotorsion pairs let abelian category definition pair classes called cotorsion pair given class objects right orthogonal defined class objects similarly define left orthogonal call cotorsion pair hereditary extia cotorsion pair complete enough injectives enough projectives means exist short exact sequences standard references include connections abelian model categories found recall grothendieck category cocomplete abelian category set generators direct limits exact injective cotorsion pair grothendieck category mean complete cotorsion pair thick class injective objects since grothendieck categories enough injectives turns cotorsion pair equivalent injective model structure mean model structure abelian sense objects cofibrant fibrant objects case exactly trivial objects exactly see injective cotorsion pairs also dual notion projective cotorsion pairs give projective model structures abelian categories enough projectives injective cotorsion pairs direct limits let cotorsion pair grothendieck category assume thick assume satisfies two three property short exact sequences section show must closed direct limits leads several interesting corollaries proof proposition adapted stovicek prop studying proof author simply realized reinterpreted yield following convenient result proposition let cotorsion pair thick closed direct limits proof first step show closed direct unions fact enough prove continuous direct unions assume module monomorphisms thickness hypothesis since diagram modules assumed continuous see colimit thing transfinite extension thus colimit ding modules complexes eklof lemma assuming limit ordinal successor ordinal colimit coincides step two show closed direct limits enough prove continuous direct limits see section especially corollary remark follows direct limits referred chains continuous direct limits smooth chains consider limi limit ordinal job show limi assume limit ordinal otherwise direct limit equals following standard way defining direct limits example see prop limi cokernel first direct sum following homomorphism xij taken pairs xij copy domain map fij map defined ith coordinate fij denotes canonical injection coproduct words direct limit image map fij note since maps fij linear set finite sums form fij range ranges short exact sequence lim since closed direct sums thick enough show show direct union well ordered still direct union modules proof follow step thinking set smaller ordinals define finite subset mapping denotes maximum element finite subset map defined ith coordinate via fij one verifies following directedlposet functor defined objects arrows taking inclusion map djj defined ith coordinate follows natural inclusion maximal element via fij djj monomorphism fact split monomorphism retraction map canonical projection also split monomorphism similar retraction image identifies submodule fij james gillespie set finite sums elements form fij ranges ranges direct system monomorphisms isomorphic via natural transformation direct system submodules direct limit limi identifies direct union submodules since conclude step completes proof course generalizations proposition categories beyond possible important applications reader note proof holds grothendieck category giving following generalization proposition let cotorsion pair grothendieck category thick closed direct limits particular class trivial objects closed direct limits whenever injective cotorsion pair proposition interesting consequences apply injective cotorsion pairs recently appearing example shown ring injective cotorsion pair class gorenstein modules next corollary says modules finite flat dimension sent zero corresponding stable homotopy category consequently gorenstein module cotorsion must injective finite flat dimension corollary let injective cotorsion pair contains objects finite flat dimension consequently fibrant object cotorsion must injective whenever finite flat dimension proof certainly contains projectives hence flat objects since precisely direct limits projectives since thick contains flat objects also contains objects finite flat dimension final statement follows since coincides class injective objects recall chain complex flat exact cycle module flat complexes categorically flat direct limits finitely generated projective complexes theorem complex called flat exact whenever exact chain complex right usual tensor product chain complexes recalled section book standard reference complexes corollary let injective cotorsion pair following hold exact complex cotorsion cycle modules hence cotorsion module every complex exact complex use notation denotes class complexes cotorsion property chain maps null homotopic whenever flat chain complex turns ding modules complexes precisely right ext orthogonal class flat complexes really class cotorsion objects category complexes proof cotorsion corollary result follows theorem tells exact complexes precisely ones cotorsion cycle modules case every complex exact projective module projective hence flat module follows prop complex retract transfinite extension complexes form flat applying well known eklof lemma conclude complexes proving going back gorenstein modules appearing corollary get following corollary corollary chain complex gorenstein modules consequently exact complex gorenstein must codg torsion cycle modules proof refer proposition shows injective cotorsion pair whose right side consists class complexes gorenstein modules rest follows already observed note corollary implies stovicek original result exact complex injectives must cotorsion cycle modules however reader familiar realize methods generalization employed stovicek methods however dualize following result quite interesting dual stovicek result assuming coherent ring theorem let coherent ring exact complex projectives absolutely pure module consequently cycle must retract transfinite extension finitely presented modules proof since exact instead show absolutely pure enough show hom homr exact whenever exact complex projectives absolutely pure let denote class absolutely pure denote class flat right since coherent forms perfect duality pair respect character modules section therefore theorem homr exact absolutely pure exact flat certainly true last statement holds since set finitely presented modules cogenerates complete cotorsion pair precisely class absolutely pure modules james gillespie remark suppose ring level modules finite flat dimension argument extend show absolutely clean modules exact complexes projectives applications ding injective modules complexes section devoted studying ding injective modules chain complexes rings ring left right coherent ring finite absolutely pure dimension refer rings ding injective modules although definitions suffice purposes definition call ding injective exists exact complex injectives ker remains exact applying homr absolutely pure module way call chain complex ding injective exists exact complex injective complexes ker remains exact applying homch absolutely pure chain complex recall chain complex injective resp absolutely pure exact cycle injective resp absolutely pure module content corollaries hom condition definitions come automatically ring fact module case following stronger statement theorem let ring ding injective exact complex ding injective exact complex ding injectives automatically ding injective cycles proof since modules finite flat dimension coincide modules finite absolutely pure dimension denote class injective cotorsion pair class ding injective modules see theorem corollary exdi prop lifts another injective cotorsion pair exdi class exact complexes ding injectives exdi flat module hence corollary exdi finite flat dimension thus exdi ext extr hence must ding injective since injective modules ding injective deduce following corollary corollary prop let ring ding injective exact complex injective therefore absolutely pure exact complex injectives remain exact applying homr ding modules complexes shown yang liu liang ding injective complexes precisely complexes ding injective chain maps null homotopic whenever absolutely pure fpinjective chain complex show null homotopy condition automatic ring corollary let ring complex ding injective ding injective proof part easy definition ding injective converse use coherent ring ding injective cotorsion pair dwdi lifts injective model structure dwdi proposition dwdi class complexes ding injective modules corollary complex finite flat case easy see dimension dwdi absolutely pure complex finite flat dimension indeed exact complex cycle module absolutely pure thus finite flat dimension taking flat resolution upper bound flat dimensions implies finite flat dimension hence dwdi absolutely pure follows hom exact absolutely whenever pure complex dwdi remark shown yang liu liang class ding whenever injective complexes precisely whenever ring shown dwdi exdi using notation complexes projective complexes main purpose prove theorem tool section characterizing ding projective complexes get generalizing results appendix modules chain complexes since abelian category course consider category chain complexes chain complexes using sign trick category identified category bicomplexes however purpose somewhat easier stick category free chain complexes free one isomorphic direct sum copies analogously say chain complex free isomorphic direct sum dni integer clearly equivalent saying isomorphic free also equivalent define exact complex free however sufficient convenient use representations dni evidently free complexes closed arbitrary direct sums also easy check free complexes projective objects projective complex retract free complex lemma eilenberg swindle given projective chain complex exists free chain complex james gillespie proof follow corollary since projective noted find another projective complex free complex setting produce free complex desired indeed need study chain complexes projective chain complexes objects component projective complex lemma let ring complex projective chain complexes direct summand complex free chain complexes furthermore exact taken exact proof lemma may find free chain complex form complex complexes degree chain complex free chain complex done since direct summand moreover exact whenever exact theorem let ring let set bounded complexes finitely generated free complexes cotorsion pair cogenerated functorially complete moreover class complexes projective complexes proof since grothendieck category enough projectives follows corollary cogenerates functorially complete cotorsion pair setting aim show precisely class complexes projective complexes note contains set projective generators example set follows corollary precisely class direct summands transfinite extensions objects lemma need show complex free complexes transfinite extension bounded complexes finitely generated free complexes let complex free complexes write dni indexing set assume isomorphic complex certainly find nonzero take one summand dnj start build bounded subcomplex setting dnj setting note dnj dni find finite subset dnj dni set dni continue way finding finite dni dni way construct subcomplex dni dni dnj note nonzero bounded complex finitely generated free complexes set finish proof argue write union continuous chain ding modules complexes consider way find bounded subcomplex consisting finitely generated free complexes degree however careful construct follows let denote indexing sets previously constructed complex dni note identify quotient complex whose degree entry dni may take dni finite continuing process continue construct increasing union corresponding nested union subsets assuming process terminate set note dni course complexes free complexes continuing process obtain ordinal continuous chain pure exact complexes complexes let short exact sequence say pure finitely presented complex sequence abelian groups homch homch homch homch also exact pure exact sequences complexes characterized terms functors hom section particular pure exact hom short exact sequence complexes finitely presented complex short exact sequence complexes finitely presented complex complex see theorem leads following notion pure exact complex complexes definition let say pure exact complex complexes following equivalent conditions satisfied homch hom exact complex abelian groups finitely presented chain complex hom exact complex complexes abelian groups finitely presented chain complex exact complex complexes abelian groups finitely presented chain complex exact complex complexes abelian groups chain complex lemma suppose bounded complex finitely presented complexes pure exact complex complexes every chain map chain homotopic proof note equivalent show hom exact hom usual section applied case instead typical application let largest degree nonzero let subcomplex degreewise split short exact sequence given induces short exact sequence hom hom hom james gillespie hom exact condition definition long exact sequence homology gives result induction let denote cotorsion pair theorem result tells pure exact complexes complexes theorem let ring let complex projective complexes pure exact complex complexes hom exact equivalently proof view theorem suffices assume bounded complex finitely generated free complexes show chain map chain homotopic construct chain homotopy ddn downwards induction since bounded take large begin induction suppose defined still idea replace new map also find map ddn first note ddn induced map composite equals ddn consider bounded complex finitely presented complexes chain map degree degree degree lemma chain map must chain homotopic gives maps upon composing becomes ddn setting still required relation moreover promised indeed relation ddn ddn ddn ddn recall complex character dual homz see section lemma let complex complexes complex right exact hom exact proof exact exact also using parts proposition get homz hom hom hom hom need one lemma proving main theorem lemma let complex complex complexes hom exact hom exact ding modules complexes proof pondering definitions see hom cochain complex complexes abelian groups whose degree zero entry complex abelian groups homch homch homch degree component complex say overall complex exact means cochain complex abelian groups homch exact hand following definitions hom isomorphic cochain complex homch abelian groups exactness hom equivalent exactness hom collection chain complexes right collection chain complexes left say duality pair immediate corollary absolutely clean level complexes give rise two duality pairs one class absolutely clean complexes right another class level complexes right theorem suppose duality pair closed pure quotients let complex projective complexes exact hom exact particular exact absolutely clean complexes hom exact level complexes exact level complexes hom exact absolutely clean complexes proof view lemma hom exact exact conversely suppose exact exact lemma tells hom exact conclude hom exact note since duality pair complex natural map pure monomorphism complexes proposition quotient also since closed pure quotients therefore create resolution elements find pure exact resolution complexes complexes gives short exact sequence pure exact bounded complex entries theorem tells hom exact hence hom hom lemma remains show hom exact whenever bounded components showing hom exact complexes projective complexes equivalent showing lemma first paragraph conclude whenever bounded complex entries seen inverse transfinite extension spheres dual eklof lemma james gillespie know closed inverse transfinite extensions theorem whence hom exact complexes projective complexes completes proof noted absolutely clean complexes level complexes give rise two duality pairs moreover class closed pure quotients propositions applications ding projective modules complexes wish prove duals corollaries theorem corollary definition call ding projective exists exact complex projectives ker remains exact applying homr flat module way call chain complex ding projective exists exact complex projective complexes ker remains exact applying homch flat chain complex recall chain complex projective resp flat exact cycle projective resp flat module theorem let ring module ding projective exact complex projective way chain complex ding projective exact complex projective complexes proof first look case let exact complex projectives need show homr remains exact flat left module theorem since coherent enough show exact absolutely pure right modules note clearly exact flat right module follows exact finite flat dimension since ring absolutely pure module finite flat dimension done note similar alternate proof could given using theorem instead next see proof holds chain complexes due work section let exact complex projective complexes need show homch remains exact flat left chain complex however looking definition hom clear equivalent statement hom remains exact flat complex theorem since coherent equivalent statement exact absolutely pure right complexes note exact flat right complex proposition follows exact complexes finite flat dimension since ring absolutely pure complex exact complex upper bound flat dimensions follows complex finite flat dimension done ding modules complexes corollary let ring complex ding projective ding projective proof part easy show converse theorem assures need show equals zero cycles exact complex projective complexes certainly find exact complex projective complex left extend complex right first note obvious degreewise split short exact sequence ding projective certainly find short exact sequence projective also ding projective gives another short exact sequence projective complex denote notice furthermore let composite monomorphism since composite two monomorphisms moreover setting cok follows snake lemma sits particular short exact sequence extension must ding projective degree since since properties may continue inductively obtain desired resolution finally paste resolution together setting done ding flat modules complexes pointed also natural notion ding flat module turns equivalent notion gorenstein flat module recall definition definition call ding flat exists exact complex flat modules ker remains exact applying absolutely pure right way call chain complex ding flat exists exact complex flat complexes james gillespie ker remains exact applying absolutely pure complex proposition let ring ding flat module nothing gorenstein flat module similarly chain complex ding flat gorenstein flat proof proposition modules goes back lemma provide new proof quick easy proof give proof complexes proof works modules note ding flat complexes clearly gorenstein flat since injective complexes absolutely pure conversely suppose gorenstein flat complex definition means exists exact complex flat complexes ker remains exact applying injective complex show complex fact remain exact applying absolutely pure right chain complex let given note equivalent show map complexes monomorphism let embedding injective complex note must pure monomorphism since absolutely pure complex commutative diagram two horizontal arrows monomorphisms since pure right vertical arrow also monomorphism since exact follows left vertical arrow must also monomorphism proposition let ring module ding flat exact complex flat way chain complex ding flat exact complex flat complexes proof proof easy similar last sentences proof theorem briefly say exact complex flat complexes wish show exact absolutely pure complex certainly exact flat complex since ring absolutely pure complex finite flat dimension argue exact corollary let ring complex ding flat ding flat proof using proposition proof corollary carries replace word projective word flat throughout proof ding modules complexes references locally presentable accessible categories number london mathematical society lecture note series cambridge university press daniel bravo james gillespie absolutely clean level gorenstein complexes preprint daniel bravo james gillespie mark hovey stable module category general ring submitted ding chen flat dimensions injective modules manuscripta math vol ding chen coherent rings finite dimension comm algebra vol ding mao reletive modules comm algebra vol ding mao envelopes covers modules finite flat dimensions comm algebra vol ding mao gorenstein gorenstein flat modules algebra appl ding mao strongly gorenstein flat modules aust math soc vol enochs tensor products chain complexes math okayama univ enochs estrada iacob gorenstein projective flat complexes noetherian rings mathematische nachrichten vol edgar enochs overtoun jenda relative homological algebra gruyter expositions mathematics vol walter gruyter berlin enochs iacob jenda closure transfinite extensions illinois journal mathematics vol covers envelopes category complexes modules research notes mathematics chapman boca raton james gillespie flat model structure trans amer math soc vol james gillespie kaplansky classes derived categories math zeit vol james gillespie model structures modules rings homology homotopy appl james gillespie gorenstein complexes recollements cotorsion pairs preprint arxiv james gillespie construct hovey triple two cotorsion pairs fundamenta mathematicae vol jan trlifaj approximations endomorphism algebras modules gruyter expositions mathematics vol walter gruyter berlin mark hovey cotorsion pairs model category structures representation theory mathematische zeitschrift vol iwanaga rings finite dimension comm algebra vol iwanaga rings finite dimension tsukuba math vol lam lectures modules rings graduate texts mathematics vol new york jan purity applications coderived singularity categories coherent rings modules london math soc rings quotients die grundlehren der mathematischen wissenschaften einzeldarstellungen band new york james gillespie charles weibel introduction homological algebra cambridge studies advanced mathematics vol cambridge university press gang yang zhongkui liu liang ding projective ding injective modules algebra colloquium gang yang zhongkui liu liang model structures categories complexes rings communications algebra vol gang yang zhongkui liu cotorsion pairs model structures proc edinb math soc vol ramapo college new jersey school theoretical applied science ramapo valley road mahwah address jim gillespie jgillesp url http
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beyond highway dimension small distance labels using tree adrian kosowski laurent viennot dec inria paris irif paris diderot france abstract goal distance labeling scheme network assign small subset node way pair nodes intersection hub sets contains node shortest existence small hub sets consequently efficient shortest path processing algorithms road networks empirical observation theoretical explanation phenomenon proposed abraham soda network parameter called highway dimension captures size hitting set collection shortest paths length least intersecting given ball radius work revisit explanation introducing tractable directly comparable parameter based solely structure spanning trees call skeleton dimension show skeleton dimension admits intuitive definition directed undirected graphs provides way computing labels efficiently using highway dimension leads comparable stronger theoretical bounds hub set size key words distance labeling highway dimension shortest path tree skeleton dimension introduction task efficiently processing shortest path queries graph studied plethora settings one interesting observation many graphs small degree geometric geographical setting road networks possible design compact data structures schemes efficiently answering shortest path queries general principle operation approach consists detecting storing subsets transit nodes appear shortest paths many node pairs attempt explain efficiency variants transit node routing tnr algorithm abraham introduced concept highway dimension parameter captures intuition map partitioned regions significantly long shortest paths region hit small number transit node vertices value presumed small constant road networks however definition highway dimension relies notion hitting set shortest path sets within network neighborhoods hence exact computation parameter known even unweighted networks motivates look measures locally defined computationally tractable capturing essentially characteristics network amenability shortest path queries supported inria project gang anr project descartes ncn grant looking precisely tnr algorithm one observes built around idea every source node set transit nodes first encountered going long way source small weaker assumption existence small hitting set set shortest paths given network neighborhood since different source nodes could use different transit nodes resulting overall large number transit nodes around given region approach leads definition skeleton dimension devote remainder paper informally skeleton dimension maximum taken nodes graph radii number distinct nodes distance set shortest paths originating length least transit node parlance states paths extend least outside disk radius pass transit nodes disk border property ensures spanning tree built around core skeleton branches given distance range rest branches relatively short bounding tree skeletons turns encompass larger class graphs shortest path cover approach used definition highway dimension still ensuring existence efficient labeling schemes motivated applications distributed algorithms distributed data representation display link small skeleton dimension graph efficient processing shortest path queries using framework distance labeling distance labeling schemes popularized gavoille among fundamental distributed data structures graph data within distance labeling work basic framework based schemes namely hub labelings framework first described name covers also referred landmark labelings setting node stores set distances subset nodes graph computed distance value queried pair nodes returned min denotes shortest path distance function pair nodes computed distance pairs nodes exact set contains least one node shortest path property family sets known shortest path cover method distance computation practice effective two reasons first networks possible show bounds sizes sets follow network structure notably considering networks bounded highway dimension abraham show appropriate cover shortest paths graph achieved using sets size conceals logarithmic factors studied graph parameters moreover order elements sets browsed performing minimum operation relevant schemes operation interrupted certain minimum found probing elements set principle numerous heuristics exact problem contraction hierarchies algorithms arc flags results organization paper section formally define skeleton dimension show continuous representation graph skeleton dimension highway dimension satisfies bound cases graphs bounded maximum degree hand show one may also define skeleton dimension different choice constants considering set shortest paths length least absolute constant choice subsequently necessary establishing relations highway dimension skeleton dimension provides better explanation small hub set size networks highway dimension particular provide natural example weighted grid log section show construct efficient hub labelings networks small skeleton dimension hub set sizes obtain graph weighted diameter bounded log average log log log worst case corollaries respectively compared previous best bounds log log labels computable polynomial time based highway dimension labeling technique based picking hubs random selection process subtree tree allows node compute hub set independently time appears independent interest particular extension technique provide section improved bounds label size general unweighted graphs distance labeling problem considered distance queries restricted nodes distance least hub sets constructed using method average size bounded size also bounded threshold log log log general theorem improves upon previous schemes including previously best result hub sets size log constructed direct application probabilistic method sets randomly sampled vertices finally sections provide concluding remarks computability proposed parameter skeleton dimension well possible generalizations applications related work distance labelings distance labeling problem undirected graphs first investigated graham pollak provided first labeling scheme labels size decoding time labels size subsequently improved log log gavoille weimann peleg finally alstrup present scheme general graphs decoding time using labels size bits matches low order terms space currently best know distance oracle time total space centralized memory model due nitto venturini specific classes graphs gavoille described log distance labeling planar graphs together lower bound class graphs additionally upper bound trees lower bound sparse graphs given distance labeling hub sets given graph computational task minimizing sizes hub sets exact distance decoding relatively well understood log approximation algorithm minimizing average size hub set sought shortest path cover property presented cohen whereas log minimizing largest hub set node given recently babenko rather surprisingly structural question obtaining bounds size hub sets specific graph classes graphs bounded degree unweighted planar graphs wide open labeling notion distance labeling first introduced describes labeling scheme correctly encoding every distance least presents scheme size recently improved alstrup scheme size together observation distances smaller stored directly results labeling scheme size sparse graphs road networks highway dimension guarantees existence distance labels size log weighted diameter graph however restricting polynomial time algorithms labels approximated within log factor using shortest path cover algorithms log factor involved procedure based case requires shortest path computation large networks labels practically computed classical heuristics contraction hierarchies performed low highway dimension guarantees exists elimination ordering graph produced bounded size however ensure running time faster pair shortest path computation besides highway dimension skeleton dimension also related notion reach introduced also used algorithm reach node path minimum distance extremity reach maximum reach shortest paths containing efficient algorithms obtained pruning nodes small reach dijkstra search similarly obtain skeleton tree pruning nodes whose reach tree less half distance root notation parameters consider connected undirected graph length function let denote number nodes let denote length path given length function given two nodes assume unique shortest path puv common assumption made without loss generality one perturb input ensure uniqueness given two nodes distance puv let maxu denote diameter ball radius centered set nodes paper assume integral notions presented easily extend real lengths use integer lengths cleaner exposition algorithms theorems also recall two structural parameters application networks geometric setting topological embedding highway dimension doubling dimension let denote collection shortest paths consider collection shortest paths around hitting set set nodes path contains node highway dimension defined smallest hitting set size definition slightly less restrictive allowing prove similar results improved bounds notion highway dimension related doubling dimension recall graph ball covered balls half radius exists shown geometric realization graph highway dimension informally geometric seen continuous graph edge seen infinitely many vertices realization degree two infinitely small edges node distance edge proof consists proving node holds also highway dimension distance hitting set pge definition presentation skeleton dimension start providing standalone definition skeleton dimension based size cuts shortest path trees show relation previously considered parameters highway doubling dimension definition parameter tree skeleton given tree rooted node length function treat directed root leaves consider geometric realization directed graph define reach reach dte define skeleton subtree induced nodes reach least half distance root precisely subtree induced reach dte width tree width tree root defined maximum number nodes points given distance root precisely width width maxr cut cut set nodes dte skeleton dimension skeleton dimension graph defined maximum width skeleton shortest path tree width denotes shortest path tree obtained union shortest paths remark assumption graph different cuts tree skeleton similar width definition skeleton dimension meaningful measure structure tree smoothed integrated variant skeleton dimension also discussed skeleton dimension geometric highway dimension graph highway dimension skeleton claim geometric realization dimension proof consider node skeleton shortest path tree consider cut cut sufficiently small cut cut size cut consider node dtu reach shortest path pvx intersects length pvx reach pvw thus pge node cut get similar path pge paths pairwise belong disjoint number thus size hitting set pge get skeleton dimension note discrete graph highway dimension highway dimension road networks expected continuous discrete versions geometric realization highway dimension coincide almost exactly particular due constant maximum degree bounded length edges graphs general setting one easily show maximum degree star example indeed hitting set may miss shortest path pge making longer extremities transforms path hit thus possible hit pge adding one node per edge adjacent node remark extended version introduces modified notion highway dimension way closely related geometric variant denote modified parameter first inequality follows analysis similar proof claim latter two shown section low skeleton dimension implies low doubling dimension known graph geometric realization highway dimension however relation need tight turns link skeleton dimension doubling dimension holds slightly weaker form proposition graph skeleton dimension proof show stronger requirement ball radius covered balls radius consider shortest path tree consider set edges containing node cut let far extremities edges cutting distance skeleton node distance greater descendant node skeleton definition thus considering obtain node distance node similarly node distance node ball thus covered balls radius centered nodes separating skeleton dimension highway dimension provide family graphs exhibit exponential gap skeleton highway dimensions setting directly inspired road networks idea consider usual square grid define edge lengths give priority certain transit arteries example paths using edges whose coordinates multiples high powers slightly lower transit times let denote grid length function defined follows identify node coordinates consider small length perturbations pxy every horizontal edge qxy every vertical edge define maxx max pxy qxy integers chosen ensure uniqueness shortest paths odd define qxy odd define pxy possible choice perturbations ensuring uniqueness shortest paths pxy qxy clear later proposition grid highway dimension skeleton dimension log number nodes proof first prove shortest paths also shortest path grid unit edge lengths paths use minimum number edges path edges length least pqdl given two nodes let denote minimum number edges path let denote number edges shortest path puv since implies puv thus shortest path grid note balls must also almost identical bgl bul implies highway dimension since ball radius centered intersects least horizontal shortest paths length qdl least define rectangle odd set nodes coordinates border set nodes one inequality least indeed equality nodes said interior main argument bounding skeleton dimension shortest path traverse interior shortest path passing inner node necessarily ends inside reason path necessarily passes lower left corner shorter reach border node following border rather using edges inside rectangle note two possible choices shortest paths could possible going corner rectangle corner diagonally opposed however choice perturbing lengths decreasing length vertical edge position qxy ensures path rightmost side preferred consider node radius qdl qdl set qdl according first part proof ball bgl sandwich balls radius bul bgl bul first consider upper right quadrant bul border set nodes bound number nodes reach denote shortest path tree qdl node interior rectangle shortest paths interior rectangle length number nodes whose coordinate multiple bounded similarly number nodes whose coordinate multiple also bounded apart nodes consider nodes resp less smallest multiple greater resp node interior resp rectangle obtain two nodes whose coordinate multiple repeating argument finally bound number nodes reach greater nodes upper right quadrant distance must edges outgoing nodes cut thus size log symmetry bound holds quadrants skeleton dimension log remark exist different lengths functions grid skeleton dimension also large case example grid unit lengths edges except edges intersecting major diagonal configured fast transit artery suffices set complement result experimental observation real grid like networks encountered brooklyn computed skeleton dimension new york graph proposed dimacs challenge turns average skeleton tree width maximum width encountered skeleton tree rooted manhattan order estimate highway dimension graph implemented heuristic finding large packing paths near given ball set disjoint paths intersecting ball length greater half radius could find packing paths brooklyn proves highway dimension graph least comparison skeleton tree center corresponding ball width branches cut radius distance see figure hub labeling using tree skeletons section assign hub sets set terminal nodes considered network assume length function edges integer weighted emulate geometric realization graph subdivide edges sufficiently short fragments inserting set additional nodes network convenience subsequent analysis assume edge integer length subdivided edges length edges length subsequently treat graph unweighted parameter definitions carry directly geometric setting sake precision formally state assumptions studied setting figure openstreetmap view brooklyn packing paths black intersecting ball radius seconds white border left skeleton tree black center ball right consider unweighted graph distinguished set terminal nodes nodes degree denote assume every node associated fixed unweighted tree throughout section denote unique path pair nodes tree concisely lead confusion identify path edge set also use symbol denote length path number edges belonging write require collection trees satisfies following property pair nodes also assume integer multiple remark graph obtained subdivision nodes graph node set distance metric tree corresponds shortest path tree node original distance metric assumption corresponds assumption uniqueness shortest paths original metric edge hub labeling assignment set edges node following property fulfilled every pair nodes exists edge set known edge hub set remark notion hub sets slightly stronger analogous notion indeed knowing edge also conclude endpoints edge belong choose work edge hub sets rather node hub sets section compactness arguments restate setting family trees notion skeleton subtree induced node set reach width width skeleton may written width cut cut finally note skeleton dimension graph may written width construction hub sets edge hub sets obtained following randomized construction assign edge real value uniformly independently random condition subsequent considerations event values distinct holds probability define central subpath subpath consisting middle edges formally nodes given next define hub edge edge minimum value central subpath arg min finally node adopt natural definition edge hub set set edge hubs node paths nodes proof correctness taking account observe symmetry central subpath respect two endpoints also follows directly hence also completes proof correctness edge hub labeling devote rest section bounding size hub sets bounding average hub set size subsequent considerations fix node restrict considerations tree assume tree oriented root towards leaves call path descending path one endpoints descendant particular every path descending path edge denote two endpoints one away root likewise descending path denote two extremal vertices closest furthest root respectively also denote distance edge root order bound expected size hub set observe elements necessarily belong skeleton satisfy certain minimality constraints respect descending paths sufficiently large length contained entirely within skeleton lemma let following claims hold reach exists descending path arg furthermore following claims hold edge satisfying claims exists descending path one two extremal edges exists descending path satisfying arg one two extremal edges proof select arbitrarily let node recall descending path arg let node recall assumption definition skeleton clearly note moreover reach hence claims follow show claim put observe arg definition next show claim observe claim arg arg moreover since claim thus follows choice descending path respectively finally show claim consider separately two cases proof claim set choose hence arg claim follows set choose choice always possible since moving along path value increases every step moreover lower end node path obtain arg claim follows remark remainder analysis valid construction hub sets satisfies claims lemma bound average hub set size precisely introduce node parameter called integrated skeleton dimension defined sum inverse distances nodes tree skeleton cut equivalence two definitions follows directly definition cuts cut taking account cut width width log even roughly log recall one may particular consider alternative construction hub set defined set edges satisfy claims lemma bounds hub set size also hold case definition results larger labels practice always correctness results observation hand hub sets may sometimes constructed efficiently definition requires scan entire tree whereas hub set may constructed based smaller skeleton lemma expected hub set size node satisfies bound proof arbitrarily define unique node path define random variable number extreme edges path satisfy arg min claim lemma follows summing random variables exhaustively vertices count element least hence linearity expectation follows equ direct application markov inequality bound lemma combined gives following corollary corollary average hub set size satisfies log probability least choice random values obtaining concentration bounds maximal size hub set requires care proceed analysis following subsection concentration bounds hub set size fixed consider size hub set given random variable indicator variable event random variables need general independent negatively correlated subsequent analysis fixed make use claim lemma bound random variable claim lemma decompose contributions descending paths located towards root away root respect define indicator event unique descending subpath length ending edge holds arg min variable event exists descending path length starting edge arg min moreover claim lemma may edges reach denote reach rewrite proceed bound sums separately order able manipulate sums conveniently first introduce partition tree according geometrically increasing scales distance partition llayers mwe consider sequence increasing integer radii given last layer corresponds index imax cutting edge set tree vertices located distances root yields following partition layers denote subset layer restricted edges lemma edge set admits partition paths cut descending path internal nodes paths degree exactly edge set considered layer path proof define partition maximal descending path whose internal nodes degree exactly let oriented induced edges let number leaves number connected components elementary relation number leaves number nodes degree forest gives moreover definition reach leaf extended along follows paths descending path distance follows leaves extended along independent descending path radius inclusive thus cut completes proof bounding sum denote set descending paths stretch precisely endpoints layer fixed path denote unique path extends consider arbitrary edge belong layers tree partition taking account decomposition set layers layers paths exists unique path observe event hold necessary two conditions jointly fulfilled must satisfy prefix minimum condition path arg min moreover must min indeed considering definition unique descending subpath length ends edge endpoint arg min path includes subpaths entire prefix path denote set edges satisfying min denote edge ordering edges increasing distance root finally denote indicator random variable event edge satisfies prefix minimum condition path follows note ranges sum indices depend random choice setting rewrite expression roughly bounding first sum cardinality splitting second double sum according even odd values cut even odd subsequently consider bounds summed expression even bounds expression follow identical arguments odd observe fixed random variables depend choice random values conditioning choice random values observe set independent random variables probability independence follows directly characterization probability element uniformly random permutation ordering prefix minimum even neven application simple chernoff bound sum variables even neven remains provide bounds concentration random variable neven upper tail goal bound hub set size log log log log obtaining bounds becomes relatively straightforward exercise chernoff bounds individual paths work pedestrian approach type process optimizing bounds larger path sets eventually give slightly tighter bounds including bound log log log denote following bound figure illustration paths edges marked solid lines remaining edges marked dashed lines lemma fix forming subforest proof fixed consider edge set contained entirely within layers see figure illustration choose arbitrarily set necessarily distinct values appear within let couple sampling values following process first fix set given choice select uniformly random permutation perform assignment values edges latter permutation defined iteratively assigning successive values yet unoccupied edge site value given number elements placed sites smallest index itj refer index representing moments time say path cut time successive moments time denote set surviving path indices time obtain subforest restricting surviving part prove claim consider random variable increases time couple sampling process first deciding time step whether place forest forest afterwards fixing specific free site uniform probability within chosen subforest observe placed given step considered process value remains unchanged time thus eliminate process time steps slight abuse notation relabel time indices steps never occurred thus time step assume free site picked uniformly random consider random variable representing number paths cut time step expectation regardless history process claim proof claim fix forest assign edge weight given reached descending path number edges put chosen edge process follows completes proof claim moreover taking bounded range obtain concentration result number steps stopping process completeness provide standalone proof claim proof claim define submartingale follows choose dominated latter condition always satisfied claim put observe necessarily hence probability event remark bounds imply following bound variance process using standard martingale bound thm applied process obtain exp substituting obtain cli taking account assumption claim follows directly recalling time step value random variable increases obtain directly claim completes proof next let random variable defined smallest integer since depends random values chosen random variables independent moreover lemma may stochastically dominated independent geometrically distributed random variable parameter follows bimax parameters negative binomial distribution represent number trials success probability successes reached application rough tail bound bimax gives recalling pli may write concavity logarithm function taking account definition therefore bound variable neven neven apply union bound two events given hold following event holds least neven max positive integers satisfying condition returning respect nodd analogous technique gives least nodd max likewise positive integers satisfying condition combining union bound substituting eventually obtain least cut max satisfy condition bounding sum random variables main arguments required establish bound similar case confine exposition differences main difference path instead unique predecessor path layer deal multiple possible descendant paths layer hand structure tree means show tighter concentration bounds case recall set descending paths stretch precisely endpoints layer denote set paths extensions event hold necessary two conditions jointly fulfilled must satisfy suffix minimum condition path arg min moreover min next denote set edges satisfying min denote edge ordering edges decreasing distance root finally denote indicator random variable event edge satisfies suffix minimum condition path subsequent analysis proceeds obtain direct analogues replacing superscripts ofpall random variables next denote obtain following analogue lemma lemma fix proof proof follows along lines lemma sli similar fixed let consider edge set forming subforest contained entirely within layers choose arbitrarily set necessarily distinct values appear within let proof lemma couple sampling values following process first fix set given choice select uniformly random permutation perform assignment values edges latter permutation defined iteratively assigning successive values yet unoccupied edge site value given number elements placed sites smallest index paths exists refer index representing moments time let say path cut time smallest time successive moments time denote set surviving indices paths cut time obtain subforest restricting surviving part treat path extends least one surviving path exactly proof lemma consider random variable increases time couple sampling process first deciding time step whether place forest forest afterwards fixing specific free site uniform probability within chosen subforest observe placed given step considered process value remains unchanged time thus eliminate process time steps slight abuse notation relabel time indices steps never occurred thus time step assume free site picked uniformly random consider random variable representing number paths cut time step expectation regardless history process claim proof claim fix forest inserting number free sites layer least hand since surviving path extends surviving path total number free sites insertion since insertion layer means insertion layer means obtain completes proof claim moreover taking bounded range obtain concentration result number steps stopping process directly martingale inequality thm parameter obtain transformations recalling time step value random variable increases obtain directly completes proof rest argument proceeds case applying lemma place lemma eventually obtain following analogue least cut max satisfy condition combining bounds introducing bounds union bound obtain following statement least cut max satisfy condition recalling bounds lemma bound imax setting applying union bound vertices obtain main technical result section present first strongest form provide two useful corollaries theorem probability least nodes satisfy following bound hub set size cut max cut maximum taken tive integers satisfying condition provide two convenient corollaries theorem case considered trees close simply bound size cuts cut skeleton dimension cut bound takes asymptotic form imax log max latter sum bounded using concavity logarithm function log imax max log max imax max imax log log log max log log also observe following link parameters since proposition graph doubling dimension bounded follows ball may contain nodes hence obtain log log log thus log additive factor bound dominated notation last factor sum stated least log combining obtain following corollary corollary probability least hub set size every node bounded log log max log log particular graph sufficiently large diameter hub set size nodes bounded log general case introducing corollary obtain following statement corollary probability least hub set size every node bounded log log log considering case trees width tree far uniform different scales distance tighter bounds obtained relating size top integrated skeleton dimension apply rough bound log log log log leaves expression form log log log log log log log log used bound rameter follows easily definition corollary probability least hub set size every node bounded log log log log log application distance labeling slight extension results note technique based analyzing tree skeletons shortest path trees direct application distance labeling problem unweighted graphs parameter recall scheme called queried pair nodes value returned decoder equal analogy integrated skeleton dimension given introduce variant parameter considers cuts distance cut claims lemma corollary give bounds average hub size log log log unweighted setting naturally translate labeling techniques directly applicable suffices subdivide edge graph path vertices distances pairs divisible choose shortest path trees pair nodes intersection contains shortest path may achieved example enforcing unique choice shortest paths node pair choosing length edge range entire analysis holds eventually replace statement claims remark elementary property tree skeleton cut since node distance continues along independent path length least performing latter sum obtain thus obtain following proposition proposition exists hub labeling scheme distance labeling problem hubs average size worst case size log log log size bit representations corresponding labels log log log log respectively size obtained labeling scheme almost optimal since holds lower bound average size hub sets fact scheme modified slightly obtain hub sets size certain threshold present details modified scheme following subsection value modified labeling scheme section present independent family distance labeling schemes property whereas scheme presented analysis valid value parameter obtain improvement previously discussed scheme threshold value construction labeling fix value parameter basic building block labeling construction hub sets node allow handle distance queries pairs nodes whose distance range providing details constructions sets first introduce auxiliary notation pair nodes denote fixed shortest path definition ties different shortest paths broken consistent manner whole graph set shortest paths rooted node spanning tree node denote shortest path subtree rooted leading nodes distance range denote subtree skeleton tree also rooted truncated first levels root remark descending paths reach least set constructed similarly include vertices central part path tree vertices however wish control number possible bad events descending path tree branches level many subpaths representative node need chosen partition vertex set tree two subsets known heavy light vertices respectively vertex belongs subtree rooted least leaves last level belongs otherwise remark possibly empty subtree whereas connected component tree less figure hub set selection distance range corresponding shortest path tree vertex shown figure set heavy vertices shaded around vertex remaining vertices distance belong vertices also marked corresponding descending paths shaded leaves considered trees maintain ancestry relation particular speak descending subpath tree one endpoints ancestor respect tree rooted ready define hub sets following randomized construction assign node real value uniformly independently random put defined set vertices exists descending subpath minimal value along path arg arg min see fig illustration correctness start showing sets desired hub property regardless choice random values may affect size sets lemma pair nodes min proof consider path moreover denoting subpath belongs trees follows prove claim lemma showing least one vertex belong achieve case analysis depending portions path belong sets exists least one vertex completes proof exists least one descending subpath length completely contained setting arg min follows obtain result applying analogous considerations previous case finally cases must follows exists least one subpath length descending subpath setting arg min obtain hence analysis consider size sets size set independent choice random variables easily bounded taking account tree leaves lemma proof let leaf node definition subtree rooted least leaves every leaf depth leaves tree depth least follows subtree contains least disjoint descending paths length least nodes since size tree obtain tree leaves moreover distance node root hence size set depends choice random variables start bounding expected number elements belonging specific connected components suppose forest consisting trees let partition vertex set represents connected component let denote number leaves tree finally let clearly partition following showing expectation consider random variable obtaining concentration results around expectation first remark descending path tree contributes elements expectation set consequently expected size set related number leaves considered connected component lemma proof let set maximal descending paths tree remark path let event exists subpath arg min use following folklore probability estimation hold one two descending subpaths length one endpoints must satisfy arg min follows linearity expectation obtain bound max linearity expectation apply claim lemma connected components obtaining following result lemma proof lemma indeed sum represents total number claim follows observe leaves leaf located distance upper endpoint distinct descending path length least tree hence obtain bound order apply chernoff bounds sum random variables start bounding range variables lemma proof definition set tree less leaves nodes distance root follows upper bound provides estimate maximum value random variable however order able perform concentration analysis range fairly large roughly also need bound tightly concentration around expected value let start showing high probability elements sets belong lemma denote bad event exists node proof consider first probability fixed node satisfies arg min fixed path nodes contains last inequality holding probability event occurring performing union bound nodes descending paths nodes event arg min occurring node paths consider less possible nodes path overall obtain next show high probability connected component contains log nodes lemma denote bad event exists node proof let denote indicator variable node set otherwise clearly random variables independent fixed used bound next proceed apply simple multiplicative chernoff bound considered random variable applying union bound gives claim ready apply type analysis random variable lemma let proof define random variable fix otherwise pku define independent random variables since functions disjoint sets random variables moreover application simple multiplicative chernoff bound gives used bound following lemma putting taking account get sufficiently large applying union bound nodes obtain taking account lemmas also overall obtain view definition proposed hub set labeling lemmas complete analysis case showing randomized construction yields high probability hub sets size nodes graph proposition exists hub labeling scheme correctly decodes distance pair nodes lying distance range using hubs size improved distance labeling arbitrary distance arbitrary instance distance labeling problem construct hub set combining results propositions large small scales distance respectively formally put first part expression value max suitably chosen threshold parameter hub set constructed following proposition thus providing distance labeling second part expression take care smaller distances range applying proposition specifically chosen distance sequence obtain hub sets set intersects shortest path nodes obtain coverage entire distance range put choose largest integer since sequence geometrically increasing view proposition obtain following bound taking account definition bounding proposition directly obtain main result section theorem exists distance labeling scheme based hub sets hub sets nodes size corresponds distance labels size log per node hub sets nodes size log log log corresponds distance labels size log log log log per node iii average size hub set taken nodes corresponds distance labels average size log per node furthermore corresponding labels constructed expected polynomial time computing skeleton dimension distance labels discrete skeleton representation given tree rooted node length function discrete representation skeleton obtained edges reach equipped length function defined reach reach otherwise idea node leaf corresponds point edge satisfies reach dte see let descendant reach thus get reach whereas dte skeleton dimension computation given tree reach node computed scan vertices reverse topological order obtaining discrete skeleton representation straightforward width computed scanning vertices distance root using priority queue storing edges containing nodes cut done log log time using dedicated integer priority queue skeleton graph thus simply obtained pair shortest path computation integer lengths dedicated priority queues done log log time remark faster computation tree skeletons could obtained practice using classical heuristics bounding reach nodes algorithm proposed alternates partial tree computation introduction shortcuts obtain efficiently reach bounds graph plus added shortcuts computation partial trees given radius allows prune nodes reach less shortcuts allow reduce reach nodes degree node two neighbors shortcut length added algorithm results reach bounds graph shortcuts reach bounds original graph obtained removing shortcuts reverse order updating reach bound node shortcut max min denote reach bounds obtained respectively subtree containing tree skeleton node obtained partial dijkstra prune nodes whose reach known less half current distance practice believe would allow compute skeleton tree time comparable query labeling algorithm adapted take resulting family trees input distance label computation computing hub set tree intricate emulate subdivision edge length unweighted edges conform analysis section sake notation number unweighted edges subdivision let denote associated random number generated edges subdivision given sample let denote set indices edges prefix minima suffix minima sequence purpose selection algorithm need generate set associated start generating elements prefix minima slight abuse notation initialise process set generate uniformly random rand successive generate index first edge edge index also first index value less follows geometric distribution rand parameter done constant time setting loglog see generate uniformly generate way indices reach bound proceed similarly reverse order edges index greater generate edges suffix minimal value note time sample values greater rather greater adapt ranges accordingly consistency choice prefix minima way perform log sampling operations per edge length obtaining log values together positions per edge expectation also respect since random choices made edges original graph independent quick chernoff bound total amortized sampling time whole graph log denotes maximum length edge remark constructing hub set subset nodes node relies random choices made tree evaluated time log selection algorithm edge minimal value middle window pair necessarily select edge generated window contains real edge extremity time virtual unweighted edge selected hub indeed select real edge belongs also manage special case middle window entirely falls inside long edge case long edge selected hub computation hub set matter scanning tree distance generated vertices maintaining sliding middle window branch reaching distance using balanced binary search tree per window storing virtual edges contains obtain hub set log log log time distance labels thus computed expected log log log log time note labels computed independently parallel log log log log time per label long randomness shared using random generators seeds generalizing definition skeleton definition skeleton corresponding notion skeleton dimension generalized two ways using different distinct distance metric compute reach point tree well modifying threshold value reach required retain point skeleton using two metrics suppose graph associated two length functions edges example road networks one typically consider travel time geographic distance resulting time distance metrics respectively another metric may interest hop count corresponding constant function edge shortest path tree node geometric realization teu computed according length function distance reach within teu computed according formally extending definition section skeleton defined subtree teu induced teu reach reach denote distance reach respect skeleton dimension width advantage approach sometimes results smaller skeleton dimension depending choice metric without affecting correctness hub labeling schemes designed paper example respective skeleton dimensions dimacs new york graph different choices metrics turn denote traveltime length functions respectively considering shortest path trees metric three cases remark similar phenomenon also taking advantage two metrics observed used approach modifying reach threshold choice reach threshold definition skeleton arbitrary indeed fixed define skeleton subtree teu induced teu reach teu dteu skeleton dimension given width values skeleton dimension different values reach threshold related following proposition proposition two constants following bounds hold proof first relation immediate since subtree second relation obtained observing cut cut indeed cut consider teu point distance branch leading reach teu least reach teu thus belongs moreover distance tev reach least implying case second relation proposition gives also implies generally derive following bounds repeatedly applying proposition dlog shows given graph skeleton dimension grows polynomially naturally one also apply generalizations together obtaining new skeleton dimension parameter reach metric reach threshold results paper hub labelings computation graphs low skeleton dimension easily generalized use instead long ensuring two skeleton trees share constant fraction shortest path particular choice made objective clarity also account simple relationship highway dimension conclusion paper proposed skeleton dimension measure network amenability shortest path schemes based nodes intend parameter easy describe computed efficiently computations hub sets based skeleton dimension allow node individually efficiently define hub set subject universal choice random construction always correct gives small hub sets remark weighted network node compute appropriate labeling log log log log time length longest integer weight network definition hub sets obtained bounds size hold undirected directed graphs directed graphs skeleton dimension appears parameter directly usable highway dimension possible extensions skeleton dimension discussed section include variants skeleton dimension values reach threshold well skeleton dimension defined using two separate distance metrics graph one corresponding needs shortest path queries used construct shortest path trees another potentially independent metric used internally computation hub labelings chosen empirically minimize width skeleton studying parameters network integrated skeleton dimension given well natural generalizations weighted graphs appear natural parameter may related average highway dimension could also use integrated skeleton dimension averaged nodes get even accurate bound average label size finally remark interplay skeleton highway dimension skeleton dimension always greater geometric highway dimension also shown clear case separation weighted network skeleton dimension asymptotically much smaller geometric highway dimension remark skeleton dimension appears particularly worthy theoretical study context models random graphs discussion context highway dimension reach geometric percolation graphs skeleton dimension displays close link coalescence exponent geodesics consequently may easier show rigorous theoretical bounds skeleton dimension highway dimension acknowledgment authors thank przemek olivier marty inspiring discussions closely related problems also thank zuzanna help figures references dimacs implementation challenge shortest path problem ittai abraham daniel delling amos fiat andrew goldberg renato werneck highway dimension provably efficient shortest path algorithms technical report september ittai abraham daniel delling amos fiat andrew goldberg renato werneck vcdimension shortest path algorithms icalp volume lecture notes computer science pages springer ittai abraham daniel delling andrew goldberg renato werneck labeling algorithm shortest paths road networks sea volume lecture notes computer science pages springer ittai abraham daniel delling andrew goldberg renato werneck hierarchical hub labelings shortest paths proceedings annual european conference algorithms esa pages berlin heidelberg ittai abraham daniel delling andrew goldberg renato werneck hierarchical hub labelings shortest paths esa volume lecture notes computer science pages springer ittai abraham amos fiat andrew goldberg renato werneck highway dimension shortest paths provably efficient algorithms moses charikar editor proceedings annual symposium discrete algorithms soda austin texas usa january pages siam ittai abraham cyril gavoille approximate distance labels routing schemes affine stretch international symposium distributed computing disc pages david aldous karthik ganesan true random spatial networks proceedings national academy sciences stephen alstrup dahlgaard mathias tejs knudsen ely porat sublinear distance labeling piotr sankowski christos zaroliagis editors annual european symposium algorithms esa august aarhus denmark volume lipics pages schloss dagstuhl fuer informatik stephen alstrup cyril gavoille esben bistrup halvorsen holger petersen simpler faster shorter labels distances graphs robert krauthgamer editor proceedings annual symposium discrete algorithms soda arlington usa january pages siam maxim babenko andrew goldberg anupam gupta viswanath nagarajan algorithms hub label optimization fedor fomin rusins freivalds marta kwiatkowska david peleg editors automata languages programming international colloquium icalp riga latvia july proceedings part volume lecture notes computer science pages springer bast stefan funke domagoj matijevic peter sanders dominik schultes transit constant time queries road networks alenex siam holger bast stefan funke peter sanders dominik schultes fast routing road networks transit nodes science reinhard bauer daniel delling sharc fast robust unidirectional routing exp algorithmics january coppersmith michael elkin sparse distance preservers additive spanners siam journal discrete mathematics fan chung lincoln survey concentration inequalities martingale inequalities survey internet mathematics edith cohen eran halperin haim kaplan uri zwick reachability distance queries via labels siam may luc devroye random variate generation andreas emil feldmann wai shing fung jochen ian post low highway dimension graphs bounded treewidth graphs icalp volume lecture notes computer science pages springer cyril gavoille david peleg ran raz distance labeling graphs algorithms october andrew goldberg haim kaplan renato werneck reach efficient shortest path algorithms alenex pages siam graham pollak embedding graphs squashed cubes alavi lick white editors graph theory applications volume lecture notes mathematics pages springer berlin heidelberg ronald gutman routing new approach shortest path algorithms optimized road networks pages siam ekkehard rolf heiko schilling fast shortest path computations dimacs implementation challenge colin mcdiarmid concentration michel habib colin mcdiarmid jorge bruce reed editors probabilistic methods algorithmic discrete mathematics pages springer berlin heidelberg igor nitto rossano venturini compact representations matrices paolo ferragina gad landau editors combinatorial pattern matching annual symposium cpm pisa italy june proceedings volume lecture notes computer science pages springer mikkel thorup integer priority queues decrease key constant time single source shortest paths problem journal computer system sciences oren weimann david peleg note exact distance labeling inf process
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proceedings machine learning research vol minimax distribution estimation wasserstein distance shashank singh sss cmu edu bapoczos cmu edu feb machine learning department carnegie mellon university pittsburgh usa abstract wasserstein metric important measure distance probability distributions several applications machine learning statistics probability theory data analysis paper upper lower bound minimax rates problem estimating probability distribution wasserstein loss terms metric properties covering packing numbers underlying sample space keywords wasserstein distance density estimation minimax theory covering number packing number introduction wasserstein metric important measure distance probability distributions based cost transforming either distribution mass transport metric sample space originating optimal transport wasserstein metric owing intuitive general nature utilized diverse areas probability theory statistics economics image processing text mining robust optimization physics villani fournier guillin esfahani kuhn gao kleywegt analysis image data wasserstein metric used various tasks texture classification face recognition sandler lindenbaum reflectance interpolation color transfer geometry processing solomon image retrieval rubner image segmentation analysis text data tasks document classification kusner machine translation zhang contrast number popular notions probability distributions distances morimoto ali silvey require distributions absolutely continuous respect base measure wasserstein distance pair probability distributions sample space equipped particularly important consequence wasserstein distances discrete empirical distributions continuous distributions informative often finite partly reason wasserstein distances widely used probability theory prove central limit related approximation theorems johnson wasserstein metric variously attributed monge kantorovich rubinstein gini mallows others see chapter villani detailed history use term distance refer proper metrics well similar notions information divergences fail satisfy symmetry triangle inequality reason use distribution estimation paper rather popular density estimation singh istribution stimation wasserstein istance chatterjee rio chen reitzner within machine learning statistics property inspired class minimum wasserstein distance estimates del barrio bassetti bernton distributions ranging exponential distributions exotic models restricted boltzmann machines rbms montavon generative adversarial networks gans arjovsky clustering similarly clustering also fall class estimators hypothesis class taken discrete distributions supported points flexible clustering algorithms also expressed way using elaborate hypothesis class pca also expressed generalized manifolds using wasserstein distance minimization boissard principle estimators equivalent empirical risk minimization taking advantage fact wasserstein distances empirical distribution distributions relevant hypothesis class finite moreover estimates often perform well practice free tuning parameters strong distributional assumptions paper study minimax sense problem estimating distribution using wasserstein distance loss function make minimal assumptions examples varied metric spaces results apply given section specifically assume sample space totally bounded metric space make assumptions distribution prove bounds minimax convergence rates distribution estimation utilizing covering numbers sample space upper bounds packing numbers lower bounds may first surprising positive results obtained mild assumptions highlights wasserstein metric quite weak metric see lemma subsequent remark detailed discussion moreover results imply without assumptions population distribution empirical distribution often minimax rganization remainder paper organized follows section provides notation required formally state problem interest results section reviews previous work studying convergence distributions wasserstein distance section provides several lemmas proofs main results rely lemmas proven appendix sections respectively contain main upper lower bound results along proofs finally section apply upper lower bounds identify minimax convergence rates number concrete examples section concludes summary contributions suggested avenues future work notation problem setting integer denotes set first positive integers sequences real numbers equivalently used indicate existence constant lim abnn indicates simultaneously istribution stimation wasserstein istance problem setting remainder paper fix metric space denotes borel let denote family borel probability distributions main object study paper wasserstein distance defined follows definition distance given two borel probability distributions distance defined inf denotes couplings set joint probability measures marginals proper metric namely symmetric arguments satisfies triangle inequality intuitively quantifies total cost transforming mass distributed according distributed according cost moving unit mass note sometimes defined terms equivalent dual formulations especially however necessary paper give formal statement problem studied paper formal problem statement suppose metric space suppose unknown iid borel probability distribution observe iid samples interested studying minimax rates estimated terms power loss specifically interested upper lower bounding quantity wrr inf sup xniid infimum taken estimators functions data definitions stating results give notation definitions needed state theoretical results sections intermediate lemmas section let denote power set let denote family borel partitions first define metric notions later useful bounding wasserstein distances definition diameter separation set resolution partition set diameter diam defined diam supx separation sep defined sep inf partition resolution res defined res diam istribution stimation wasserstein istance definition refinement partition nested partitions suppose partitions said refinement every exists sequence partitions called nested refinement define covering packing number metric space classic widely used measures size complexity metric space dudley haussler zhou zhang main convergence results stated terms quantities well packing radius acts approximately inverse packing number definition covering number packing number packing radius metric space covering number defined min res packing number defined max sep finally packing radius defined sup sep sometimes use covering packing number metric space say cases write rather respectively specific also refer number remark covering packing numbers metric space closely related particular always packing number packing radius also close approximate inverse relationship particular always however may need general case remark defined covering number slightly differently usual using partitions rather covers however given definition equivalentsto usual definition since partition cover set countable cover sthere exists partition defined recursively often called disjointification istribution stimation wasserstein istance related work long line work dudley ajtai dereich boissard fournier guillin weed bach studied distribution estimation wasserstein distance papers focusing analyzing convergence empirical distribution population distribution wasserstein distance terms upper bounds general tight upper bounds weed bach provide results general metric spaces terms covering numbers spaces main results expressed terms particular notion dimension call wasserstein dimension allows derive convergence rates order matching rate achieved unit cube general upper bounds function covering number somewhat easier apply weed bach use section derive convergence rates even cases infinite dimensional cases best knowledge work deriving minimax lower bounds distribution estimation wasserstein loss noted previous works focused studying convergence rate empirical distribution true distribution wasserstein distance rate several lower bounds established matching known upper bounds many quite general cases however many distribution estimators besides empirical distribution considered example tempting especially given infinite dimensionality distribution estimated try reduce variance techniques smoothing importance sampling bucklew however lower bound results given section imply empirical distribution already minimax optimal constant factors many cases preliminary lemmas begin providing basic lemmas lemmas fundamentally novel used subsequent proofs main upper lower bounds also help provide intuition behavior wasserstein metric connections metrics probability distributions first lemma relates wasserstein distance notion resolution partition lemma suppose countable borel partition let borel probability measures every res next lemma gives simple lower upper bounds wasserstein distance distributions supported countable subset terms diam sep since main results utilize coverings packings approximate finite sets lemma provide first step towards approximating wasserstein distance distributions distributions finite sets indeed lower bound inequality suffice prove lower bounds although tighter upper bound based upper bound necessary obtain tight upper bounds lemma suppose metric space suppose borel probability distributions countable support exists countable set wrr diam sep istribution stimation remark recall term wasserstein istance inequality distance densities respect counting measure quantity twice total variation distance sup hence lemma equivalently written sep diam sep diam bounding distance terms total variation distance noted example equality holds precisely unit discrete metric given metric spaces discrete sep wasserstein metric topologically least strong total variation metric metric convergence wasserstein metric implies convergence total variation respectively hand bounded metric spaces converse true either cases rates convergence may differ metrics although metric spaces discrete bounded finite space lemma gave simple upper bound wasserstein distance factor diam turns large obtain tight rates number cases interest ddimensional unit cube discussed example following lemma gives tighter upper bound based hierarchy nested partitions allows obtain tighter bounds diam distance mass must transported note lemma reduces trivial combination lemmas indeed lemmas starting point proving lemma induction note idea upper bound utilized extensively numerous versions proven see fact lemma fournier guillin proposition weed bach however versions specific euclidean space best knowledge proposition weed bach applies general metric spaces lemma let positive integer suppose nested sequence countable borel partitions borel probability measures wrr res res istribution stimation wasserstein istance lemma requires sequence partitions also nested number implies existence small partitions small resolution partitions need nested becomes small reason give technical lemma given sequence partitions constructs nested sequence partitions cardinality small increase resolution lemma suppose partitions suppose countable exists partition res res res refinement upper bounds section utilize covering number bounds previous section develop error bounds density estimation using empirical distribution wasserstein loss expectation bounds use lemmas prove bounds expected wasserstein distance empirical true distributions begin simple technical lemma bounding expected deviation multinomial random variable mean lemma suppose multinomial let theorem let metric space borel probability measure let denote iid empirical distribution iid samples give sequence diam istribution stimation wasserstein istance proof recursively applying lemma exists sequence partitions satisfying following conditions res nested note vector npb indexed follows distribution categories means given npb npb multinomial thus lemma thus lemma lower bounds section prove minimax lower bounds family densities metric space density estimation wasserstein distance quantity wrr inf sup iid infimum estimators functions bounds terms packing radius istribution stimation wasserstein istance theorem suppose unknown borel probability measure suppose iid observe iid samples exists universal constant chjw chjw sup inf sup iid proof proof based reduction minimax lower bound estimating mean parameter vector multinomial distribution norm specifically use corollary han implies existence universal constant chjw independent chjw inf sup denotes standard probability simplex distributed iid according multinomial distribution parameter vector infimum taken estimators functions let denote class discrete distributions corollary han implies kpb chjw inf sup infimum taken estimators functions recalling lemma implies sep kpb wrr inf sup wrr sep inf sup kpb sep inf sup kpb chjw sep chjw theorem follows taking supremum sides istribution stimation wasserstein istance example applications conclude exploring applications results several metric spaces following examples unknown borel probability measure specified observe iid samples upper bounds denotes empirical distribution samples example finite space consider case finite set discrete metric given covering number thus setting sending theorem gives hand thus setting theorem gives chjw inf sup iid example unit cube euclidean metric consider case unit cube euclidean metric suppose borel probability measure let denote empirical distribution iid samples setting recalling one show theorem gives gives setting wrr log gives rate wrr log otherwise sending gives rate wrr setting log gives rate wrr summarize wrr log istribution stimation wasserstein istance reproducing results fournier guillin hand easy check packing radius satisfies thus applying theorem inf sup wrr chjw max combining upper lower bounds gives minimax rate density estimation loss wrr inf sup except case upper lower bounds separated factor log moreover rate achieved using empirical distribution estimate consider high dimensional problem size grow example binary hypercube hamming metric suppose binary hypercube suppose hamming metric given covering number easily upper bounded thus setting sending theorem gives famous bound see lemma rigollet interpreted statement thus setting long log theorem applying bound gives wrr sup inf sup iid log min summarize case consistent estimation possible log log log finally consider distributions infinite dimensional space smooth functions example ball metric suppose class unit functions unit cube given sup istribution stimation wasserstein istance covering packing numbers order exp devore lorentz specifically exist positive constants exp exp also follows inequality log since diam setting theorem gives log log log wrr log hand setting theorem gives inf sup log iid showing distribution estimation wrr extremely slow minimax rate log although considered due notational complexity defining spaces analogous rates hold also since rates depend covering packing numbers identical rates derived related sobolev besov classes one might wonder interested studying wasserstein convergence distributions spaces smooth functions example main motivation comes fact historically smooth function spaces widely used modeling images complex naturalistic signals mallat far recently empirical breakthroughs made generative modeling particularly images based principle minimizing wasserstein distance empirical distribution large class models encoded deep neural network montavon arjovsky gulrajani however little known theoretical properties methods work studying optimization landscape models nagarajan kolter know work exploring statistical properties given extremely slow minimax convergence rate derived must case class distributions encoded models far smaller sparser important avenue work thus explicitly identify stronger assumptions made distributions interesting classes signals images bridge gap empirical performance theoretical understanding istribution stimation wasserstein istance conclusion paper derived upper lower bounds distribution estimation wasserstein loss upper bounds generalize prior results tighter certain cases lower bounds best knowledge first minimax lower bounds problem also provided several simple examples upper lower bounds agree future work studied minimax rates entire class distributions metric space would useful understand minimax rates improve additional assumptions smoothness assumptions made see fournier guillin somewhat improved upper bounds assumptions euclidean space given rather slow convergence rates found many cases studying minimax rates assumptions may help explain relatively favorable empirical performance popular distribution estimators based empirical risk minimization wasserstein loss moreover rates interest weak metrics wasserstein distance stronger metrics may infinite undefined studying minimax rates additional assumptions allow better understanding wasserstein metric relation commonly used metrics acknowledgments work partly supported nsf graduate research fellowship nsf darpa program afrl grants references ajtai optimal matchings combinatorica syed mumtaz ali samuel silvey general class coefficients divergence one distribution another journal royal statistical society series methodological pages martin arjovsky soumith chintala bottou wasserstein gan arxiv preprint amparo javier konstantin getman estimation wasserstein zolotarev distances class exponential variables arxiv preprint federico bassetti antonella bodini eugenio regazzini minimum kantorovich distance estimators statistics probability letters espen bernton pierre jacob mathieu gerber christian robert inference generative models using wasserstein distance arxiv preprint emmanuel boissard thibaut gouic mean speed convergence empirical occupation measures wasserstein distance annales institut henri statistiques volume pages institut henri istribution stimation wasserstein istance emmanuel boissard thibaut gouic loubes distributions template estimate wasserstein metrics bernoulli james bucklew introduction rare event simulation springer science business media sourav chatterjee new method normal approximation annals probability louis chen larry goldstein shao normal approximation steins method springer science business media imre eine informationstheoretische ungleichung und ihre anwendung auf beweis der ergodizitaet von markoffschen ketten magyer tud akad mat kutato int eustasio del barrio evarist carlos central limit theorems wasserstein distance empirical true distributions annals probability pages eustasio del barrio evarist carlos correction central limit theorems wasserstein distance empirical true distributions annals probability steffen dereich michael scheutzow reik schottstedt constructive quantization approximation empirical measures annales institut henri statistiques volume pages institut henri ronald devore george lorentz constructive approximation volume springer science business media luc devroye equivalence weak strong complete convergence kernel density estimates annals statistics pages khanh huy nguyen huy nguyen ronitt rubinfeld sublinear time algorithms earth movers distance theory computing systems joseph doob measure theory volume springer science business media richard dudley sizes compact subsets hilbert space continuity gaussian processes journal functional analysis dudley speed mean convergence annals mathematical statistics peyman mohajerin esfahani daniel kuhn distributionally robust optimization using wasserstein metric performance guarantees tractable reformulations arxiv preprint nicolas fournier arnaud guillin rate convergence wasserstein distance empirical measure probability theory related fields istribution stimation wasserstein istance rui gao anton kleywegt distributionally robust stochastic optimization wasserstein distance arxiv preprint ishaan gulrajani faruk ahmed martin arjovsky vincent dumoulin aaron courville improved training wasserstein gans advances neural information processing systems pages yanjun han jiantao jiao tsachy weissman minimax estimation discrete distributions loss ieee transactions information theory david haussler sphere packing numbers subsets boolean bounded vapnikchervonenkis dimension journal combinatorial theory series nhat xuanlong nguyen mikhail yurochkin hung hai bui viet huynh dinh phung multilevel clustering via wasserstein means arxiv preprint oliver johnson richard samworth central limit theorem convergence stable laws mallows distance bernoulli matt kusner sun nicholas kolkin kilian weinberger word embeddings document distances international conference machine learning pages mallat wavelet tour signal processing academic press montavon marco cuturi wasserstein training restricted boltzmann machines advances neural information processing systems pages tetsuzo morimoto markov processes journal physical society japan vaishnavh nagarajan zico kolter gradient descent gan optimization locally stable arxiv preprint kangyu xavier bresson tony chan selim esedoglu local histogram based segmentation using wasserstein distance international journal computer vision gabriel numerical tours signal processing computing science engineering matthias reitzner matthias schulte central limit theorems poisson point processes annals probability phillippe rigollet statistics lecture notes course emmanuel rio upper bounds minimal distances central limit theorem annales institut henri statistiques volume pages institut henri emmanuel rio asymptotic constants minimal distance central limit theorem electronic communications probability istribution stimation wasserstein istance yossi rubner carlo tomasi leonidas guibas earth mover distance metric image retrieval international journal computer vision ludger wasserstein distance approximation theorems probability theory related fields roman sandler michael lindenbaum nonnegative matrix factorization earth mover distance metric image analysis ieee transactions pattern analysis machine intelligence justin solomon fernando goes gabriel marco cuturi adrian butscher andy nguyen tao leonidas guibas convolutional wasserstein distances efficient optimal transportation geometric domains acm transactions graphics tog villani optimal transport old new volume springer science business media jonathan weed francis bach sharp asymptotic rates convergence empirical measures wasserstein distance arxiv preprint meng zhang yang liu luan maosong sun tatsuya izuha jie hao building earth mover distance bilingual word embeddings machine translation aaai pages tong zhang covering number bounds certain regularized linear function classes journal machine learning research mar zhou covering number learning theory journal complexity istribution stimation wasserstein istance appendix proofs lemmas lemma suppose countable borel partition let borel probability measures every res proof fact intuitively obvious clearly exists transportation map moves mass within therefore without moving mass completeness give formal construction let denote coupling conditionally independent given set easy verify since countable partition supported lemma suppose metric space suppose borel probability distributions countable support exists countable set wrr diam sep proof term precisely unweighted amount mass must transported transform hence result intuitively fairly obvious mass moved cost least sep diam however completeness give formal proof prove lower bound suppose coupling similarly existence measure verified theorem similarly usual product measure see section doob istribution stimation wasserstein istance since follows therefore since whenever sep whenever sep sep taking infimum sides gives wrr sep prove upper bound since upper bounded diam suffices construct coupling moves mass given point min one way follows fix ordering elements define define min min move mass yji move mass xki construction total mass moved way lim lemma let positive integer suppose sequence nested countable borel partitions borel probability distributions res wrr res istribution stimation wasserstein istance proof proof follows ideas slightly generalizes proof proposition weed bach intuitively prove lemma suffices find transportation map recursively define borel measures defined construction measure furthermore consequently although probability measures slightly generalize definition wasserstein distance writing wrr inf wrr particular convenient one easily show construction sequences wrr wrr lemma implies wrr diam res res wrr furthermore lemma gives wrr res istribution stimation wasserstein istance plugging last two inequalities inequality gives desired result wrr res res lemma suppose partitions suppose countable exists partition res res res refinement proof enumerate elements define recursively define set clearly equality need hold may triangle inequality diam diam finally since partition write refinement proving lemma recall simple concentration inequality bounding deviation multinomial random variable mean lemma lemma devroye suppose multinomial let exp lemma follows easy corollary istribution stimation wasserstein istance lemma consider setting lemma proof since almost surely lemma exp exp exp exp exp istribution stimation wasserstein istance
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intelligent device discovery internet things enabling robot society james sunthonlap phuoc nguyen zilong jan california state university los angeles state university drive los angeles usa email internet things iot continuously growing connect billions smart devices anywhere anytime structure enables variety applications services interactions human objects future smart devices supposed able autonomously discover target device desired features thus yield computing service network service data fusion leads generation set entirely new services applications supervised even imagined human beings pervasiveness smart devices well heterogeneity design functionalities raise major concern smart device efficiently discover desired target device paper propose distributed sand scheme achieves fast scalable efficient device discovery iot proposed sand scheme adopts novel device ranking criteria measures device degree social relationship diversity clustering coefficient betweenness based device ranking criteria discovery request guided travel critical devices stand major intersections network thus quickly reach desired target device contacting limited number intermediate devices conduct comprehensive simulations random networks networks evaluate performance sand terms discovery success rate number devices contacted number communication hops simulation results demonstrate effectiveness sand help intelligent device discovery sand iot devices well computing facilities software data internet autonomously establish new social connections human formulate computing groups perform required computing tasks facilitate fusion variety computing service network service data generate novel applications services evolve individual aritificial intelligence collaborative intelligence eventually enable birth robot society index things distributed device discovery computing network data fusion robot society ntroduction internet things iot continuously growing connect billions smart devices anywhere anytime structure recent forecast international data corporation idc shows iot involve billion objects iot associated ecosystem predicted trillion market iot devices capable sensing analyzing evaluating surrounding objects people collaborate work together provide set new applications services smart home intelligent transportation system iot envisioned dramatically change enhance interactions human objects bringing transformative benefits lives human beings iot evolves promising future smart devices may capability discover target device desired features autonomously collaborate accomplish certain missions tasks intelligent device discovery strategy may allow iot devices efficiently establish new connections devices based need connections set gather certain number computing powers network functions program source codes raw datasets etc enable new services applications variety computing service network service data fused following guidance intelligent device discovery strategy example glucose level monitor send request find glucose analyzer collaborate evaluating given patient glucose level glucose analyzer may also need search national wide database compare given patient results patients given advisement alert given patient intelligent device discovery strategy could guide request appropriate destinations obtaining required computing facilities functions data provide evaluation service natually leading computing service network service data fusion addition iot devices also form social connections human beings collaborate generate entirely new kinds applications services help collaborative intelligence smart devices data rather supervision human beings future scenarios one key challenges achieve fast scalable efficient device discovery millions billions devices iot intelligent device discovery forwarding strategy guide discovery request quickly arrive desired target device minimum detours addition since iot devices power constrained device discovery avoid involving many intermediate devices process data exchange existing works topic iot device discovery work identified differences user search search presented challenges requirements iot device search authors studied device discovery different network topologies including centralized network regular decentralized network hierarchical network focused analyzing effect network topology discovery success rate rather designing new device discovery strategies study proposed centralized iot search engine accurately interpret context discovery request thus make proper management search usage iot middleware environment however centralized approach may scalable network size large vulnerable single point failure recent work investigated distributed device discovery strategy based networking icn however approach may applicable iot system since relies features icn cache broadcast capabilities could introduce extra cost iot devices work focus investigating autonomous intelligent device discovery iot treat smart devices independent objects limit study artificial intelligence single device explore humanlike social behaviors collaborative intelligence smart devices envisioned next episode research iot robotics particular propose novel iot device discovery scheme called distributed sand discover desired target device fast efficient scalable manner sand iot devices fair equal peers interact autonomous distributed manner human maintains information local neighboring peers iot device ranked measuring device degree social relationship diversity clustering coefficient betweenness based sand opts forward discovery request neighboring device highest rank likely stand major intersection network fast efficiently lead request desired target device conduct comprehensive simulations evaluate proposed sand scheme random network network simulation results show sand achieve high discovery success rate short discovery path contacting small number intermediate devices rest paper organized follows first introduce traditional centralized distributed device discovery schemes section describe design details proposed sand scheme section section present performance evaluation section discuss research challenges opportunities related iot device discovery finally conclude paper propose future work section raditional evice iscovery chemes two traditional solutions addressing device discovery iot centralized scheme distributed scheme former leverages centralized controller resolve discovery request latter uses simple broadcast mechanism discover required target device search manner detailed description comparison two approaches presented following subsections centralized iot device discovery scheme achieve fast efficient iot device discovery simple yet effective approach introduce centralized controller keeps track general information device functionality devices maintains shortest paths device pairs exists source device requests discover target device provides desired function discovery request sent centralized controller resolve centralized controller find location desired target device reply source device shortest path information target device centralized scheme fast accurate due centralized controller global view devices network also efficient since centralized controller provide shortest path source target devices involves minimum number intermediate devices transmitting discovery request centralized scheme designed robustness dynamic changes network new device joins network needs register general information controller device leaves network also needs unsubscribe centralized controller addition centralized controller periodically send heartbeat messages registered devices maintain list alive devices network device fails reply heartbeat three attempts considered mode lost removed list alive centralized controller see centralized scheme highly relies centralized controller manages whole system communication iot devices distributed iot device discovery scheme centralized scheme fast efficient however may scalable number devices large contrast distributed scheme offers scalable device discovery iot distributed scheme device maintains local neighbor list consists neighbors general information new device joins network exchanges general information nearby devices within transmission range device leaves network needs inform neighbors update local neighbor list heartbeats exchanged devices maintain fresh live lost status devices neighbors distributed discovery scheme works search manner source device makes request discover target device desired features distributed scheme simply broadcasts discovery request neighboring devices process iterates desired target device found compared centralized scheme distributed scheme resilient possible failures since centralized control risk single point failure also robust network dynamically changing since device needs maintain local neighbor information also infer distributed scheme find communication path source target devices since search used limitation large number intermediate devices may involved transmitting discovery request broadcast used energy consuming harmful iot system devices power constrained iii ocial istributed evice iscovery sand famous experiment travers milgram found people tied short chain connections two persons connected six hops simply exploring social networks recent facebook research confirmed observation concluded among billion active users average person separated hops away another person iot evolves devices could also exhibit social relationships example devices considered family manufacturer considered colleagues work together provide service social aspects therefore used intelligently navigate device discovery requests device social network connect two devices hops exhibited human social network based motivation propose distributed sand scheme achieve scalable fast efficient device discovery iot sand iot devices autonomously establish meaningful relationships devices form overlay device social network addition communication network may faster efficient devices find desired target device searching among friends overlay device social network oppose simply broadcasting communication network distributed scheme works following subsections present details sand architecture device ranking criteria forwarding strategy respectively sand architecture sand iot devices form two layers networks consists communication network overlay device social network communication network bottom layer abstracts data communication links iot devices long two iot devices within transmission range communication link exists actually distributed scheme introduced previous section considers communication network performs simple broadcasting device discovery sand addition communication network overlay device social network constructed help achieve effective iot device discovery two iot devices communicate exchange device general information manufacturer functionality ownership location information overlay device social network exists link two iot devices valid social relation similar human relationships device social relationships typically include family friends neighbors colleagues etc devices manufacturer considered family iphone apple device friendship described objects interact frequently one another tend share common theme example bob smartphone body sensors considered friends since interact frequently serve key components providing services iot devices locate room floor considered neighbors iot devices considered colleagues work together provide specific service example temperature sensors humidity sensors air conditioner colleagues work together offer comfortable living environment smart home worth noting simulation introduced section abstract device general information number device features exists social link two iot devices common feature illustrative example sand smart home shown fig show overlay device social network assume devices connected wifi communication network complete mesh network shown overlay social network refrigerator washing machine connected since manufacturer samsung neighbor relationship exists vacuum washing machine locate storage room friendship exists telephone refrigerator involved frequent interactions serve home owner refrigerator microwave boiler considered colleagues work together prepare meals sand iot device joins network exchange general information nearby devices within transmission range consequently social links established valid social relationships exist assume iot device intelligent sense establish social relationships autonomously like human social aspects iot devices give ability form overlay device social network automatically worth noting overlay device social network static dynamically changing device movements device relationship changes colleague relationship may change frequently hence iot device needs periodically update social connections sand sand iot devices supposed act human beings social aspects devices allow dynamically create new social connections form new colleagueship work together generate new services social aspects fig sand smart home sand devices become visible aware thus leading fast effective device discovery process device discovery rather simply broadcasting discovery message neighbors distribute schemes sand send discovery message neighbors preferred order based rank neighboring device higher rank device likely faster able forward discovery message target device requested source device social aspects sand devices become visible devices thus making sand scalable efficient baseline distributed scheme device ranking criteria sand sand device discovery adopts search strategy discovery request forwarded neighboring devices preferred order based ranks rank device determined four factors device degree diversity local clustering coefficient local betweenness iot device first three factors help sand select device may reach broad community betweenness leads discovery request device stands major intersection multiple shortest paths network thus device ranking criteria makes sand intelligent accurate fast device discovery strategy present calculation details four factors follows device degree overlay social network degree device denoted defined number social links example fig device degree since five social connections associated higher device degree likely associated routing path desired target device since outlets device diversity sand diversity device denoted defined number types social links device associated example fig device diversity since three different types social links family links colleagueship links friendship link iot device could high device degree connections within type social relationship device still assigned relatively low rank since diverse enough reach distinct communities devices contrast iot device high diversity involved many different types social relations may broad reachability thus higher chance connect desired target device clustering coefficient clustering coefficient shows likely device neighbors form clique local clustering coefficient device defined number social links neighborhood including device neighbors divided total number possible links neighborhood given device local clustering coefficient calculated est est est social link device device set devices neighborhood device set social links neighborhood device degree device higher local clustering coefficient likely device neighbors forming clique fully mesh thus diameter device neighborhood smaller lead faster wider dissemination discovery request betweenness betweenness reflects probability given device stands critical intersection multiple shortest paths network defined number shortest paths traverse given device divided total number possible shortest paths sand local betweenness measures probability neighborhood includes given device neighbors local betweenness device calculated set devices neighborhood device pair devices degree device shortest path traverses device otherwise higher local betweenness higher chance given device located intersection connects major shortest paths neighborhood discovery requests arrive intersection disseminated anyplace network easily quickly shortest paths sand rank device defined multiplication mentioned four factors higher rank likely device forward discovery request desired target device fast efficient manner sand forwarding strategy based architecture device ranking criteria described propose sand device discovery scheme subsection rather simple broadcast used distributed scheme sand forwards discovery request neighbor device ranks highest example fig discovery request forwarded highest rank degree diversity local clustering coefficient local betweenness device highest rank supposed effective one leads desired target device discovery process sand performs search limited search depth search exhaust deepest level depth level tunable parameter desired target device found searching levels sand step back examine unchecked neighbors depth pseudocode sand discovery algorithm shown algorithm algorithm sand device discovery algorithm input discovery request rst source target output communication path pst pst pst empty pst desired target device return reverse path pst else checked continue else pst depth continue else rank sort neighbor devices pst highest rank end end similar distributed scheme sand scalable since iot device maintains information local connections sand also resilient robust since centralized controller iot device fair equal peer compared distributed scheme sand efficient former applies simple broadcast thus significant large number devices involved discovery process latter sand performs depthfirst search limited search depth guidance intelligent device ranking criteria number devices involved relatively small advantage sand may lead low energy consumption process transmitting discovery request critically beneficiary iot system devices power constrained hence see sand inherits advantages distributed scheme improves communication efficiency erformance valuation section focus evaluating performance proposed sand scheme compared distributed scheme uses simple broadcast conduct simulations random network network random network irregular mesh network social links randomly generated iot devices network generated considering power law fraction devices connections follows distribution usually meaningful reasonable test sand network since social networks sand works considering overlay device social network simulation generate network consists iot devices iot device associated number connections neighboring devices number connections follows uniform distribution power law random network network respectively forms communication network physical link communication network transmission latency uniformly distributed within distributed scheme runs communication network sand iot device randomly generate three features two iot devices physically connected share features equipped social link forms overlay device social network total number features network set vary simulation results shown generate discovery requests randomly selected source devices randomly chosen desired features obtain average results discovery request time live regards evaluation focus three performance metrics success rate average number devices contacted success average number hops discovery path success present simulation results findings following parts success rate simulation target device desired feature found time live expires considered successful discovery otherwise considered failure discovery request dropped success rate defined number successful discovery divided total number discovery requests show success rate function number features fig fig given fixed number iot devices default system number features increases system become diverse variety types devices fig see random network success rate distributed scheme sand decreases network becomes diverse distributed scheme performs better sand since distributed scheme uses simple broadcast sand uses search may yield long discovery period exceeds required time live distributed broadcast sand success rate success rate broadcast distributed sand features success rate random networks features success rate networks fig simulation results random network network however network shown fig see sand achieves similar success rate distributed scheme high compared results random network success rate experiences significant increase primarily network superhub devices large number connections could help discovery request reach target device short time random network device similar node degree discovery request may take long time reach target bad cases time live requirement violated thus failures occur another reason device ranking criteria sand effective successfully navigate discovery request desired target device given social networks infer sand practically solid solution since success rate average number devices contacted one main interests research measure device discovery process many devices contacted desired target device found energy efficient less number devices involved transmitting discovery message critically important iot system devices power constrained fig fig show number devices contacted different number features random network network respectively count measures successful discovery simulation results observe random networks number features increases sand outperforms distributed scheme average performance improvement respectively also seen number features increases performance improvement sand distributed scheme becomes significant networks indicates sand efficient iot system diverse heterogeneous thanks adoption search intelligent device ranking criteria sand efficiently discover desired target device without involving many unnecessary intermediate devices thus sand potentially achieve low energy consumption process transmitting discovery request critically beneficiary iot devices power constrained communication hops another interest research find many communication hops separate source device desired target device discovery process done source target devices communicate discovery path cooperate perform computing tasks communication number hops discovery path small table show average number hops discovery path random network network respectively seen sand achieve small communication hops distributed scheme latter supposed optimal one using search finding validates effectiveness device ranking criteria sand intelligently navigate discovery desired target device minimum detours also see number features increases number hops increases network scenarios reasonable becomes difficult needs hops query find desired target device network becomes diverse comparing results two tables also observe random network network average number hops exhibits average increase two hops reason behind random network regular mesh network devices relatively fair similar number connections network superhub devices huge number connections network given discovery request usually forwarded superhub devices first finds path desired target device thus resulting larger number hops random network furthermore evaluate distribution devices number hops discovery path network shown fig plot device distributions sand distributed broadcast sand avg devices contacted avg devices contacted broadcast distributed sand features avg devices contacted random networks features avg devices contacted networks fig simulation results random network network table hops random networks broadcast sand table hops networks features broadcast sand devices features hops using features result confirms past research social networks shows cases two iot devices separated hops also observe average number hops small network less diverse see center peak distribution features average number hops increases network becomes diverse features also infer device discovered hops probability discovery unsuccessful increases dramatically reason behind unsuccessful discovery involves devices relatively isolated devices quite common discovery isolated devices violate time live requirement get dropped contrast successful discovery usually involves devices cluster connected superhubs esearch hallenges pportunities efficient iot device discovery may lead generation new set applications services well research opportunities computing service data fusion malware source traceback formation robot society etc meanwhile many open challenges regards applications following parts propose research opportunities challenges related topic intelligent device discovery iot beyond fig device distribution hops sand nodes computing intelligent device discovery enable new format computing services number smart devices client initiate computing task device smart watch set specified requirements computing power demand cpu memory function platform demand data demand source code raw dataset source iot device responsible deliverying requirements appropriate computing facilities internet gathering results back case intelligent device discovery strategy used establish connections iot devices computing facilities internet example user wants issue request check glucose level smart watch smart watch use intelligent device discovery search nearest server enough computing memory power perform computing tasks server may use intelligent device discovery find source code glucose level analyzer download install analyzer perform analysis intelligent device discovery grants iot devices computing facilities capability fetch data functions need performing given computing tasks thus iot devices computing facilities perform computing tasks manner process one open challenges standardize experssion computing requirements devices machines initiate standard requests understood machines another challenge optimize request reply flow routes network resource consumption minimized latency minimized computing network data fusion computing scenario variety computing facilities network data programming source code patient raw data interconnected fused running intelligent device discovery strategy given service request intelligent device discovery help gathering computing switching storage hardware specific site fetching pool patients data different site obtaining source codes particular computing functions network functions another site computing switching storage hardware served placeholders source codes computing network functions raw dataset discovered fetched intelligent device discovery strategy served input hardware placeholders data software hardware distributed different sites coupled intelligent device discovery approach hence intelligent device discovery strategy promising enable computing network data fusion achieve efficient communication fused devices computing network data fusion becomes true devices machines easily launch new services specifying needs internet internet resolve needs standard format service function chains efficiently connect required devices funcitons intelligent device discovery strategy provide required new services specified devices machines computing network data fusion promising generate novel services supervised human beings fusion considered intermediate phase prepares enables formation robot society achieve open challenge develop platform address iot interoperability issue facilitate computing network service data fusion platform another challenge efficiently discover deliver required data source codes selected hardware sites timely manner seamless service provided users traceback cyber attack source another application intelligent device discovery area network security sources malicious attacks usually spoofed avoid detected intelligent device discovery used traceback actual source malware attacks based characteristics features malicious attacks security protection system issue traceback request malware source generator certain key features malicious traffic overlay social network top communication network constructed based model section enhanced model accurately predict hidden machines constructed using method machine learning deep learning techniques also applied improve accruacy prediction proposed intelligent device discovery strategy run enhanced overlay social network hidden links based prediction traceback malicious source attacker even malicious source spoofed hidden behind links relationships intelligent device discovery still guide traceback route based pattern features malicious attacks robot society intelligent device discovery used initiate connection given device desired device first time first contact two devices considered know maintain certain type social relationship human communications different devices continue grow variety computing service network service data fused robot society could formed consists iot devices servers robots machines internet device relationship robot society human relationship human society friend colleague family relationships considered paper also high chance unique machine type relationships may exist robot society one challenging meaningful work analyze large dataset network traffic data categorize relationships internet useful give insight relationships communication purpose machines also helpful formulate foundamental rules establish social relationship another open challenge ensure security connection two establish social relationship necessary avoid establishing relationships potential malicious devices achieve may need develop machine learning based solutions decision tree classification convolutional neural networks recurrent neural networks train identify roles contact roles robot society reduce chance establishing connections malicious devices general promising research area investigate social behavior collaborative intelligence robot society addition artificial intelligence single onclusion uture ork internet things iot grows continuously connects billions smart devices heterogeneous functionalities purposes platforms iot evolves promising social relationships could autonomously established smart devices help intelligent device discovery strategy iot devices easily discover resources source codes devices computing facilities meet needs associated one resources capabilities either computing power network function source code data intelligent device discovery used glue couple resources distributed sites achieve computing network service data fusion acts like robot establish social connections autonomously collaborate create new applications services could eventually lead formation robot society goal research address impending scalability issue process device discovery iot paper proposed novel approach leverages social aspects iot devices achieve scalable efficient iot device discovery proposed distributed sand scheme applies search forwards discovery request neighboring devices preferred order defined novel social device ranking criteria specifically ranking criteria takes consideration device degree diversity clustering coefficient could potentially navigate discovery request reach broad community furthermore local betweenness considered ranking criteria prioritizes devices stands critical intersections multiple shortest paths network guidance intelligent device ranking criteria discovery request find desired target device fast efficient manner conducted comprehensive simulations validate effectiveness sand random network network simulation results shown sand achieve success rate communication hops distributed scheme uses broadcast addition found sand scheme contacts much smaller number intermediate devices distributed scheme discovery process thus potentially leading much less energy consumption future work plan extend study following aspects first plan extend sand multicast feature allows one discovery request fetch multiple replies could useful wireless sensor network applications temperature monitor system addition rather specifying one feature desired target device enhance sand scheme allowing claim multiple features achieve accurate discovery also plan implement sand physical testbed conduct experimental studies secondly plan analyze various network traffic datasets understand meanings purposes behind communications use machine learning techniques categorize social relationships iot used improve discovery speed discovery accuracy sand also provide basic insights exploring formation robot society thirdly plan investigate interoperability issues iot explore standard platform enables variety computing network service data fusion eferences zaslavsky jayaraman discovery internet things acm ubiquity magazine october article sun chang yang liao deployment replica servers virtual content distribution networks data fusion information sciences http sun liao zhao sun chang towards provisioning hybrid virtual networks federated cloud data centers future generation computer systems https darema chang distributed behavior model orchestration cognitive internet things solution journal enterprise information systems barnaghi sheth searching internet things requirements challenges ieee intelligent systems vol ccori biase zuffo silva device discovery strategies iot proc ieee international symposium consumer electronics lunardi matos tiburski search engine industrial iot discovery search selection usage devices proc ieee conference emerging technologies factory automation dong ravindran wang icn based distributed iot resource discovery routing proc ieee international conference telecommunications travers milgram experimental study small world problem sociometry vol edunov diuk filiz burke three half degrees separation online available https watts steven strogatz collective dynamics networks nature vol june newman networks introduction oxford university press isbn barabsi albert emergence scaling random networks science vol zhen ling junzhou luo kui wei xinwen torward discovery blocking traceback malicious traffic tor ieee transactions information forensics security vol berlusconi calderoni parolini verani piccardi link prediction criminal networks tool criminal intelligence analysis plos one https chauhan kaur chang advancement applicability classifiers variant exponential model optimize accuracy deep learning journal ambient intelligence humanized computing
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sofic entropy gaussian actions dec ben hayes abstract associated orthogonal representation countable discrete group probability action called gaussian action using polish model formalism developed compute entropy sense bowen gaussian actions group sofic computations entropy gaussian actions done acting group abelian thus results new even amenable case fundamental approach methods noncommutative harmonic analysis replace fourier analysis used abelian case contents introduction preliminaries sofic entropy generating sets tightness entropy gaussian actions definition gaussian actions preliminaries group von neumann algebra embedding sequences preliminaries real subspaces left regular representation sofic entropy gaussian actions references introduction paper concerned giving new computations sofic entropy specifically computing entropy gaussian actions entropy actions classical goes back work kolmogorov entropy roughly measures chaotic action kieffer showed one generalize entropy actions amenable groups amenable group group sequence nonempty finite sets almost invariant translation elements group abelian groups nilpotent groups solvable groups amenable free group letters entropy amenable groups studied many people useful invariant ergodic theory computed many cases positivity entropy implies interesting structural properties fundamental examples led many believe good entropy theory beyond realm amenable groups groundbreaking seminal work bowen defined notion entropy actions sofic groups class sofic groups considerably larger amenable groups contains amenable groups residually finite groups linear groups closed free products amalgamation amenable subgroups see since subject fairly young much known sofic entropy entropy actions amenable groups structure beginning emerge calculated interesting examples bernoulli shifts see well algebraic actions see additionally recent work shows one deduce structural properties action assumptions positive sofic entropy goal paper add list computations sofic entropy computing entropy gaussian actions gaussian action construction way associate functorial way orthogonal representation probability action called gaussian action date april mathematics subject classification key words phrases sofic entropy gaussian actions noncommutative harmonic analysis ben hayes refer reader section precise definition intuition mention finitedimensional action induced action gaussian measure gaussian action natural generalization construction case representations recall left regular representation given known gaussian action associated representation bernoulli action base gaussian measure thus gaussian actions class actions similar class bernoulli shifts best knowledge entropy gaussian actions computed acting group abelian see state result need introduce following decomposition representations define singularity orthogonal representations exactly unitary case given orthogonal representations say singular respect write nonzero subrepresentation embeds say absolutely continuous respect write embeds general pairs orthogonal representations always write singular abelian reduces lebesgue decomposition measure theory thus regard noncommutative analogue lebesgue decomposition theorem let countable discrete sofic group sofic approximation let orthogonal representation separable let induced gaussian action write singular since appears new specifically mention amenable case corollary let countable discrete amenable group let orthogonal representation induced gaussian action write embeddable gave formula entropy probability action polish space borel probability measure homeomorphisms given arbitrary probability action probability action polish borel measure homeomorphisms called polish model probability actions definition entropy terms polish model took account topology nontrivial way like definition entropy presence compact model developed computation entropy gaussian actions goes polish models model associated family generators representation measure canonically defined terms representation although one write compact model unnatural measure expressed nicely terms representation mention entropy sofic case roughly measure many finitary simulations probability action typical way prove existence simulations probabilistic argument gaussian action probabilistic argument uses gaussian measures spaces consequence methods associated sofic approximation group natural way describing gaussian measure subrepresentation left regular representation limit gaussian measures let sketch able handle case singular respect nonabelian case problem showing zero entropy reduces fact two singular measures close close zero representation theory group captured universal natural replace nonabelian case statement singularity sofic entropy gaussian actions representations becomes statement singularity measures prove analogous characterization singularity representations terms elements short abstract harmonic analysis present abelian case noncommutative harmonic analysis using framework mention amenable infinite conjugacy class group every nontrivial conjugacy class infinite easy proof embeds namely one show case contains consequence theory factors case factor onto isomorphic gaussian measure since entropy amenable groups decreases factor maps done proof fails disastrously case far true sofic entropy decreases factor maps groups fact shown every nonamenable group measure space factors onto every nontrivial bernoulli shift see corollary even bernoulli shift infinite entropy moreover contains free group taken positive number fact even true recent result seward see theorem implies every nonamenable sofic group probability action factor action sofic entropy less thus simple proof nonamenable case based factors must use direct proof even amenable case argument relies group infinite conjugacy class group one needs general methods handle general case preliminaries sofic entropy use symmetric group letters set use sym set bijections definition let countable discrete group sofic approximation sequence sdi functions assumed homomorphisms lim udi lim udi call sofic sofic approximation amenable groups residually finite groups sofic also known soficity closed free products amalgamation amenable subgroups see additionally residually sofic groups locally sofic groups sofic thus malcev theorem know linear groups sofic known graph products sofic groups sofic subgroup sofic amenable sense mean sofic one argue methods theorem example consider observations definition recall definition entropy presence polish model given let polish space let countable discrete group homeomorphisms say continuous pseudometric dynamically generating every every neighborhood exists finite max notice definition includes hypothesis continuous use banach space bounded continuous functions norm sup given pseudometric space say write say use minimal cardinality subset say use maximal cardinality subset leave exercise show ben hayes use metric defined polish space use prob space borel probability measures countable discrete group homeomorphisms use space elements prob ready state definition sofic entropy defined counting exponential growth maps approximately preserve structure approximately equivariant call maps microstates heuristic term defined rigorously definition let countable discrete sofic group sofic approximation sdi let polish space homeomorphisms fix bounded continuous pseudometric finite let map max notice map accounts group action structure recall denotes space bounded continuous functions prob finite let prob sets form basis open sets topology called weak topology use topology account structure microstates definition let countable discrete sofic group sofic approximation sdi let polish space homeomorphisms fix finite set set map udi definition let countable discrete sofic group sofic approximation sdi let polish space homeomorphisms let fix bounded dynamically generating pseudometric finite set lim sup log inf finite finite sup call entropy respect shown agrees entropy defined generating sets tightness since separable would like reduce checking approximate property microstates functions smaller class functions example separable topology uniform convergence compact sets require family functions dense topology give sufficiently small family functions deal however work need modify microstates uniform tightness proceed definitions definition let polish space family said generating every every compact every span akgk sofic entropy gaussian actions proceed modified version sofic entropy case generating set functions definition let countable discrete group function let polish space homeomorphisms preserving borel probability measure let dynamically generating pseudometric open subset finite let mapu set suppose sofic dynamically generating pseudometric sofic approximation sdi set log mapu lim sup inf open sup compact inf finite finite sup definition necessary trick add quantifiers definition sofic entropy reader may concerned already number quantifiers involved original definition sofic entropy compute sofic entropy gaussian actions section clear correct tradeoff difficulty involved computation dealing quantifiers instead show approximate property class functions norm dense however one easily see class functions generating use definition theorem let countable discrete sofic group sofic approximation let polish space homeomorphisms preserving borel probability measure dynamically generating pseudometric generating particular proof particular part follows theorem first show let since polish may find compact fix open subset containing urysohn lemma may find note prob see finite mapu map thus finite ben hayes taking infimum find taking infimum find take supremum infimum find show let definition generating let given finite sets let sufficiently small depending upon manner determined later since polish may find compact let compact given let find span kgj kgj akfj may find open neighborhood kgj let finite sets depend upon manner determined shortly let open neighborhood contained mapu udi udi udi udi kfj kfj udi udi udi kfj kfj udi choose sufficiently small sufficiently large may force udi choose max kfj max kfj thus mapu map since arbitrary may take supremum see sofic entropy gaussian actions taking infimum proves entropy gaussian actions gaussian actions natural class actions induced orthogonal representations group representation left regular representation gaussian action simply bernoulli action gaussian measure prove results give structural results koopman representation probability action sofic group assumption action positive entropy example could show positive entropy must contain piece left regular representation precise statements see theorem corollary exploit connections representation theory apply spectral consequences positive entropy gaussian actions also exploit similarity bernoulli shifts compute entropy gaussian actions let countable discrete group recall orthogonal representation real hilbert space homomorphism group orthogonal transformations set let complexification equipped unique sesquilinear inner product extending one let complexification unique unitary transformation definition gaussian actions natural way define gaussian actions von neumann algebras definition let complex hilbert space von neumann algebra containing identity closed weak operator topology say vector cyclic suppose standard probability space let given map allows view von neumann algebra turns see theorem commutative von neumann algebra cyclic vector separable standard probability space unitary additionally linear functional weak operator topology continuous complex measure leave reader verify ben hayes definition let real hilbert space gaussian algebra associated denoted commutative von neumann algebra cyclic vector generated unitaries satisfying exp let suppose countable discrete group representation action determined uniquely action called gaussian action gaussian algebra exists unique isomorphism see also existence uniqueness gaussian action let sketch alternate construction gaussian algebra first consider case dimensional say case simply take exp general consider universal generated unitaries satisfying relation one make sense unitary saying element one check linear function defined exp checking reduces showing take exp exp replacing span reduces verification case already shown explicitly exhibiting integration respect gaussian measure explained similar reasoning one checks one runs gns construction let completion inner product induced homomorphism given sot let straightforward verify desired properties remarks definition definition standard probability space action furthermore action uniquely determined isomorphism note two orthogonal representations sofic entropy gaussian actions definition via von neumann algebras may abstract let mention simple version definition case cyclic representation recall use proposition let countable discrete group orthogonal representation suppose vector span gaussian action isomorphic shift action measure determined exp exp proof choose realization proposition sequence converges strong operator topology thus strongly continuous group fix stone theorem closed operator exp sense functional calculus recall viewed von neumann algebra multiplication operators hard see elements whose multiplication operators unitaries almost everywhere equal measurable functions thus identify measurable function similar reasons identify measurable function thus every true almost every exp define hard see every exp almost every define since unitaries form generate see gives isomorphism ben hayes additionally exp exp exp preliminaries group von neumann algebra embedding sequences purposes need linearize sofic approximation approximations algebras associated let ring finite formal linear combinations elements addition defined naturally multiplication defined also define involution given sofic approximation sdi define mdi order talk asymptotic properties extended sofic approximation need analytic object associated let left regular representation defined continue use linear extension group von neumann algebra defined wot denotes weak operator topology use denote group von neumann algebra define leave exercise reader verify following properties equality weak operator topology continuous call third property tracial property typically view subset particular use well functional restriction order state linearization sofic approximation properly give general definition definition complex algebra equipped involution conjugate linear antimultiplicative definition tracial pair equipped linear functional sofic entropy gaussian actions equality let let define hilbert space completion inner product condition definition representation defined densely let make tracial using usual trace particular use denote operator norm let free call elements indeterminates elements unique sending use image definition let tracial embedding sequence sequence mdi sup frequently use following fact see notice since suffices handle case since proved proof next two propositions left reader proposition let countable discrete sofic group sofic approximation sdi extend maps mdi linearly embedding sequence proposition let tracial mdi embedding sequence mdi another sequence functions sup embedding sequence fact need extend sofic approximation group von neumann algebra use following lemma lemma let countable discrete group embedding sequence extends one use preceding lemma sofic combination proposition ben hayes preliminaries real subspaces left regular representation define fourier algebra functions linear functional weak operator topology continuous call continuous extension note continuity kaplansky density theorem must unique let consist continuous extension positive linear functional let theorem norm makes banach space theorem consists functions form moreover inf infimum intuition leave reader verify abelian consists consists state basic properties following proposition lastly given set proposition let countable discrete group unitary representation particular define additionally dense ybi proof assumption implies may extend weak operator topology continuous thus continuous extension given particular part follows discussion preceding proposition write conclusion follows easily equality continuous extension given conversely let continuous extension xyy since see norm equality density statement contained lemma sofic entropy gaussian actions inequality see proposition proof generalizes situation order compute entropy gaussian actions need discuss real subspaces need real version let recall convolution defined similarly define use note thus sup thus define observe replacing see sup thus unique bounded operator extending norm equal write image operator use theorem proposition sup bar denotes complex conjugation follows weak operator topology continuous operator lemma let countable discrete group orthogonal representation real separable hilbert space suppose cyclic vector orthogonal projection proof let cyclic vector define preceding proposition may find letting using find let approximating square root function polynomials see note ben hayes inequality preceding proposition cauchy sequence hence converges also lim lim lim thus span let projection onto span let denote complexification operator commutes theorem shows unique orthogonal projection moreover thus need extend sofic approximation embedding sequence lemma however also want mdi proposition let countable discrete group sdi sofic approximation exists embedding sequence mdi mdi mdi orthogonal projection orthogonal projections mdi proof lemma may extend embedding sequence mdi proposition suffices show mdi sup kxi kxi define aij aij suffices show indeed assuming convergence may redefine let let let since sofic entropy gaussian actions letting using sofic approximation find lim sup letting proves since embedding sequence functional calculus thus setting completes proof lastly need analogous definition singularity unitary case two orthogonal representations use space real linear bounded maps say mutually singular written whenever nonzero subrepresentation isomorphic similarly say embeddable lemma let countable discrete group two orthogonal representations following equivalent iii proof proof equivalent iii implies proposition copy proof implies proposition provided prove analogue polar decomposition let bounded operator let complexification using real banach space transpose thus approximating square root function polynomials see let polar decomposition sot lim find rest proposition show case left regular representation concepts singularity absolute continuity real case related complex case ben hayes lemma let countable discrete group let orthogonal representation unitary sense similarly proof suppose applying zorn lemma write direct sum cyclic representations applying lemma see converse even easier suppose closed linear subspace embeds complexification embeds conversely suppose bounded map real representation see complex vector space see previous lemma since spans following proved way proposition proposition let countable discrete group two orthogonal representations sofic entropy gaussian actions section compute entropy gaussian actions let first start simple corollary theorem corollary let countable discrete sofic group sofic approximation let orthogonal representation separable hilbert space suppose corresponding gaussian action proof proposition find unique exp almost every uniqueness almost everywhere fact span generated von neumann algebra exp hard argue generated borel measurable sets measure zero thus theorem lemma know turn computation sofic entropy gaussian actions case need following general lemmas sofic entropy gaussian actions lemma let countable discrete sofic group sofic approximation sdi let polish space bounded compatible necessarily complete metric fix let bernoulli action give dynamically generating pseudometric finite define finite finite finite set containing large map proof let diameter udi soficity setting completes proof lemma let countable discrete sofic group sofic approximation sdi lemma extend approximation sequence mdi fix finite let mdi sup kxi following statements hold sequence udi lim sup sequence udi udi lim sup ben hayes proof first handle case may choose udi soficity may also force supp still udi case easy see handle general case let choose since approximation sequence lim thus large may find udi large thus therefore large first part find udi thus large since udi large use diagonal argument complete proof let note sofic entropy gaussian actions look enough find lim sup udi udi prove ktk lim clearly prove existence first let set use expression tends lim udi hek udi udi proves inequality general case follows approximation notation finite measurable let defined say finite schwartz function standard fourier analysis exp prove choose rdi respect gaussian measure rdi high prob ability microstate approximately preserve measure integrated schwartz functions need following notation define ben hayes lemma let countable discrete sofic group sofic approximation sdi let orthogonal projection fix sequence orthogonal projections mdi kpi define gaussian measure lebesgue measure let defined lemma let defined proposition let finite subsets compact hausdorff space sequence udi udi proof define rdi udi show rdi udi rdi rdi udi rdi lemma follow chebyshev inequality write exp note proposition fact exp exp rdi exp rdi exp rdi interchanges integrals valid bounded sofic entropy gaussian actions thus rdi exp rdi exp rdi exp using rdi exp exp obvious generalization proposition preceding lemma kpi lim sup sup sup exp exp use expression goes zero additionally sup exp since equations fact bounded imply exp udi rdi proved turn proof computations kpi interchanges integrals valid preceding lemma udi udi ben hayes lim kpi sup thus sup kpi additionally sup kpi equations fact bounded imply exp rdi since proved udi lemma let infinite set space generates sense definition proof let compact say function depends upon finitely many coordinates finite theorem function depending upon finitely many coordinates kcb let let compact continuous image projection map function whenever well known sofic entropy gaussian actions let since schwartz function finally completes proof theorem let countable discrete sofic group sofic approximation sdi let orthogonal representation real separable hilbert space let corresponding gaussian action let standard probability space action proof shall first reduce case cyclic vector span suppose prove theorem cyclic case let cyclic since assuming theorem cyclic case suffices show cyclic since separable may write cyclic let cyclic case induction using lim hard show use preceding lemma theorem proposition may regard measure defined exp exp orthogonal projection extend approximation sequence still denoted mdi proposition exists sequence orthogonal projections mdi kpi let gaussian measure defined choosing compact model may assume compact metrizable space homeomorphisms let compatible metric let dynamically pseudometric defined min ben hayes shall use generating set use dynamically generating pseudometric defined let finite given let enumeration elements inductively find positive real numbers define set note compact let open neighborhood let finite sets since infinitely many lemma finite set containing identity rdi also assume may done enlarging compact choose may assume note case udi udi thus log suppose every exists lim sup sofic entropy gaussian actions using lebesgue measure ball ball ball corollary stirling formula ball set log log rdi exp last line following stirling formula thus lim sup log log log log log taking infimum supremum infimum find log log let see prove general formula entropy gaussian actions corollary let countable discrete sofic group sofic approximation sdi let orthogonal representation proposition write let corresponding gaussian actions ben hayes proof statement direct corollary corollary fact sofic entropy always nonnegative first case statement follows general fact two actions standard probability spaces implies second case statement also special case statement last case statement follows theorem give examples show let countable discrete sofic group sofic approximation sdi say ergodic whenever lim udi lim udi udi following folklore result include proof completeness proposition let countable discrete sofic group ergodic sofic approximation nonergodic probability action standard probability space proof let sdi let set let finite observable given let diagonalization argument easy see finite containing lim sup max udi lim sup udi udi sufficiently small since set max udi always sufficiently small udi udi udi thus see thus using kerr definition entropy via partitions combining theorem following corollary let countable discrete sofic group ergodic sofic approximation let orthogonal representation weakly mixing could compact associated gaussian action sofic entropy gaussian actions mention example ergodic sofic approximation free group two generators choose sofic approximation randomly namely let chosen uniformly random let unique homomorphism known high probability sofic approximation see well lemma also known theory expanders high probability ergodic sofic approximation see remarks section take orthogonal representation weakly mixing take references bowen measure conjugacy invariants actions countable sofic groups amer math soc bowen entropy expansive algebraic actions residually finite groups ergodic theory dynam systems bowen weak isomorphisms bernoulli shifts israel math bowen every countably infinite group almost ornstein dynamical systems group actions volume contemp math pages amer math soc providence harmonic models spanning forests residually finite groups funct brown ozawa approximations cambridge university press ciobanu holt rees sofic groups graph products graphs groups pacific math november conway course operator theory graduate studies mathematics american mathematical society providence dooley golodets spectrum completely positive entropy actions countable amenable groups funct dykema kerr pichot orbit equivalence sofic approximation dykema kerr pichot sofic dimension discrete measurable groupoids trans amer math soc elek szabo sofic groups group theory folland real analysis modern techinques applications john wiley sons hoboken second edition friedman proof alon second eigenvalue conjecture related problems memoirs ams hayes determinants sofic entropy hayes polish models sofic entropy inst math jussieu appear hayes von neumann dimension banach space representations sofic groups funct kerr bernoulli actions sofic groups completely positive entropy israel math appear kerr topological entropy variational principle actions sofic groups invent math kieffer generalized theorem action amenable group probability space ann prob entropy gaussian actions countable abelian groups fund meyerovitch positive sofic entropy implies finite stabilizer nica asymptotically free families random unitaries symmetric groups pacific math ornstein weiss entropy isomorphism theorems actions amenable groups anal paunescu sofic actions equivalence relations funct november paunescu convex structures revisited ergodic theory dynam systems appear peterson sinclair cocycle superrigidity gaussian actions ergodic theory dynam systems popa independence properties sublagebras ultraproduct factors funct seward every action group factor small action mod takesaki theory operator algebras york takesaki theory operator algebras volume encyclopaedia mathematical sciences springer new york stevenson center nashville address
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aug extended plus closure complete local rings raymond heitmann linquan abstract full extended plus closure developed replacement tight closure mixed characteristic rings shown adapting perfectoid algebra techniques complete local rings closure property fact generally possibly ramified complete regular local ring mixed characteristic ideals local domain finite epf consequence ideals regular local rings closed fact implies validity direct summand conjecture theorem mixed characteristic introduction first author introduced several closure operations mixed characteristic rings notably full extended plus closure closures proposed possibilities play role mixed characteristic tight closure characteristic hoped might allow prove homological conjectures remained unresolved mixed characteristic indeed full extended plus closure figure proof direct summand conjecture dimension course direct summand conjecture resolved without use closure operation however still remains interest note extent full extended plus closure fills void left open absence tight closure mixed characteristic paper note number properties shown add several others particular deal two fundamental questions left open shall see full extended plus closure property complete local rings ideals regular local rings closed latter fact sufficient imply direct summand conjecture truly alternate proof work makes use notation techniques part article full extended plus closure complete local rings since know whether completing closing contracting occasionally gives larger closure theorem gives new information rings complete primary result article sense full extended plus closure captures obstructions flatness principal theorems theorem corollary let complete local domain mixed characteristic let parameters epf date august raymond heitmann linquan theorem theorem let complete regular local ring mixed characteristic let integral domain finite extension ideal epf equivalent formulation theorem divorced language extended plus closure theorem theorem let complete regular local ring mixed characteristic let integral domain finite let module exist natural map torr tori zero every consequently every torr exist every paper organized follows section give brief introduction extended plus closure results discussed generally known although proposition actually new also give short introduction terminology conventions section devoted new results basics notation ideal ring involved always clear ideal need cumbersome notation first author introduced four closure operations mixed characteristic rings subsequently one full extended plus closure received attention also case article fact shall preserve notation epf full shall mostly refer closure extended plus closure noted original definition early results allow either mixed characteristic characteristic however characteristic case less interesting full extended plus closure trivially contains tight closure definition let integral domain ideal element said full extended plus closure designated epf provided exists every every positive integer remark extended plus closure actually defined noetherian rings tight closure one may compute extended plus closure computing modulo minimal prime ideal taking intersection liftings back original ring general setting hold great interest article focus case integral domains first shall highlight basic results first see behaves reasonable closure operation proposition proposition let ideals noetherian ring epf ideal epf epf epf epf epf epf extended plus closure complete local rings epf epf epf epf epf epf particular closed closed remark known local henselization epf epf theorem however result unfortunately known even excellent domain replace completion reason main theorems extended plus closure obviously generalize rings finally two interesting theorems earlier work two theorems match consequential theorems tight closure theory first full extended plus closure gives type theorem theorem theorem let ring let ideal generated elements suppose integral closure integer epf note type theorem simple consequence theorem include result next serves lemma need proof lemma theorem proposition let integral domain ideal suppose exists integral extension recall classical theorem says regular ring ideal generated elements integral closure contained every integer recover result proving regular local ring every ideal closed extended plus closure see theorem give two proofs theorem obtained earlier work vanishing conjecture maps tor also follows main result article following result theorem theorem let regular local ring maximal ideal full extended plus closure property finite extensions complete regular local rings epf every ideal remark actual statement result presumes coloncapturing property holds finite extensions regular local rings however statement lemma makes clear one may restrict complete regular local rings faithful flatness fact epf holds complete regular local rings implies holds regular local rings one may observe results well known tight closure fact alternate definition tight closure closely resembles extended plus closure instead requiring tight closure requires irf integral closure largest purely inseparable extension quotient field superficially means tight closure could smaller however hochster huneke introduced dagger closure must contain extended plus closure showed theorem characteristic complete local rings tight closure dagger closure equal thus proposition complete local ring equal characteristic extended plus closure coincides tight closure ideals raymond heitmann linquan perfectoid algebras freely use language perfectoid spaces almost mathematics paper always work following situation perfect field characteristic let ring witt vectors coefficients let completion perfectoid field sense ring integers perfectoid banach set powerbounded elements bounded frobenius surjective called integral perfectoid complete free satisfies frobenius induces isomorphism two categories equivalent theorem via functors almost mathematics article measured respect flat ideal two cases use nonzero element stated explicitly throughout main results noetherian integral domain mixed characteristic let denote absolute integral closure let denote completion nonzero element use denote integral closure lemma let integral domain mixed characteristic ideal exists every every positive integer epf proof need show howpps ever psyi limit cauchy sequence uij follows uij cauchy sequence converges zero thus sufficiently large uij uij lemma let universally catenarian noetherian integral domain mixed characteristic suppose height two ideal proof pick write write repeat process write keep going note thus element since universally catenary height two ideal integral hence almost isomorphic respect thus practice often ignore distinction since one always pass without affecting issue extended plus closure complete local rings extension since direct limit normal integral extensions nonzerodivisor therefore finishes proof lemma let universally catenarian noetherian integral domain mixed characteristic dimension least two ideal exists nonzero every every positive integer epf proof find element height two ideal likewise every integral extension since universally catenary hypothesis lemma clearly holds place note rpg larger may assume lemma follow immediately lemma show injection almost isomorphism respect suppose integer completion may write integral forces integral follows integrally closed without loss generality may assume reduce case fix let complete proof suffices show positive integers lemma fix may write pnn dpt integral integral pnn means exists integer pnn pnn pnn since ideals contain power easy see pnn pnn pnn thus integral pnn follows absolutely integrally closed proposition gives therefore pnn desired suppose complete unramified regular local ring mixed characteristic local extension complete local domain cohen structure theorem coefficient ring complete unramified dvr residue field let denote algebraic closure let let completion complete formally smooth mod next point actually noetherian though need lemma notation noetherian ring proof first general complete noetherian noetherian suppose infinitely generated prime ideal since noetherian contain finitely generated say ptp suppose write ptj thus gives repeat process find raymond heitmann linquan aij since complete aij therefore finitely generated see noetherian enough show noetherian previous paragraph prove induction case clear prove every prime ideal finitely generated since integral since domain minimal prime cohen structure theorem hence finite extension noetherian induction hypothesis therefore finitely generated thus finitely generated returning development next set since generically exists nonzero finite thus finite let completion let completion integral perfectoid furthermore let construction integral perfectoid example see theorem integral perfectoid functions zariski closed subset spa defined ideal explicitly described completion integral closure inside proved almost faithfully flat mod every example see theorem importantly proved following remarkable result theorem theorem let perfectoid algebra perfectoid field residue characteristic suppose nonzerodivisor contains compatible system roots let finite algebra exists larger perfectoid algebra inclusion continuous contained integral closure isomorphism every finite remark easy see isomorphic integral closure clearly integral closure contained suppose equation belongs multiplication equation tells integral thus contained integral closure notation frequently use following notation throughout rest article complete unramified regular local ring modulefinite domain extension fix nonzero element extended plus closure complete local rings finite construct paragraph lemma construct paragraph theorem perfectoid algebra theorem next lemma experts record completeness lemma using terminology notation flat proof theorem know flat hence flat construction since flat flat also know complete since integral perfectoid argument lemma simply replace shows flat lemma using terminology notation exists map rpg proof completion integral get embedding hence map recall pletion next fix embedding turn induces embedding completions rings embedding next extend phism sends particular since get induced homomorphism finally since integral perfectoid least isomorphic integral closure hence description get map induces map extends almost map integral closure isomorphic integral closure remark get map proof complete ready prove main theorem unramified case theorem using terminology notation ideal consequently epf proof exist chain maps later one isomorphic flat lemma thus next lemma map rpg implies general map complete rings noetherian flat flat see remark raymond heitmann linquan every every epf lemma thus prove usual form corollary using terminology notation suppose parameters consequence epf proof first prove assume elements actually belong claim true flat lemma prove induction case done suppose counterexample minimal first suppose clearly contained minimal prime exists eters minimality assumption counterexample thus know without loss generality may assume equation argument used nowp shows consider since decreased decreased thus hence finishes proof first conclusion finally every every lemma thus epf lemma next objective extend theorem ramified case theorem let possibly ramified complete regular local ring mixed characteristic let integral domain finite extension ideal epf proof finite extension unramified complete local ring choose finite height two ideal use standard framework notation first note proof theorem lemma lemma give desired result provided prove thus done show flat follows formally fact regular every system parameters fact regular sequence corollary give detailed argument claim flat proof claim enough show torr zero finitely generated every use descending induction clearly true dim since regular suppose torr zero finitely generated want show torr extended plus closure complete local rings zero finitely generated considering prime cyclic filtration enough prove let pick regular sequence associated prime long exact sequence tor gives torr tork tork torr zero bek cause regular sequence corollary torr zero induction follows torr zero remark analog theorem theorem equal characteristic true replace extended plus closure tight closure fact even plus closure suppose extension excellent local domains characteristic regular let ideals see let since know since regular balanced big algebra characteristic faithfully flat thus hence theorem let possibly ramified complete regular local ring mixed characteristic let integral domain finite let module exists natural map torr tori zero every consequently every torr exists every proof finite extension unramified complete regular local ring choose finite height two ideal use standard framework notation claim proof theorem flat thus tori follows torr torr torr zero since factors lemma point note isomorphic lemma thus map tori torr also zero finally tensoring commutative diagram induces commutative diagram torr torr torr raymond heitmann linquan injectivity map first line diagram order show image torr zero tori suffices prove image zero torr clear factors tori already know image zero torr remark assert torr almost zero almost flat requires next theorem simple consequence results following directly corollary theorem however proof relies heavily deep perfectoid abhyankar lemma would like offer direct proof based earlier work vanishing conjecture maps tor theorem let regular local ring mixed characteristic let ideal epf proof first claim every exists natural map sending pure prove contradiction suppose exists every map sending pure let regular system parameters since pure map sends socle element exists depending next claim every see suppose exists domain say represents class torr since valuation criterion integral closure know exists mixed characteristic dvr means represents nonzero class torr contradicts vanishing conjecture maps tor mixed characteristic theorem applied shows clearly impossible computing valuations finally suppose epf every since map sending pure know every implies hence epf also completed proof theorem extended plus closure complete local rings theorem let regular ring let ideal generated elements integral closure contained proof suffices prove result locally assume either equicharacteristic regular local ring mixed characteristic regular local ring hochster huneke proved result equal characteristic case see theorem equal characteristic case equal characteristic case section generalizations equal characteristic case using multiplier ideals handle mixed characteristic case theorem know integral closure contained epf theorem completes proof references lem abhyankar perfectoide conjecture facteur direct bhatt direct summand conjecture derived variant bhatt morrow scholze integral hodge theory gabber romero almost ring theory lecture notes mathematics vol berlin heitmann plus closure mixed characteristic algebra heitmann extensions plus closure algebra heitmann direct summand conjecture dimension three ann math heitmann big algebras vanishing conjecture maps tor mixed characteristic hochster huneke tight closure invariant theory theorem amer math soc hochster huneke tight closure equal characteristic zero preprint hochster huneke tight closure elements small order integral extensions pure appl algebra lazarsfeld positivity algebraic geometry ergebnisse der mathematik und ihrer grenzgebiete folge series modern surveys mathematics results mathematics related areas series series modern surveys mathematics vol berlin lipman sathaye jacobian ideals theorem michigan math schwede perfectoid ideals regular rings bounds symbolic powers scholze perfectoid spaces publ math inst hautes sci
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nov lattice envelopes uri bader alex furman roman sauer abstract introduce class countable groups abstract grouptheoretic conditions includes linear groups finite amenable radical finitely generated residually finite groups betti numbers virtually product two infinite groups includes acylindrically hyperbolic groups group class determine general structure possible lattice embeddings compactly generated locally compact groups contain lattice leads precise description possible lattice embeddings groups class applications include determination possible lattice embeddings fundamental groups closed manifolds pinched negative curvature introduction motivation background let locally compact second countable group hereafter denoted group carries radon measure unique scalar multiple known haar measure subgroup lattice discrete carries finite measure equivalently admits borel fundamental domain finite haar measure compact one says uniform lattice otherwise nonuniform lattice inclusion called lattice embedding shall also say lattice envelope classical examples lattices come geometry arithmetic starting locally symmetric manifold finite volume obtain lattice embedding isom fundamental group isometry group universal covering via action deck transformations real lie group isom euclidean direct factors lattice isom uniform compact fact real lie groups algebraic expressed zero sets polynomials allows construct arithmetic lattices sld sld mathematics subject classification primary secondary key words phrases locally compact groups lattices geometric group theory part supported bsf grant nsf grant dms three authors thank msri support geometric group theory program addition totally disconnected call tdlc group uri bader alex furman roman sauer analogous constructions exist products real algebraic groups sld sld sld sld qpk notations conventions arithmetic lattices defined central theme study lattices connections lattices ambient group mostow strong rigidity uniform case case asserts irreducible lattice real lie group determines ambient lie group among lie groups determines embedding ambient lie group uniquely automorphism natural ask extent lattices determine lattice envelopes among lcsc groups countable groups trivial lattice envelopes make question precise introduce notion virtual isomorphism lattice embeddings definition generalizes notion weak commensurability virtual isomorphism abstract groups see definition lattice embedding called trivial virtually isomorphic identity homomorphism countable discrete group regarded lattice embedding virtual isomorphism equivalence relation refer information problem given countable group describe possible lattice envelopes virtual isomorphism study lattices lattices often harder work compared uniform ones example condition integrability lattices required different purposes spectral gap condition also known weak cocompactness required automatically satisfied uniform lattices proven examples lattices often using elaborate arguments motivates following problem prevalent lattice embeddings main result state main result theorem towards problem need put conditions group definition say countable group property upper bound order finite subgroups irr virtually isomorphic product two infinite groups caf every amenable commensurated subgroup finite nbc every normal subgroup every commensurated subgroup finite index subgroup commutes abbreviations stand bound torsion irreduciblity commensurated amenable finite normal commensurated definition commensurated subgroup see prove following result shows large classes groups satisfy conditions also describe classes detail lattice envelopes theorem following countable groups property caf groups number degree linear groups finite amenable radical groups class dreg introduced class contains acylindrically hyperbolic groups thus convergence groups following countable groups property nbc groups first number linear groups finitely generated residually finite groups groups class dreg groups also property irr statement nbc groups case due corollary course linear groups fall case provided finitely generated next theorem announced slightly stronger formulation nbc caf condition gives partial answer problem concerns possible lattice envelopes given group impose restriction lattice envelope except compactly generated condition also removed assumed finitely generated see lemma theorem structure possible lattice envelopes let countable group properties irr caf nbc let lattice embedding compactly generated lcsc group lattice embedding virtually isomorphic one following irreducible lattice connected real lie group without compact factors inclusion subgroup commensurable possibly trivial tree extension finite set places compatible lattice totally disconnected locally compact group without normal amenable subgroups addition lattice uniform notion tree extension see definition three cases distinct see remark show example necessity condition nbc theorem classical case lattices real lie groups includes arithmetic examples lattices appear case lattices statement refers irreducible lattices product finitely many real lie groups ones factors types present case contains large class examples includes lattices groups fundamental groups finite cubical complexes whose universal cover group automorphisms last case remains quite mysterious although structure theory simple totally disconnected groups emerged last decade regard caprace survey recommended however assuming property last case allow lattices hence obtain following partial solution problem uri bader alex furman roman sauer corollary classification lattices let countable group properties irr caf nbc every lattice embedding virtually isomorphic either irreducible lattice connected center free real lie group without compact factors possibly trivial tree extension lattice real factors present particular every lattice embedding weakly compact exception lattice embeddings lie groups isogenous every lattice embedding square integrable direct corollary except last statement consequence results weak compactness square integrability classical lattices weak compactness lattices connected lie groups stated proved lemma weak compactness lattices groups setup follows stronger property statement square integrability taken shalom work see higher rank case theorem rank case applications applications main result present precise classification results lattice envelopes specific groups proofs found theorem mostow rigidity locally compact targets let either irreducible lattice real lie group without compact factors lattice setup lattice embedding lcsc group virtually isomorphic theorem may viewed generalization mostow strong rigidity corresponding special case theorem real lie group also known real lie group accounts one even assumes focus aligning two given lattice embeddings automorphism fact mostow considered case uniform lattice embeddings later obtained prasad margulis second author proved theorem case simple real lie group rkr general lattice envelope case real lie group uniform lattice envelope case excluded theorem two reasons first strong rigidity hold setting moduli space embeddings surface group group multidimensional secondly lattices virtually free groups finite rank embedded lattice completely different lcsc groups automorphism group aut locally finite tree next results show examples related ones mentioned theorem lattice embeddings free groups let finite extension finitely generated free group let lattice embedding lattice envelopes virtually isomorphic lattice uniform virtually isomorphic lattice closed cocompact subgroup automorphism group tree note second possibility theorem includes examples groups uniform lattices surface groups possibilities lattice embeddings even restricted theorem lattice embeddings surface groups let uniform lattice lattice embedding virtually isomorphic uniform lattice embedding let manifold admits riemannian metric strictly negative sectional curvature homeomorphic locally symmetric one conjecture fundamental groups manifolds nontrivial lattice envelopes additional pinching assumption able prove theorem lack lattice embeddings pinched negative curvature fundamental group closed riemannian manifold dimension whose sectional curvatures range admit lattice embedding unless homeomorphic closed hyperbolic manifold remarks result order gromov thurston construct infinitely many examples negatively curved manifolds pinching dimensions homeomorphic hyperbolic manifolds taking connected sums hyperbolic manifold exotic sphere farrell jones construct closed smooth manifolds homeomorphic hyperbolic manifold whose smooth structure support hyperbolic riemannian metric recommend survey farrell jones ontaneda issues finally obtain following surprising characterization free groups theorem lattice embeddings groups let group first number upper bound order finite subgroups possesses compactly generated lattice envelope free subgroup finite index structure paper devote proof theorem discuss example burger mozes shows necessity condition nbc one drops nbc still keeping caf exotic lattice embeddings covered three cases theorem definition virtually isomorphism lattice embeddings tools trade proof theorem provided difficult proof result outer automorphisms lattices theorem result known expected hold experts proof far required generality hope proof provides useful reference uri bader alex furman roman sauer bulk paper devoted proof theorem paper announced results paper provided proofs special cases might helpful reader first step proof rely property caf positive solution hilbert problem show given lattice embedding virtually isomorphic one product lie group totally disconnected group may assume let subgroup second step proof split way projection dense depending finiteness distinguish three cases third step correspond cases statement theorem second case sophisticated identify lattice certain closed cocompact subgroup lattice corresponding factor identification arithmeticity theorem proved companion paper heavily relies margulis commensurator rigidity actually prove paper result greater generality assume compact generation ambient group version needed slightly general arithmeticity theorem caprace monod theorem basic setup groups detailed explanation arithmeticity theorem provided use margulis normal subgroup theorem aforementioned result outer automorphism groups conclude second case condition nbc appears whole proof namely third step proof lemma final step proof identifies difference tree extension sense setup important ingredients fact rigidity theorem kleinerleeb work proof applications theorems major step always identify mysterious case namely theorem end appeal ideas geometric group theory final step proof theorem based fact lattice property totally disconnected lattice envelopes properties irr caf nbc describe classes theorem detail prove theorem describe exotic lattice embedding group property nbc commensurated commensurable groups recall following well known definitions definition subgroup commensurated subset finite index every commensurated say commensurated definition two groups commensurable subgroups finite index case write weakly commensurable virtually isomorphic lattice envelopes subgroups finite index finite normal subgroups notion virtual isomorphism generalized lattice embeddings definition proof following easy lemma left reader lemma commensurated preserved following situations preimages commensurated subgroups homomorphisms commensurated intersections commensurated subgroups commensurated finite index subgroup commensurated subgroup commensurated caf nbc necessarily preserved one passes weakly commensurable group however preserved passing quotients finite kernels lemma let finite normal subgroup property caf caf proof let projection let amenable commensurated subgroup commensurated lemma since finite extension amenable property caf group finite thus finite lemma let finite normal subgroup property nbc nbc proof let subgroups commensurated satisfy properties see lemma hence finite index subgroup commutes projection finite index subgroup commutes groups class dreg acylindrically hyperbolic groups class dreg introduced thom closely related class creg dreg creg attempt define negative curvature groups cohomological way definition dreg analytical let unitary representation map uniformly bounded vector space modulo bounded ones forms group class dreg class groups sense section concrete description groups dreg proposition corollary group dreg unbounded following theorem proved classes groups later identified acylindrically hyperbolic groups osin uri bader alex furman roman sauer theorem theorem class dreg strictly contains class acylindrically hyperbolic groups also mention following result sun theorem convergence groups acylindrically hyperbolic remark sequel show properties nbc caf groups dreg using difficult theorem implies nbc caf acylindrically hyperbolic groups emphasize however one deduce caf nbc directly quite easily acylindrically hyperbolic groups example caf follows directly corollary acylindrically hyperbolic groups show groups dreg satisfy nbc property trivial reason situation apply nbc happen means show following stronger property groups dreg definition group property nbc satisfies following normal subgroup commensurated subgroup finite finite obvious nbc implies nbc lemma let group normal subgroup commensurated subgroup every finite subset assign subgroups way following holds normalized normal finite index finite index commutes thus implies particular finitely generated finite index subgroup commutes proof every let normal core subgroup since commensurated hence finite index words largest normal subgroup property let map well defined clearly thus since normal commutator lies also hence commutes subset let normality group commutes subset well subgroups hsi satisfy required properties lattice envelopes theorem every group dreg irr caf nbc proof properties irr caf shown theorem let dreg let normal commensurated subgroups respectively trivial intersection assume contradiction subgroups infinite according loc cit let enumeration refer notation lemma applied situation group increasing union subgroups nfi hence nfk thus infinite group commutes finite index subgroup mfk since commensurated also commensurated lemma restriction maps commensurated subgroups injective hence restriction map injective reach contradiction latter module vanishes theorem fact product two infinite groups conclusion proof theorem proof property caf refer cases statement theorem since numbers infinite amenable group vanish result every group number caf according corollary exactly theorem see theorem classes included dreg see theorems proof property nbc refer cases statement theorem let group positive first number assume contradiction nbc let subgroups normal commensurated asssume neither finite show since commensurated vanishing would imply according corollary lemma group ascending union groups virtually product infinite groups product infinite groups vanishing first number formula numbers fact zeroth number infinite group vanishes true virtual product large follows fact vanishing numbers invariant virtual isomorphism easily deduced basic properties one may cite much general result vanishing numbers lcsc groups coarse invariant finally implies theorem let gln linear group let subgroups trivial intersection normal commensurated let assignment finite subsets subgroups lemma noetherian property algebraic group gln uri bader alex furman roman sauer exists finite set zariski closure minimal among finite subsets thus every zariski dense since get particular finite index subgroup commutes implies nbc linear groups mentioned exactly corollary see theorem proof property irr formula numbers fact previous proof vanishing unaffected passing virtually isomorphic groups follows groups positive first numbers property irr groups dreg irr contained theorem examples groups without property nbc among conditions required main result nbc property opaque present two examples groups property nbc first example wreath product second one comes lattice envelope shows necessity nbc condition theorem example infinite residuallyqfinite group countable group consider product group endowed natural shift action let subgroup consisting periodic elements elements finite index clearly countable subgroup acts let observe commensurated subgroup commute finite index subgroup yet finitely generated subgroup lemma example let irreducible lattice sln sln let symmetric space sln let building associated sln action extended action extension denoted fundamental group infinitely generated free group universal covering note tree moreover irreducible lattice aut sln type example originates work general constructions direction discussed group properties irr caf lacks property nbc let show property caf first let commensurated subgroup image projection amenable commensurated subgroup follows theorem caf thus finite since satisfies caf loc intersection finite well implies finite let aut stabilizer vertex compact open subgroup easy verify normal subgroup subgroup sln commensurated lemma violate property nbc finite index subgroup commutes lattice envelopes general facts locally compact groups lattices introduce analog weak commensurability virtual isomorphism groups lattice embeddings look broader classes subgroups lcsc groups lattices discrete subgroups closed subgroups finite covolume recall definition amenable radical study behaviour passage closed subgroups finite covolume finally study automorphisms lcsc groups lattices different categories outer automorphism groups groups virtual isomorphism lattice embeddings definition two lattice embeddings virtually isomorphic open finite index subgroups compact normal subgroups topological isomorphism restricts isomorphism lattices commutative square lattice embedding trivial virtually isomorphic identity homomorphism countable discrete group particular virtually isomorphic virtually isomorphic sense definition justify definition one needs horizontal maps diagram lattice embeddings follows lemma lemma proposition virtual isomorphism equivalence relation class lattice embeddings proof claim transitivity proof similar proof weak commensurability among groups transitive let lattice embeddings virtually isomorphic finite index subgroups compact normal subgroups isomorphisms restrict isomorphisms respectively subgroup finite index let projection let kernel composition quotient maps uri bader alex furman roman sauer note contains since compact compact normal subgroup consider compositions quotient maps isomorphisms similarly using inverses isomorphisms obtain homomorphisms set let preimages quotient maps finite index thus also compact normal subgroup homomorphism isomorphism inverse map since preserve lattices isomorphism restricts isomorphism finishes proof transitivity discrete subgroups locally compact groups collect important albeit easy facts used proof main result lemma let lattice lcsc group let open subgroup lattice lemma theorem let lcsc group normal closed subgroup projection lattice discrete lattice projection also lattice following statement main source commensurated subgroups lemma let lcsc group compact open subgroup commensurated subgroup intersection commensurated proof open subgroup compact group resp hence finite index resp second statement follows taking taking intersections lemma lattice envelope finitely generated group compactly generated proof let lattice lcsc group let generated finite set let open neighborhood identity compact closure claim compact set generates subgroup finite index indeed contains group generated hence open thus countable discrete space hence finite hence adding finite set one obtains compact generating set lattice envelopes closed subgroups finite covolume notion lattice discrete subgroup finite covolume generalized closed necessarily discrete subgroups follows closed subgroup lcsc group said finite covolume carries finite borel measure finite covolume unimodular unimodular lemma lemma lemma let lcsc group closed subgroups finite covolume finite covolumes well known lattice tdlc group uniform mild generalization given following lemma corollary let unimodular tdlc group let closed subgroup finite covolume let haar measure upper bound haar measures compact open subgroups sup compact open subgroup compact lemma let connected real lie group without compact factors every closed subgroup finite covolume form direct factor lattice proof lie algebra connected component since zariski dense borel density theorem follows ideal semisimple lie algebra let decomposition simple lie algebras let let lie subgroup lie algebra since simple either implies discrete since finite covolume lattice amenable radical definition amenable radical lcsc group maximal closed normal amenable subgroup denote radam lemma let lcsc group let closed subgroup finite covolume radam radam lemma deduced following result uri bader alex furman roman sauer lemma furstenberg let lcsc group continuous action minimal strongly proximal let closed subgroup finite covolume restriction also minimal strongly proximal furstenberg interested restriction lattices proof applies general closed subgroups finite covolume proof lemma lemma definition radam yields inclusion radam radam converse inclusion use equivalent characterization amenable radical common kernel minimal strongly proximal actions compact metrizable spaces proposition thus converse inclusion suffices show given arbitrary minimal strongly proximal homeo restriction also minimal strongly proximal would show radam acts trivially every minimal strongly proximal content lemma outer automorphism groups describe various results groups automorphisms groups outer automorphisms certain groups confident main result section theorem known experts seem literature appropriate generality theorem let number fields let connected noncommutative adjoint absolutely simple let finite compatible sets places group abstractly commensurable finite outer automorphism group proof theorem given end subsection theorem used together lemma split certain extensions groups subgroup group let aut autb denote automorphism group subgroup aut preserving outer automorphism group lcsc group denote group continuous automorphisms autc group continuous outer automorphisms outc autc note continuous automorphism homeomorphism open mapping theorem begin quoting two auxiliary results elementary lemma corollary exercise let normal subgroup group assume trivial center right hand square commutative diagram aut lattice envelopes whose rows exact whose vertical homomorphisms induced conjugation pullback diagram groups thus trivial group extension splits direct product lemma let normal subgroup group let center sequence groups autb aut aut exact left group abelian group maps cocycle following three results immediate consequences lemma lemma every topological automorphism product gtd connected group tdlc group gtd product topological automorphism topological automorphism gtd proof apply lemma situation map obvious right inverse preserve continuity automorphisms let continuous automorphism since trivial center injective lemma let group let characteristic subgroup finite index finite center restriction map aut aut finite kernel proof denote kernel aut autb aut center since aut finite enough show kernel aut finite lemma isomorphic finite since finite corollary group characteristic subgroup finite index finite center finite outer automorphism group also finite outer automorphism group proof let characteristic finite center finite outer automorphism group since group aut surjects onto aut enough show former finite follows exactness ker aut aut aut finite assumption kernel finite lemma purposes subsection introduce following definition definition group said strongly irreducible every homomorphism finite cokernel either ker ker group said strongly outer finite outer automorphism group finite index subgroups finite lemma zariski dense subgroup connected noncommutative adjoint group strongly irreducible proof let connected adjoint let zariski dense subgroup uri bader alex furman roman sauer let homomorphism finite cokernel since zariski dense zariski closure finite index subgroup connected group zariski closure define commute together generate simplicity conclude trivial follows ker theorem let group abstractly commensurable product finite family finitely generated strongly irreducible strongly outer finite groups finite outer automorphism group proof theorem preceded preparation first make following definition standard definition standard subgroup product groups subgroup form finite index subgroup lemma let finite family strongly irreducible groups let finite family groups monomorphism finite cokernel bijection monomorphism finite cokernel product maps particular image standard subgroup proof let product respectively proof induction nothing prove let qwe decompose set let ker projection strong irreducibility partition hence composition projection factors thus giving homomorphism observe injective homomorphisms finite cokernels without loss generality induction hypothesis therefore applying induction hypothesis statement follows applying lemma automorphisms obtain following corollary let finite family strongly irreducible groups obvious embedding aut aut finite cokernel lemma let finite family strongly irreducible groups let product let subgroup finite index characteristic subgroup finite index lattice envelopes proof let call subgroup simultaneously standard subgroup substandard subgroup lemma every aut preserves substandard subgroups hence characteristic unique maximal element collection substandard subgroups proof theorem assumption finite index subgroup isomorphic finite index subgroup product finite family finitely generated strongly irreducible strongly outer finite groups identify various subgroups images isomorphism finitely generated thus finitely generated thus finite index subgroup characteristic intersection subgroups index lemma characteristic subgroup finite index standard finite index strongly irreducibility passes finite index subgroups strongly irreducible characteristic transitive characteristic corollary obvious embedding aut aut finite cokernel strong outer finite assumption finite hence finite strong irreducibility assumption finite center thus corollary implies finite following proposition well known proposition let number field let connected adjoint absolutely simple let finite compatible set places finite group continuous outer automorphisms proof lemma group strongly irreducible corollary natural embedding aut aut finite cokernel continuous automorphism product continuous automorphisms implies restriction autc autc finite cokernel thus enough show outc finite every since every continuous field automorphism trivial closure finite field finite group continuous field automorphisms moreover group finite group algebraic outer automorphisms given dynkin diagram automorphisms thus outc indeed finite chapter proposition let number field let connected adjoint absolutely simple let finite compatible set places finitely generated strongly irreducible strongly outer finite uri bader alex furman roman sauer proof chapter theorem finitely generated lemma strongly irreducible left prove strongly outer finite end consider subgroup finite index group irreducible lattice let autc conjugation homomorphism injective finite cokernel proposition embeds lattice claim normalizer discrete conjugation action aut isomorphism let first conclude finiteness since contains lattice discrete also lattice hence isomorphic hence finite regarding discreteness follow show countable closed since finitely generated aut countable hence countable denotes centralizer factor trivial density projection factor since finite index finite countability follows verify enough show every automorphism induces continuous automorphism consequence strong rigidity theorem theorem vii proof theorem proposition groups finitely generated strongly irreducible strongly outer finite thus proof follows theorem finally need classical result work tits theorem let number field let finite place let connected adjoint absolutely simple automorphism group building associated contains subgroup finite index proof higher rank automorphism group associated building isomorphic aut isomorphic extension group aut algebraic automorphisms subgroup field automorphisms follows results see proposition elaboration group aut contains subgroup finite index finitely many field automorphisms since every field automorphism continuous trivial closure rationals finite implies statement locally compact groups recall map metric spaces every two maps bounded distance sup group group composition equivalence classes bounded distance lattice envelopes lemma let lcsc group containing closed subgroup compact projection locally continuous compactly generated let isom properly discontinuous cocompact action proper geodesic metric space natural homomorphism isom extends homomorphism following property exist constants represented every bounded set neighbourhood identity remark projection lcsc group coset space closed subgroup locally continuous finitedimensional respect covering dimension theorem tdlc groups ones apply lemma proof lemma since compactly generated compact also compactly generated thus possess word metrics unique inclusion construct unique bounded distance following way let projection let local continuous crosssection defined compact neighbourhood extend global relatively compact image necessarily continuous assignment yields isomorphism let theorem map pick similarly obtain isomorphism let homomorphism sends group element left translation similarly easy see map coincides composition define composition clearly extends let lgg denote left translation similarly let lhh left translation represented lgg clearly constants depending let bounded subset let sup sup uri bader alex furman roman sauer set bounded thus relatively compact let closure continuous map sends continuity compactness identity neighbourhood mapped function ghs continuous maps continuity compactness identity neighborhood let let ghs upon replacing constant max statement follows arithmetic lattices tree extensions arithmetic core introduce notion tree extensions lattices appears main result theorem discuss arithmeticity result lattices product semisimple lie group tdlc group identifies second case theorem arithmetic lattices tree extensions setup let number field ring integers let connected noncommutative absolutely simple adjoint let set inequivalent valuations places let denote archimedean ones fin ones finite places denote completion respect local field let finite subset places compatible sense explicitly means following every contains contains least one finite one infinite place let fin fin let ring let normal subgroup defined section perfect case situation subgroup generated unipotent elements finite index connected component identity real lie group define quotient finite lattice envelopes reduction theory borel diagonal embedding realizes lattice note fin splitting real lie group totally disconnected locally compact group thus make following remark remark finite set places compatible lattice product least two lcsc group one lie group one tdlc group exhibit generalization previous type lattices definition let finite let denote finite places fin denote associated tree aut tdlc group automorphisms simplicial tree closed intermediate group aut called tree extension since cocompact inclusion subgroup commensurable tree extension lattice embedding lemma remark denote buildings symmetric space group isom contains subgroup finite index theorem isom finite index subgroup isom follows proposition thus closed intermediate subgroup isom passing finite index subgroup tree extension arithmetic core theorem next state arithmeticity result lattices products provides key step proof theorem theorem arithmetic core theorem let connected real lie group without compact factors tdlc group lattice assume projection dense image projection dense image projection injective compactly generated exist number fields connected adjoint absolutely simple finite sets places compatible following properties topological isomorphism uri bader alex furman roman sauer continuous epimorphism compact kernel closed intermediate subgroup sifin fin image commensurable image sifin result quite close theorem paper caprace monod fact assumed simple instead theorem essentially theorem companion paper prove theorem deducing general statement assumed compactly generated general case sets places might infinite gap becomes large current paper use arithmetic core theorem step case iii point proof know yet called finitely generated know compact generation called gtd inherited compact generation assumption reader convenience sketch idea proof theorem appears apart step approach differs taken caprace monod following sketch ignore important details sake transparency choose compact open subgroup observe projection lattice commensurated projection irreducible margulis commensurator arithmeticity theorems provide number field simple group real lie group isogenous commensurable arithmetic lattice set infinite places reducible commensurable product irreducible lattice leading fields groups clarity continue case view subgroup commensurator commenh arithmetic lattice one glosses difference adjoint forms sld pgld commensurator subgroup rational points therefore define fin set places fin image unbounded example sld psl set fin would consist primes appear arbitrarily high powers denominators entries commensurable subgroup fin image tdlc group precompact showed using theorem alternatively use implies image dense open subgroup finite lattice envelopes index one even show image dense open subgroup finite index fin one considers closure diagonal imbedding fin using fact projections factors dense closure projections open compact one shows graph continuous epimorphism fin compact kernel finally fact lattice implies image contained lattice lattice fin thus compactly generated fin implies fin hence also finite also follows finitely generated proof theorem starting point proof theorem consequence hilbert problem observed theorem theorem every locally compact group contains open normal finite index subgroup containing radam quotient radam isomorphic direct product connected real lie group without compact factors tdlc group trivial amenable radical step reduction lattice product first step proof shall take advantage property caf theorem amenable radical lattice envelope group property caf compact proof let open normal finite index subgroup radam gss gtd gss theorem namely connected real lie group without compact factors gtd tdlc group radam pick compact open subgroup consider following commutative diagram radam radam gss arrows obvious inclusions projections moreover defined requiring commutativity first show commensurated equivalent commensurated radam lemma let radam since normal thus radam normal radam conjugation uri bader alex furman roman sauer continuous automorphism gss lemma automorphism product continuous automorphisms css hence conjugation maps since subgroup ctd open compact intersection finite index implies commensurated radam conclude commensurated let preimage gss let ker gss group extension amenable group radam compact group hence amenable group lattice lemma invoke deep result breuillard gelander follows theorem closed amenable subgroup projection lattice discrete result breuillard gelander generalization auslander theorem solvable lie group lemma group lattice thus amenable note used follows diagram hand commensurated hence finite implies closed subgroup radam compact let summarize situation first step proposition let property caf let lattice embedding amenable radical radam compact passing open normal finite index subgroup containing finite index subgroup taking quotients compact finite normal subgroups one obtains lattice embedding gss virtually isomorphic product connected center free semisimple real lie group without compact factors gss tdlc group trivial amenable radical gtd furthermore compactly generated compactly generated step projection factor irreducible let gss lattice product proposition since compact generation assumed theorem group compactly generated real lie group gss splits direct product simple factors gss subset denote sjk viewed subgroup factor group gss set given subset consider image projection prj gss lattice envelopes note possible dense prj discrete proper subset lemma unique maximal subset projection prj discrete proof suffices show collection subsets discrete projection closed union let subsets prj discrete prk discrete let open neighborhoods identity prj prk view subgroup open neighborhood identity therefore discrete proves claim let maximal subset claim denote gss consider projection prl gtd prl define kernel projection ker prl gtd lattices lemma consider projections gtd gtd lemma projection dense proof let closure projection gtd closed subgroup gtd containing lattice lemma closed subgroup finite covolume follows finite covolume lemma form direct factor lattice setting splitting trivial indeed otherwise group splits gss projection lies discrete contradicts maximality factor completes proof lemma define lcsc group gss gss gtd closed subgroup tdlc group gtd group tdlc group connected real lie group group closed subgroup gtd containing lattice lemma forms lattice finite covolume subgroup gtd thus subgroup finite covolume since uri bader alex furman roman sauer radam gtd deduce radam using lemma summarize proposition let gss proposition splitting closed subgroup gtd setting projection prl lattice prl kernel ker lattice gss projection dense projection prtd dense closed subgroup finite covolume gtd trivial amenable radical gtd compactly generated conditions needed application theorem satisfied exception injectivity projection prss next topic concern step identifying lattice embedding still denote projections lie tdlc factors prss prtd images subgroup prss pri indicated superscripts respectively subgroup let denote centralizer let gtd compact open subgroup tdlc group define following groups gtd gss projection gss gtd gss remark since normalized dense image gtd normalizer closed subgroup closed subgroup normal gtd justifies last definition moreover since gss commutes follows normal note since compact generation passes quotients gtd compactly generated remark point claim triviality amenable radical gtd even though follow later analysis lemma group commutes finite index subgroup lattice envelopes proof point want apply property nbc original group note property nbc pass finite index subgroups general argue specifically refer notation proposition note subgroup show normal commensurated claim follows since nbc lemma group gtd topologically characteristic subgroup lemma since normal normal since normal subgroup group normal subgroup gss commensurated since normal every topological automorphism product automorphism gtd one lemma since normal commensurated previous lemma upon making smaller may assume centralizes follows centralizes record particular subgroup isomorphic lemma inclusions miss gss lattice embeddings proof lemma lattice gss since open subgroup since also compact image lattice gss well similarly gss lattices since definition one easily sees images gss coincide particular lattice since also normal see remark quotient lattice gss lemma finally position identify lattice embedding virtually isomorphic original lattice embedding distinguish three cases depending finiteness groups case finite lattice tdlc group case connected real lie group gss without compact factors finite group lattice thus gss must trivial thus tdlc group trivial amenable radical contains lattice assumed property applies lemma uniform lattice either discrete means original lattice embedding trivial case main theorem case infinite finite lattice lie group case finite lattice gss conclude trivial therefore gtd last equality follows discreteness uri bader alex furman roman sauer since prss lattice subgroup lattice see proposition must finite index short exact sequence prl prl equals finite index subgroup since infinite irr condition forces finite gtd finite finite covolume latter compact fact trivial trivial amenable radical conclude classical lattice connected real lie group gss lattice irreducible due assumption irr corresponds case main theorem case iii infinite lattice recall projection gss lattice assumption infinite means inclusion gss lattice embedding lemma point may apply arithmetic core theorem deduce dividing gtd compact normal subgroup one product lattices nontrivial totally disconnected factors become clear trivial one irreducible lattice precisely number fields absolutely simple groups finite sets places compatible denoting connected real lie groups gss certain closed intermediate groups fin sifin moreover commensurable product lattices finite index subgroups finite index upon passing smaller finite index subgroup may assume inclusion normal subgroup lemma gss lattice contains product irreducible lattices finite index subgroup relying nbc condition showed see commutes abelian subgroup characteristic subgroup gtd normal since trivial amenable radical lattice envelopes hence restriction quotient map injective image normal subgroup normal contains subgroup finite index normal commutative diagram summarizes various relations groups since commensurable normal subgroup infinite since irreducible lattice margulis normal subgroup theorem implies finite index thus finite index injectivity group thus abstractly commensurable product lattices theorem finite outer automorphism group center trivial since lies amenable radical trivial proposition lemma gss gss sifin since intersection subgroup normal implies normal lemma finite index subgroup ker splits direct product assumption irr implies one factors finite infinite obtain finite index hence finite index finite obtain finite group lattice envelope compact thus trivial gss using condition irr also deduce one irreducible factor hence furthermore group known finite trivial normal gtd radam follows radam particular compact normal subgroup gtd actually trivial uri bader alex furman roman sauer deduce number field connected adjoint absolutely simple finite set fin places irreducible lattice lcsc group gss fin remains identify lcsc group established triviality hence gss remains identify totally disconnected component gtd contains fin fin closed subgroup finite covolume step lattice envelope gtd fin let enumerate elements fin way least two extreme cases course possible let building associated let write notation gtd spaces irreducible euclidean buildings cocompact affine weyl group moufang tits boundary group acts automorphisms simplicial complex action vertex set finitely many orbits vertex stabilizers maximal compact subgroups building tree theorem let tdlc group trivial amenable radical contains gtd closed subgroup finite covolume compact open subgroup finite index contains natural embedding isom extends injective continuous homomorphism isom closed image apply result gtd whose amenable radical trivial according remark und upon passing closed subgroup finite index group tree extension sense definition finishes proof theorem based theorem proof theorem proceeds several steps heavily use techniques geometric group theory inspired approach taken section compactness theorem see corollary general result cat every compact subgroup fixes vertex since finitely many vertices finitely many vertex stabilizer groups conjugation particular endowed haar measure upper bound haar measure open compact subgroups sup open compact subgroup lattice envelopes thus lemma applies implies compactness definition subgroup applying lemma combination remark acting one obtains homomorphism quasiisometry group constants every represents class following property every bounded set neighborhood identity apply following splitting theorem kleiner leeb theorem theorem every every within distance product factors permutation symn expense increasing constants may hence assume product permutation symn another consequence theorem product embeds subgroup index define finite index subgroup note contains homomorphisms every represents restriction agrees homomorphism isom pri projection factor view quantitative statement theorem expense increasing following holds true every every bounded set neighborhood identity openness group defined kernel homomorphism finite group know point homomorphism continuous would imply open thus lcsc group next provide direct argument shows openness let dxk denote metric respectively let bounded subset whose diameter exceeds let let neighborhood identity satisfies next show contained open subgroup suppose let let points whose distance least pick points either dxj uri bader alex furman roman sauer dxj without loss generality assume first case let dxj contradicting therefore mapping isom higher rank factors rigidity theorem higher rank irreducible buildings kleiner leeb next important ingredient theorem theorem let every constant every within distance unique isometry moreover two distinct isometries within bounded distance hence natural homomorphism isom isomorphism composing inverse obtain homomorphisms isom restriction homomorphism isom statement theorem applied show constant every bounded subset neighbourhood identity dxi lemma map continuous proof let isom open neighborhood identity show open neighborhood identity contained rely following geometric fact buildings follows example every constant open neighborhood identity isom bounded set depending isom sup dxi apply general fact constant identity neighborhood let fix bounded subset isom sup dxi applying statement specific subset provides neighborhood identity dxi since mean continuity follows lattice envelopes mapping homeo rank factors next turn tree factors every tree bounded degree space ends compact one embedding groups homeo let composition homeo lemma map continuous closed image every proof continuity statement proved theorem argument analogous one lemma however instead geometric fact one uses following fact consequence lemma tree bounded degree constants identity neighbourhood homeo compact subset image homeo sup belongs since cocompact image homeo equals image homeo closed also image closed identifying image since continuous ker closed subgroup set ker regarding subgroup thus inclusion induces monomorphism since restricts injective map isom homeo respectively theorem isom contains closed subgroup finite index thus contains subgroup finite index let consider tree factors fix index map describes thus faithful tree since every acts within uniformly bounded distance next resort work mosher sageev whyte trees main result says theorem theorem cobounded group bushy tree bounded degree isometric action group possibly different tree since degree least acts cocompactly assumptions fulfilled hence tree acts isometries dtj sup uri bader alex furman roman sauer preliminary remarks proof lemma order rely description buildings see corollary theorem need dimension tree consists two vertices nonconjugated maximal subgroups one edges generate description using image simply connected form whole group might one orbit vertices action group tree called minimal proper subtrees invariant group action action locally finite tree minimal induced action boundary tree minimal sense dynamical systems proper closed invariant subsets consequence limit set minimal closed invariant subset boundary see general result attributed gromov context spaces conversely group acting automorphisms locally finite tree minimal set consisting two points union geodesics end points called convex hull minimal therefore contains moreover exists retraction given nearest point projection retraction equivariant respect aut returning situation recall action tree minimal hence minimal since equivariant homeomorphism action also minimal particular convex hull gives minimal tree without loss generality may assume convex hull boundary vertices degree one action defines proper action space triples distinct points thus minimal strongly proximal subgroup finite covolume thus action also minimal strongly proximal lemma minimality implies convex hull boundary minimal also tree lemma action homomorphism isom injective continuous closed image proof map induces homeomorphism since isometry group group either inject homeomorphism group respectively injective map isom injective since isom embeds closed subgroup homeo continuity closedness follow continuity closedness composition isom homeo latter conjugated via map homeo whose continuity closedness implied lemma lattice envelopes lemma cellular homeomorphism homothety stretch factor proof upon subdividing may assume acts without inversion continuity part lemma orbits compact subgroups bounded fixed point theorem corollary vertex fixed respectively choose pair vertices minimal distance geodesic segment connected thus subtree generated minimality moreover fixed map sends vertex edge onto affine homeomorphism extended cellular map clearly surjective next show locally injective enough show local injectivity vertices symmetry let consider consider neighbours since conjugated pairwise distinct let center tripod given want show implies local injectivity contained stabilizer since maximal group theoretic description tree theorem follows since chose minimal distance among vertices fixed respectively surjective locally injective map trees homeomorphism let distance obviously locally homothety stretch factor thus globally conjugating homothety lemma obtain topological isomorphism isom isom compatible two embeddings hence embedded closed intermediate subgroup isom homomorphism regarded isom composed natural embedding isom homeo also denote map isom let diagonal embedding obtain continuous homomorphism closed image isom isom whose restriction natural embedding isom since compact ker compact since amenable radical thus lemma trivial map injective completes proof theorem therefore also proof main classification result theorem proofs theorems proof theorem let lattice embedding assume uri bader alex furman roman sauer either connected real lie group without compact factors irreducible lattice setup commensurable lattice satisfies assumptions theorem indeed lattice zariski dense algebraic group borel density theorem theorem thus theorem ensures satisfies conditions caf nbc irreducible lattice satisfies irr lattices semisimple groups known finitely generated selberg lemma contains subgroup finite index particular condition applies let lcsc group lattice embedding theorem states open subgroup finite index compact normal subgroup upon replacing assume one following three possibilities connected real lie group without compact factors irreducible lattice number field finite set places including infinite finite ones commensurable lattice setup tdlc group trivial amenable radical uniform lattice cases covered following theorem strong rigidity theorem mostow prasad margulis let embeddings irreducible lattices centerfree lie groups arithmetic lattices exists continuous group isomorphism strong rigidity theorem proved mostow cases real lie lie groups uniform irreducible lattices prasad extended result lattices including lattices rank one real lie groups higher rank cases including situations uniform lattice embeddings follow margulis theorem see theorem therefore remains prove case discrete group containing subgroup finite index end use use homomorphism provided lemma shall prove claim kernel ker open subgroup let first show claim suffices complete proof theorem let denote haar measure since open therefore restriction haar measure lattice group homomorphism injective thus one may include borel fundamental domain lattice envelopes follows finite therefore compact group assumed trivial amenable radical deduce since open follows discrete lattice subgroup finite index left proving claim deduced rigidity results lattices lie groups refer farb survey background references first assume irreducible lattice case known coincide commensurator countable group commenh first proved schwartz real lie groups schwartz eskin higher rank real lie groups wortman cases hence homomorphism countable image kernel ker clearly borel subgroup since countably many cosets cover one borel subgroup positive haar measure open contains identity neighborhood proves claim case next assume uniform lattice let decomposition simple factors associated symmetric spaces buildings least one factors simple real lie group shall assume arguing step section particular theorem observe upon replacing open subgroup finite index replacing finite index subgroup may assume takes value consider homomorphism shall show factors continuous homomorphism isom considering following cases symmetric space higher rank corresponding rkr case natural homomorphism isom bijection group homomorphism isom continuous lemma quaternionic hyperbolic space cayley plane corresponding also case natural homomorphism isom bijection pansu homomorphism isom continuous lemma uri bader alex furman roman sauer real hyperbolic space corresponding boundary infinity sphere natural conformal structure preserved isom fact well known isomorphism isom conf extends isomorphism qconf group group homeomorphisms sphere result tukia subgroup qconf uniformly acts cocompactly triples points conjugate conf applied indeed lemma exist constants represented therefore lemma induce uniformly group mappings group contains acts cocompactly therefore also cocompactly triples points therefore get homomorphism isom conf qconf topology isom coincides conf inherited polish topology homeo hence continuity homomorphism isom follows lemma complex hyperbolic space corresponding case boundary sphere natural conformal class subriemannian structure analogues theory mappings result analogous tukia theorem allow prove see conjugate conf isom continuity resulting homomorphism isom follows lemma shows homomorphism factors continuous homomorphism simple real lie group isom let open compact subgroup tdlc group image profinite group subgroup isom real lie group therefore must finite ker open subgroup since ker contains open subgroup also open subgroup completes proof claim thereby proof theorem proof theorem let finite extension finitely generated free group let lattice envelope since satisfies assumptions theorem theorem need discuss three possibilities virtually isomorphic irreducible lattice semisimple real lie group first number positive olbrich work see also chapter computations lattice envelopes locally symmetric spaces book lattices positive first numbers ones case lattice since example virtual cohomological dimension one virtually isomorphic lattices setup particular virtually isomorphic lattice embedding product two lcsc groups ruled first number see argument proof theorem alternatively situation ruled fact lattices gromov hyperbolic virtually isomorphic uniform lattice tdlc group trivial amenable radical particular virtually isomorphic free group passing finite index subgroup may assume free group let cayleygraph regular tree degree least natural action homomorphism isom extends lemma homomorphism describes cobounded see remark preceding theorem next appeal work mosher sageev whyte step proof theorem theorem taken work isometric automatically also cobounded cocompact action isom possibly different tree using arguments lemma lemma verbatim one concludes injective continuous closed image open mapping theorem topologically isomorphic latter cocompact subgroup isom cocompactness action proof theorem let uniform lattice let another lattice embedding gromov hyperbolic group satisfies assumptions theorem according theorem need discuss three possibilities virtually isomorphic irreducible lattice semisimple real lie group proof positivity first number one reduced case case cocompact two embeddings need conjugate virtually isomorphic lattices setup case ruled reason case proof theorem virtually isomorphic uniform lattice tdlc group covered theorem possibility trivial lattice embedding proof theorem let closed riemannian manifold dimension least sectional uri bader alex furman roman sauer curvatures ranging let fundamental group let lattice envelope since finitely generated compactly generated lemma group gromov hyperbolic properties caf nbc irr according theorem fundamental group closed aspherical manifold torsionfree particular also property theorem following three possibilities lattice embedding virtual isomorphism since torsionfree may assume upon replacing finite index subgroup finite index subgroup quotient compact normal subgroup finite cover still isomorphic virtually one following cases irreducible lattice embedding semisimple real lie group lattice sense setup uniform lattice tdlc group trivial amenable radical need show possibility case discrete unless homeomorphic hyperbolic manifold since gromov hyperbolic rule case reason case possible real rank uniform note lattices contain free abelian subgroup rank let analyze situation uniform lattice simple lie group real rank let associated symmetric space thus real complex quaternionic cayley hyperbolic aspherical spaces fundamental group locally symmetric space homotopy equivalent real hyperbolic homeomorphic closed hyperbolic manifold following striking result farrell jones work borel conjecture possibility ruled assumption theorem topological rigidity let closed curved manifold dimension closed manifold homotopy equivalent homeomorphic possibilities complex quaternionic cayley hyperbolic ruled applying following result mok siu yeung pinching assumption theorem geometric rigidity theorem let homotopy equivalent closed riemannian manifolds assume negatively curved complex quaternionic cayley hyperbolic isometric scaling particular sectional curvatures range summing case never occurs case occur homeomorphic hyperbolic manifold finally let consider case show tdlc group discrete upon dividing compact normal subgroup consider natural homomorphism isom homeo homeo lattice envelopes course coincides natural action homeo gromov boundary refer lemma situation theorem conclude map extends homomorphism homeo latter reference also shown continuous kernel ker compact image locally compact subspace topology open mapping theorem topologically isomorphic since trivial amenable radical trivial since tdlc group acts continuously faithfully sphere positive solution conjecture would imply discrete thus finish proof general conjecture remains open situation appeal work contains following result theorem conjecture boundary actions corollary let gromov hyperbolic duality group let homeo subgroup finite dimensional locally compact subspace topology lies image natural homomorphism homeo let hausdorff dimension respect visual metric let topological dimension assume lie group apply theorem topological dimension remains verify hausdorff dimension equals volume entropy hvol see theorem hvol lim sup log vol direct consequence comparison theorem theorem obtain manifold sectional curvature hvol present situation obtain hvol assumptions theorem satisfied conclude time lie group tdlc group thus discrete proof theorem let group positive first number property assume admits compactly generated lattice envelope since theorem satisfies conditions theorem virtually isomorphic lattice embedding one first two types theorem totally disconnected case ruled first number preserved virtual isomorphism easily deduced basic properties even quasi isometry theorem hence lattice uri bader alex furman roman sauer product two lcsc groups implies first number lattice envelope positive theorem first number product two unimodular lcsc groups zero theorem excludes sarithmetic lattice setup type ruled hence lattice connected semisimple lie group without compact factors olbrich work see also chapter computations locally symmetric spaces book lattices positive first numbers ones since also group hence virtually isomorphic free group references abramenko brown buildings graduate texts mathematics vol springer new york theory applications bader caprace gelander mozes lattices amenable groups arxiv preprint bader caprace lecureux linearity lattices affine buildings ergodicity singular cartan flow arxiv preprint bader furman gelander monod property rigidity actions banach spaces acta math bader furman sauer structure arithmeticity lattice envelopes math acad sci paris adelic arithmeticity theorem lattices products arxiv preprint weak notions normality vanishing rank int math res imrn bekka uniqueness invariant means proc amer math soc bestvina fujiwara bounded cohomology subgroups mapping class groups geom topol borel density maximality arithmetic subgroups reine angew math borel tits homomorphismes abstraits groupes simples ann math french bourbaki general topology chapters elements mathematics berlin springerverlag berlin translated french reprint english translation breuillard gelander topological tits alternative ann math breuillard kalantar kennedy ozawa unique trace property discrete groups arxiv preprint bridson haefliger metric spaces curvature grundlehren der mathematischen wissenschaften fundamental principles mathematical sciences vol berlin brown cohomology groups graduate texts mathematics vol new york corrected reprint original burger monod continuous bounded cohomology applications rigidity theory geom funct anal burger mozes groups acting trees local global structure inst hautes sci publ math caprace simple locally compact groups appear proceedings european congress mathematics caprace kropholler reid wesolek residual profinite closures commensurated subgroups arxiv preprint caprace monod lattice two groups arithmetic israel math lattice envelopes isometry groups curved spaces structure theory topol isometry groups curved spaces discrete subgroups topol cheeger gromov group cohomology topology chow groups complex hyperbolic space trans amer math soc coornaert mesures sur bord espace hyperbolique sens gromov pacific math french french summary eskin rigidity nonuniform lattices higher rank symmetric spaces amer math soc farb classification lattices semisimple lie groups math res lett farb schwartz geometry hilbert modular groups differential geom farrell jones negatively curved manifolds exotic smooth structures amer math soc farrell jones ontaneda negative curvature exotic topology surveys differential geometry vol surv differ vol int press somerville farrell jones topological rigidity compact curved manifolds differential geometry riemannian geometry los angeles proc sympos pure vol amer math providence furman rigidity locally compact targets geom funct anal minimal strongly proximal actions locally compact groups israel math furstenberg rigidity cocycles ergodic actions semisimple lie groups margulis zimmer bourbaki seminar vol lecture notes vol springer berlin gallot hulin lafontaine riemannian geometry universitext springerverlag berlin gelander karlsson margulis superrigidity generalized harmonic maps uniformly convex spaces geom funct anal gromov thurston pinching constants hyperbolic manifolds invent math bounded cohomology isometry groups hyperbolic spaces eur math soc jems karube local locally compact groups math soc japan kleinbock margulis bounded orbits nonquasiunipotent flows homogeneous spaces moscow seminar dynamical systems amer math soc transl ser vol amer math providence kleiner leeb rigidity symmetric spaces euclidean buildings inst hautes sci publ math kyed petersen vaes numbers locally compact groups cross section equivalence relations trans amer math soc theory applications geometry ergebnisse der mathematik und ihrer grenzgebiete folge series modern surveys mathematics results mathematics related areas series series modern surveys mathematics vol berlin margulis lattices semisimple algebraic groups lie groups representations proc summer school group representations bolyai math budapest halsted new york finiteness quotient groups discrete subgroups funktsional anal prilozhen russian uri bader alex furman roman sauer discrete subgroups semisimple lie groups ergebnisse der mathematik und ihrer grenzgebiete results mathematics related areas vol berlin pattern rigidity conjecture geom topol mok siu yeung geometric superrigidity invent math monod superrigidity irreducible lattices geometric splitting amer math soc mosher sageev whyte trees bounded valence ann math mostow strong rigidity locally symmetric spaces princeton university press princeton annals mathematics studies olbrich locally symmetric spaces doc math osin acylindrically hyperbolic groups trans amer math soc petersen numbers locally compact groups thesis department mathematical sciences faculty science university copenhagen pansu des espaces rang ann math french english summary prasad strong rigidity lattices invent math elementary proof theorem theorem tits bull soc math france english french summary raghunathan discrete subgroups lie groups new ergebnisse der mathematik und ihrer grenzgebiete band displacement function isometries euclidean buildings indag math sauer vanishing numbers locally compact groups invariant coarse equivalence arxiv schwartz classification rank one lattices inst hautes sci publ math rigidity diophantine approximation acta math shalom rigidity commensurators irreducible lattices invent math rigidity unitary representations semisimple groups fundamental groups manifolds rank one transformation group ann math schmidt mehrfach perfekte math ann german sun convergence groups acylindrically hyperbolic arxiv preprint thom low degree bounded cohomology negatively curved groups groups geom dyn tits buildings spherical type finite lecture notes mathematics vol york tukia quasiconformal groups analyse math tits sur groupe des automorphismes arbre essays topology related topics georges rham springer wells automorphisms group extensions trans amer math soc wortman rigidity higher rank lattices geom topol yue ergodic theory discrete isometry groups manifolds variable negative curvature trans amer math soc lattice envelopes weizmann institute rehovot address university illinois chicago chicago address furman karlsruhe institute technology address
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neural network approach railway stand lateral skew control peter mark jan ivo department instrumentation control engineering czech technical university prague czech republic department automobiles internal combustion engines railway vehicles czech technical university prague czech republic abstract paper presents study adaptive approach lateral skew control experimental railway stand preliminary experiments real experimental railway stand simulations mechanical model indicates difficulties control device thus use neural networks identification control lateral skew shall investigated paper focuses realdata based modelling railway stand various neural network models linear neural unit quadratic neural unit architectures furthermore training methods neural architectures variation examined accompanied discussion produced experimental results keywords roller rig linear neural unit quadratic neural unit rtrl bptt introduction ongoing problem currently research railway industry lateral skew control railway carriage wheel sets independently rotating wheels irw particular control position leading wheel set railway carriage bogie achieve central positioning wheel set respect span rail track furthermore control lateral deviation wheel set follow given desired set point primary necessity control arises need improve behaviour irw wheel sets currently study ctu experimental railway stand roller rig particularly minimisation wheel flange rail head wearing well lateral forces furthermore optimal stability higher speeds wheel set date various methods control wheel set lateral position irw wheel sets investigation mechanical electrical means control featuring individual drawbacks ctu latest study features active control roller rig manipulation yaw torque rig wheel set via state feedback cascade pid control linearization model ctu roller rig however results initial experimentation shows method suitable real time control thus paper aims investigate possible use neural network approach lateral control railway wheel sets suitability application problem motivated promising theoretical studies higherorder neural units honus especially quadratic neural unit engineering problems studies focused use supervised learning based approaches polynomial structured neural units also known class honus adaptive identification control real engineering systems motivation arises successful implementation quadratic neural unit controller control bathyscaphe system located automatic control laboratories ctu controller adhered closely desired behaviour system conventionally used pid controller extension result recalled work study made via introduction new software adaptive identification control along testing theoretical system previously mentioned bathyscaphe system given paper aim investigate use neural network approaches following manner begin explaining depth problem behind recently employed state feedback cascade pid control linearization ctu roller rig proceeding section describe principles control schemes behind various experimented methods adaptive identification control following experimental analysis various approaches focusing firstly adaptive identification ctu roller rig system test various methods control final component paper analyse compare produced experimental results conclusion drawn end problem description section aims describe depth functionality behind previously introduced experimental railway stand ctu roller rig importantly scope paper elaborate necessity control roller rig issues conventional linear pid control state feedback control controlling lateral skew experimental setup roller rig ctu figure figure depicts experimental setup ctu roller rig along model design simulation purposes rig features motor drives two pairs diameter rollers independently driven via largest set drives simulate straight curved tracks similar present real railway networks central servo motor introduced yaw lower roller pair replicating curved track motion setup however assumes simulation rail pair without effects rail buckling manipulation wheel set yaw separate servo motor central wheel set installed scope paper crucial component action servo motor used control actuation lateral skew wheel sets discussed control setups paper analyse control via manipulation servo motor torque finally fifth drive located front roller set control differential figure ctu roller rig real left ptc model right problem state feedback cascade pid control work method lateral skew control via state feedback cascade pid setup introduced presented setup linearization roller rig used via translation state space representation achieved via matrix dynamics however according theory linearization roller rig system appeared uncontrollable unobservable via analysis common linear algebra control approaches figure control loop linearization roller rig state feedback cascade pid control figure figure depicts control scheme setup numerous testing found simulation step approximately necessary order achieve stability continuous control loop used practical application hence reason alternative approach necessary lateral skew control railway wheel set thus paper aims investigating possibility adaptive identification control stand based measured data purposes paper data generated via simulation model via software simpack linked via simat toolbox matlab simulink provide real time simulation roller rig plant system investigated control approaches data used training data neural network models simplify otherwise complex simulation models allow constant sampling real time usability furthermore investigation potentials based control applied methods section various approaches aimed adaptive identification control previously introduced lateral skew problem described approaches based well know gradient descent method used tool defining learning rule applied neural units applied neural units adaptive models trained via two methods focused paper methods namely method incremental training real time recurrent learning rtrl applicable dynamic adaptive models batch form training variation back propagation time bptt training extension rtrl training combination famous levenbergmarquardt equation preliminaries subsection aims review important theories works behind method structures neural units used within paper firstly begin introducing famous algorithm linear quadratic neural units understand must begin polynomial models linear quadratic lnu qnu neural units respectively follows represents output lnu qnu respectively regards lnu stands updatable vector neural weights represents input values engineering system case purely static model sense dynamic model combination inputs real system neural model outputs looking equation rowx representation utilised input vector colw represents weight matrix quadratic neural unit general may understand algorithm applied neural units colw colw equation depicts algorithm lnu qnu respectively output algorithm incrementally update neural weights adaptively teach lnu qnu model learn engineering system represents learning rate algorithm analogical humans setting relatively high corresponds faster learning human consequently means less detail human remember digest learning furthermore setting parameter smaller value corresponds slowing rate learning human may remember information learned quite well less information overall subject next parameter representing number sample represents current error real calculated output model final term corresponds partial derivatives neural unit output respect individual neural weights regarding qnu equation see algorithm analogical exception updating matrix neural weights opposed vector sense lnu till structures lnu qnu reviewed format rtrl method learning neural weights updated new sample engineering system data however certain engineering processes advantageous use bptt batch method training neural weights runs algorithm rather sample rtrl method focuses contemporary governing law system opposed bptt focuses main governing law input outputs engineering system bptt method achieved via extension rtrl method combination famous levenbergmarquardt equation also important note method preferable cases data may affected noise following equation denotes weight update rule bptt method neural weights updated run algorithm batch trained following way equation describes change necessary update batch trained weights represents jacobian matrix derivatives neural unit may complete partial derivatives neural model respect neural weights practical applications seems useful simply introduce jacobian matrix input vector matrix colx lnu qnu respectively furthermore important note colx rowxt often adaptive neural units apparent modification normalised learning rate may used solve issues associated instability learning practise possible employ simplified normalised learning rate presented work follows equation represents normalised learning rate sense lnu analogically represented qnu replacement sample input vector colx noted representation holds rtrl training dynamic adaptive models algorithms used paper bptt method training take learning rate adaptive identification control previous section focused usage neural units sense adaptive identification real engineering system subsection extend neural units method control brief review works extension applied lateral skew problem indeed focus within paper figure adaptive identification supervised learning neural networks figure figure depicts identification scheme previously reviewed neural network architectures scope paper simulation real roller rig used data generation investigated neural network approaches control represents input data roller rig case yaw torque servo motor system manipulating lateral skew wheel set output yreal simulated output real time simpack model output neural unit difference error figure adaptive control loop experimental study neural network controller figure shows extension discussed neural architectures application lateral skew control roller rig system neural unit model identified may utilised foundation neural network based control setup scope paper propose offline tuned control scheme goal investigate potentials applying neural network based control application however extension online control indeed ultimate aim research problem beyond paper figure depicts use second neural unit controller similarly previously mentioned architectures controller neural unit may take shape lnu qnu adaptive models however case adaptive neural weights tuned differently used neural unit model hence depicted analogically qnu structure applied neural weights would represented matrix form figure figure introduced new vector vector comprises combination outputs neural unit model difference desired behaviour case desired yaw torque lateral skew roller rig output neural model collectively variable thus serves manipulator newly feed samples neural unit identified model algorithms employed following manner achieve adaptation neural weights controller follows adaptable neural weights neural unit controller ereg error desired value real system case roller rig desired value denoted real system output value sample partial derivative output neural unit model respect individual adaptive neural weights neural unit controller extension weight update scheme bptt training would result following form change neural weights batch would analogical equation experimental analysis section aims analyse previously reviewed methods adaptive identification control via introduced neural network architectures particularly aim investigate applicability discussed neural architectures problem lateral skew control railway wheel sets section simulate results using simulation data previously described simpack model speed linked matlab simulink real time simulation data produced sampling second interval first analyse results identification roller rig system via different methods neural network models followed extension control various combinations neural network architectures tested dlnu rtrl almost exact identification roller rig data lines superimposed deviation model roller rig data sample figure testing adaptive identification plant roller rig represented blue colour green neural model trained dlnu rtrl training epochs previous samples model output previous samples real system input figure figure following figure figure illustrates adaptive identification process roller rig system data note methods dynamic lnu dlnu dynamic qnu dqnu rtrl learning methods achieved almost exact identification dqnu performed slightly faster terms convergence minima sum square errors opposed dlnu dqnu rtrl excellent identification slightly faster learning dlnu parameters error model still within excellent range sum square errors decreases quickly dqnu parameters figure testing adaptive identification plant roller rig represented blue colour green neural model trained dqnu rtrl training epochs previous samples model output previous samples real system input desired black dashed roller rig system data blue neural controller magenta applied identified roller rig data error plot illustrates deviation neural controller desired behaviour sum square errors reaches close epochs figure testing adaptively tuned control loop roller rig represented trained dlnu constant previously trained parameters figure adaptive feedback controller trained via lnu epochs adaptive identification dlnu rtrl neural unit controller lateral skew control roller rig system using lnu incremental training data sample epochs previous samples neural model output previous samples difference neural model output desired behaviour system closer desired behaviour system qnu bptt difference neural controller desired behaviour within sum square errors reaches close epochs qnu bptt training figure testing adaptively tuned control loop roller rig represented trained dqnu constant previously trained parameters figure adaptive feedback controller trained via qnu epochs adaptive identification dqnu rtrl neural unit controller lateral skew control roller rig system using qnu bptt training figure figure show application various neural units control lateral skew black dotted line illustrates chosen desired behaviour roller rig lateral skew found bptt training method used combination qnu best performance providing closer control desired behaviour compared incremental training methods lnu qnu architectures desired lateral skew black dashed line set ideal position zero model provides almost exact response desired first samples figure testing adaptively tuned control loop roller rig represented trained dqnu constant previously trained parameters figure adaptive feedback controller trained via qnu epochs adaptive identification dqnu rtrl neural unit controller lateral skew control roller rig system using qnu bptt training figure depicts theoretical test roller rig system idealised case zero lateral skew included results investigate capability used neural controller figure see several small samples beginning controller application controller provides almost exactly zero lateral skew result however must reasoned really possible terms actuation real system real conditions rig achieve ideal result discussion experimental results previous section experimental results various neural network architectures adaptive identification control presented first set figures regarding adaptive identification roller rig system found suitable learning method dlnu dqnu rtrl training method relatively epochs adequate architectures identify system data almost exact identification dqnu model slightly faster learning behaviour system thus used identified neural unit model control section extension various architectures control showed distinguishable results application lnu qnu incremental learning neural controller possible higher number epochs run higher learning model small learning rate runs algorithm unable match desired behaviour even moderately however first samples application figure reasonable behaviour controller achieved combination identification via dqnu followed extension qnu trained bptt neural controller showed follow closely desired behaviour roller rig system tested combinations setup qnu bptt neural controller featured input vector comprised several errors differences desired behaviour output neural unit model training substantially quicker incremental training method showing several epochs sufficient model achieve optimal behaviour respect desired system results two forms desired behaviour used first proposal lateral derivation wheel set would within small tolerance range realistic real application control presented desired data relative small range lateral skew simulation data within exampled range could sure wheel flanges would contact rails real roller rig desired behaviour however relative setup wheel sets rails thus desired deviation lateral skew outputs may indeed vary real railway applications second idealised situation lateral skew would occur principally unrealistic neural controller achieve zero lateral skew due combination natural factors real railway wheel sets however demonstrational purposes may note capability neural controller providing almost exact behaviour idealised lateral skew applied identified roller rig system conclusion referring back original control loop depicted figure presented work simulink model linearization roller rig system state feedback cascade pid controller recall minimum sampling seconds numerical stability necessary possible practical application investigation presented neural network approaches may conclude promising potential real application controlling lateral skew irw railway wheel sets note achieved sampling within order seconds adequate functionality neural network based adaptive identification control system applied roller rig tested models paper dlnu dqnu adaptive identification models lnu qnu architectures controller experimental results architectures showed dlnu dqnu precisely approximated complex simpack model roller rig system result however could vary using real training data measured rig terms control lateral skew qnu architecture showed better results keep lateral skew within close limits desired behaviour system also general consensus previous findings using qnu control also note significantly faster tuning control loop via qnu compared lnu particularly bptt training method applied controller compared incremental method desirable control shown much fewer epochs runs controller algorithm thus case study shows proper tuning investigated neural units application real system theoretically possible achieve adequate control investigated problem thus real application adaptive control loop neural network based architecture indeed next step research acknowledgements authors would like acknowledge following grant support work cognitive signal processing methods dynamic systems also technology agency czech republic project competence center railway vehicles references kalivoda bauer curving behaviour bogie independently rotating wheels simulations scaled roller rig tests international symposium dynamics vehicles roads tracks qingdao china august bruni godall mei tsunashima control monitoring railway vehicle dynamics vehicle system dynamics vol issue gupta bukovsky homma solo hou fundamentals higher order neural networks modeling simulation artificial higher order neural networks modeling simulation zhang igi global bukovsky bila gupta hou homma foundation classification nonconventional neural units paradigm nonsynaptic neural interaction discoveries breakthroughs cognitive informatics natural intelligence within series advances cognitive informatics natural intelligence acini wang igi publishing hershey usa isbn bukovsky homma smetana rodriguez mironovova vrana quadratic neural unit good compromise linear models neural networks industrial applications icci ieee international conference cognitive informatics tsinghua university beijing china july bukovsky redlapalli gupta quadratic cubic neural units identification fast state feedback control unknown dynamic systems fourth international symposium uncertainty modeling analysis isuma ieee computer society maryland usa isbn rodriguez bukovsky homma potentials quadratic neural unit applications international journal software science computational intelligence ijssci vol issue igi global publishing hershey usa issn bukovsky lepold bila quadratic neural unit network validation process data steam turbine loop energetic boiler wcci ieee int joint conf neural networks ijcnn barcelona spain williams zipser learning algorithm continually running fully recurrent neural networks neural computation vol backpropagation time proc ieee vol peter benes software application adaptive identification controller tuning student conference stc faculty mechanical engineering ctu prague ladislav smetana nonlinear automatic control laboratory system master thesis czech tech univ prague laboratory system batyscaphe automatic control laboratory dpt instr cont fme czech technical university prague online http short biography authors peter mark received bachelor degree honours czech technical university ctu prague currently master student expected phd studies follow research focuses neural networks adaptive identification control industrial systems including hoist mechanisms skew control mechanisms cranes railway vehicles peter work awarded local international student competitions also industrial bosch award cejnek received bachelor degree czech technical university ctu prague currently master student czech technical university prague research focuses neural networks adaptive systems novelty detection time series biomedical applications matous work awarded local international student competitions jan kalivoda graduated czech technical university ctu prague received master degree honours field machines equipment transportation currently teacher active researcher department automotive combustion engine railway engineering ctu research interests include mbs models railway vehicles mechatronics railway vehicles active control railway vehicle suspensions wheel sets ivo bukovsky graduated czech technical university prague ctu received field control system engineering became associate professor since research interests include neural networks adaptive evaluation time series systems approaches control biomedical applications visiting researcher university saskatchewan university manitoba canada visiting professor tohoku university
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sequential coding sources jun anatoly khina ashish khisti victoria kostina babak hassibi consider problem sequential transmission sources show limit large spatial block lengths greedy compression respect squared error distortion optimal tension optimizing distortion source current time instant future times extend result case time random compression rate allocated independently rate time instants turn allows derive optimal performance sequential coding channels instantaneous feedback case packet erasures delayed feedback connect problem compression side information known encoder may known decoder recent packets serve side information may erased conclude paper demonstrating loss due delay one time unit rather small index coding correlated sources successive refinement source streaming packet erasures source coding side information ntroduction sequential coding sources increasingly finding applications video streaming cyberphysical networked control systems use compressed packetbased transmission strive achieve minimum distortion given compression rates setting introduced treated case viswanathan berger users special case sources explicit expression achievable given distortions derived extended general jointly gaussian case practice however protocols prone erasures possible delays case sequential coding presence packet erasures treated various erasure models case first packet prone erasure considered general approach trades performance given previously sent packets performance given last packet proposed random independent identically distributed packet erasures hybridation modulation pcm differential pcm dpcm termed leaky dpcm proposed analyzed case low erasure probability scenario erasures occur bursts considered sequence source vectors sampled khina kostina hassibi department electrical engineering california institute technology pasadena usa khina vkostina hassibi khisti department electrical computer engineering university toronto toronto canada akhisti process temporal dimension must encoded sequentially reconstructed zero delay decoder channel introduces burst erasures certain maximum length decoder required reconstruct sequences fall erasure period recovery window following works assume feedback available encoder namely encoder know whether transmitted packet successfully arrives decoder erased process paper first consider problem sequential coding sources determine distortion region large frames specifically show greedy quantization optimizes distortion time also optimal minimizing distortion future time instants insight allows extend result case compression rate available transmission packet time determined prior transmission channel instantaneous output feedback viewed special case noiseless channel random rate allocation corresponding event optimal region sequential coding sources presence packet erasures instantaneous output feedback thereby follows simple particularization general result tackle challenging delayed feedback setting encoder know whether recently transmitted packets arrived viewing recent packets side information available encoder possibly decoder leveraging results kaspi along specialization gaussian case perron adapt transmission scheme case delayed feedback provide detailed description proposed scheme case feedback delayed one time unit demonstrate loss compared case instantaneous feedback small roblem tatement present model source channel admissible encoder decoder required causal work see fig throughout paper denotes euclidean norm random variables denoted letters scenario considered also viewed special case results heegard berger available encoder adjusting distortion measure augmenting source interestingly knowing encoder allows one improve optimal performance scenario gaussian case see rem encoder bits decoder define distortion assuming limit exists lim fig sequential coding source setup temporal subscripts random vectors frames length boldface possibly accented letters denote temporal sequences temporal subscripts set natural numbers notations represent deterministic scalars assume communication spans time interval source consider source whose outcomes vectors frames length samples along spatial dimension satisfy temporal markov relation known process coefficients satisfy outcomes along spatial dimension gaussian mutually independent across time zero mean variances assume convenience denote kst average power entries vector obtain following recursive relation channel time packet sent noiseless channel finite rate causal encoder sees time applies causal function entire source sequence observed generate packet causal decoder applies causal function sequence received packets construct estimate time distortion error distortion time defined kst specialize source process fixed parameters namely power converges definition region region closure achievable distortion tuples rate tuple however large inverse region iii istortion ate egion equential oding auss arkov ources optimal achievable distortions given rates model sec provided following theorem theorem region region sequential coding rate tuple given distortion tuples satisfy remark establishes optimal region causal decoder setting ishwar case sources note ishwar provide explicit result sumrate case torbatian yang extend result case three jointly gaussian sources necessarily constitute markov chain work hand fully characterizes distortion region case sources remark results proof provided sequel imply optimal greedy quantization every step achieved via gaussian backward forward channels becomes optimal large moreover achieves optimum simultaneously meaning tension minimizing current distortion future distortions prove theorem first construct optimal greedy scheme determine performance sec show fact globally optimal goes infinity constructing outer bound scenario sec achievable construct inner bound using optimal greedy scheme scheme quantizers assumed minimum mean square error mmse quantizers note quantized values quantizers uncorrelated resulting quantization errors scheme encoder time generates prediction error defined previous source reconstruction decoder linear recursive relation provided sequel generates quantized reconstruction prediction error quantizing using optimal mmse quantizer rate frame length sends channel decoder time receives recovers reconstruction prediction error generates estimate optimal achievable distortions scheme long frame lengths follows assertion inner bound let however small expected distortion scheme time satisfies recursion large enough proof first note error denoted equal assertion outer bound consider setting sec average achievable distortion time bounded satisfies equality proof let shall prove since independent average power entries equal using property function mean square error distortion source given variance upper bounded white gaussian source variance see obtain following recursion hence achievable within arbitrarily small sufficiently large mmse estimators respectively given past channel outputs induction sequence defined denote conditional given expectation respect random vector distributed basic step first note since vector consists gaussian entries variance satisfied equality prove use fact optimal achievable distortion gaussian source entries power rate dictated function rdf follows thus distortion also distortion reconstructing using express shall construct outer bound coincides inner bound assert large frame lengths impossible converse inductive step let suppose true shall prove holds also ksk ksk ksk follows law total expectation holds since distribution follows bounding inner expectation conditional distortion function shannon lower bound also proves due follows inequality holds since gaussian scaling property differential entropies jensen inequality andom ate udgets assertion outer bound noise consider setting sec independent noise entries average achievable distortion time bounded given recursion proof proof identical assert replaced steady state asymptotically stationary sources asymptotically stationary source steadystate average distortion follows corollary steady state let however small minimum distortion equal induction hypothesis holds definition sequence satisfies also proves concludes proof desired hence remark evident proof result corol remains true initial value follows following standard set inequalities converges exponentially fast equivalently large enough section generalize results sec iii random rates independent rate revealed encoder transmission time theorem region region sequential coding independent rates given distortion tuples satisfy remark immediate consequence theorem jensen inequality using packets fixed rate equal performs better using random rates proof achievable since achievability scheme use knowledge future transmission rates encode decode packet time kst kst taking expectation respect using independence achieve impossible revealing rates encoder decoder prior start transmission improve distortion thus distortions conditioned bounded taking expectation respect attain desired result special case asymptotically stationary source distortion given follows corollary steady state assume rates proof note fixed point let however small minimum since converges state distortion equal easily proved follows assume otherwise already fixed point large enough recall gaussian setting var proof note fixed point since converges easily proved follows assume otherwise already fixed point multiple packets per frame equivalently hence converges exponentially fast packet rasures nstantaneous eedback one packet per frame important special case budget model sec packet erasures since packet erasure time viewed assuming encoder sends packets fixed rate cognizant packet erasures instantaneously packet erasure channel cast random rate channel sec events corresponds successful arrival packet time means erased denote received output corresponds erasure otherwise assume according ber distribution remark shall concentrate case packets fixed rate simplify subsequent discussion way randomness rate comes effect nevertheless results follow easily extended rate allocations effect packet erasures added manner corollary region region sequential coding packet erasures instantaneous feedback given proof computing expectation obtain desired corollary steady state distortion given corol sec assumed one packet sent per source frame instead one may choose transmit multiple packets lower rate per one source frame repetition packet trades diversity multiplexing case potentially improve overall performance improvement scheme proposed repetitive transmission single compressed description replaced multiple descriptions compression assume availability perfect instantaneous feedback packet improvement achieved noting scenario falls randomrate budget framework sec specifically assume use packets equal rate hence total rate rate probability distribution amounts denoting number successful packet arrivals time corresponding source frame assuming erasure events packets probability implies according binomial distribution interestingly optimal number packets depends total rate packet successful arrival probability determined number minimizes demonstrated fig remark considered uniform rate allocations packets clearly one generalize approach packet rates remark practice one might expect longer packets prone higher erasure probability taken account deciding minimizes packet rasures elayed eedback section consider case packet erasures output feedback case time encoder know whether last packet arrived know knows erasure pattern preceding packets knows encoder decoder mappings written recall definition end recall following result perron specialization jointly gaussian rate required achieve distortions given rkaspi log log log denotes harmonic mean akb fig evaluation packets possible values two total rates case result kaspi established region lossy compression may may available remark kaspi result also viewed special case adjustments see theorem let gaussian source power jointly gaussian available encoder satisfies gaussian noise power independent denote reconstructions without mean squared error distortion requirements respectively smallest use backward channel represent opposed forward channel used remark surprisingly observed perron signal available encoder corresponding case considered required rate strictly higher stark contrast case never available encoder case sideinformation always available decoder studied wyner ziv knowing encoder allows correlate noise quantization error thing possible available encoder two noises must independent case allows improvement though modest one implied results dual channel problem prop case time previous packet serve note always available encoder decoder may may access depending whether previous packet arrived since feedback delayed transmission current packet encoder know whether previous packet lost tradeoff given rate determined probability successful packet arrival scheme encoder time generates prediction error generates quantizing prediction error available encoder possibly decoder depending using optimal quantizer rate frame length minimizes averaged distortion dtweighted precisely since encoder know time denote reconstruction decoder namely given corresponding distortion dtweighted denote reconstruction namely given corresponding distortion denote reconstruction namely given corresponding distortion encoder sees possible available decoder minimize dtweighted sends channel decoder time receives generates reconstruction prediction error generates estimate scheme optimal greedy scheme whose performance stated next limit large theorem let however small large enough expected distortion scheme time given satisfies recursion fig distortions function time various schemes presented section along scheme sec derive performance special case asymptotically stationary source prediction scheme use prediction source samples independent scheme achieves distortion distortions minimize dtweighted rkaspi proof ths generated remark contrast case instantaneous feedback evaluating average distortions explicit form recall corol much challenging numerically instead somewhat surprisingly loss performance scheme due feedback delay rather small compared scenario sec feedback available instantaneously values demonstrated fig perfomances schemes compared along performances following three simple schemes values close loss becomes even smaller cases using scheme sec assumes previous packet arrived erased respectively becomes optimal power entries given assumes worst case since time encoder know safe way would work achieves distortion rate satisfies assumes best case optimistic counterpart previous scheme always works scheme achieves distortion vii iscussion eedback arger elays extend scheme sec larger delays generalization needed unfortunately optimal region two decoders remains open problem known case source possible sis form markov chain degraded nonetheless achievable regions multiple decoders proposed used construction schemes accommodate larger delays acknowledgment authors thank caltech valuable discussions eferences viswanathan berger squential coding correlated sources ieee trans inf theory vol ishwar delayed sequential coding correlated sources ieee trans inf theory vol erratum delayed sequential coding correlated sources ieee trans inf theory vol june yang zheng rate distortion theory causal video coding characterization computation algorithm comparison ieee trans inf theory vol information theoretic performance comparison causal video coding predictive video coding ieee trans inf theory vol mar torbatian yang causal coding multiple jointly gaussian sources proc annual allerton conf control monticello usa eslamifar causal video coding possible loss first encoded frame master thesis university waterloo waterloo ontario canada song chen wang liu gaussian robust sequential predictive coding ieee trans inf theory vol june huang peng chiang advances scalable amendment ieee comm magazine vol huang kochman wornell causal transmission colored source frames packet erasure channel proc data comp conf dcc snowbird usa mar etezadi khisti trott sequential transmission markov sources burst erasure channels ieee trans inf theory vol etezadi khisti chen truncated prediction framework streaming erasure channels ieee trans inf theory submitted jul revised minero franceschetti dey nair data rate theorem stabilization feedback channels ieee trans auto control vol kaspi may present decoder ieee trans inf theory vol perron diggavi telatar role encoder sideinformation source coding multiple decoders proc ieee int symp inf theory isit seattle usa july heegard berger side information may absent ieee trans inf theory vol khina erez source coding composite side information decoder proc ieee conv electrical electron engineers israel ieeei eilat israel cover thomas elements information theory second edition new york wiley tse viswanath fundamentals wireless communication cambridge univ press ostergaard quevedo multiple descriptions packetized predictive control eurasip advances sig vol apr witsenhausen source networks minimal breakdown degradation bell sys tech vol wolf wyner ziv source coding multiple descriptions bell sys tech vol ozarow source coding problem two channels three receivers bell sys tech vol gamal cover achievable rates multiple descriptions ieee trans inf theory vol wyner ziv function source coding side information decoder ieee trans inf theory vol wyner function source coding side information decoder general sources information control vol zamir erez gaussian input bad ieee trans inf theory vol jun philosof zamir cost uncorrelation noncooperation mimo channels ieee trans inf theory vol
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reuse neural modules general video game playing alexander braylan mark hollenbeck elliot meyerson risto miikkulainen dec department computer science university texas austin braylan mhollen ekm risto abstract general approach knowledge transfer introduced agent controlled neural network adapts reuses existing networks learns new domain networks trained new domain improve performance routing activation selectively previously learned neural structure regardless learned neuroevolution implementation approach presented application sequential domains approach general previous approaches neural transfer reinforcement learning requires prior assumptions nature task relatedness mappings method analyzed stochastic version arcade learning environment demonstrating improves performance complex atari games success transfer predicted based characterization game dynamics introduction ability apply available previously learned knowledge new tasks hallmark general intelligence transfer learning process reusing knowledge previously learned source tasks bootstrap learning target tasks sequential control domains robotics video transfer particularly important previous experience help agents explore new environments efficiently taylor stone konidaris scheidwasser barto knowledge acquired previous tasks also contains information agent decision making learning dynamics thus useful even domains seem unrelated existing approaches transfer learning domains demonstrated successful transfer varying kinds knowledge make two fundamental assumptions restrict generality sort priori humandefined understanding tasks related separability knowledge extraction target learning first assumption limits well approach applied restricting use cases agent copyright association advancement artificial intelligence rights reserved provided additional relational knowledge learned talvitie singh taylor kuhlmann stone ammar cases task mappings useful second assumption implies known knowledge useful incorporated learning target task begins preventing agent adapting way uses source knowledge gains information target domain general reuse static modules grusm proposed paper general neural network approach transfer learning avoids assumptions grusm augments learning process allow learning networks route existing neural modules source networks selectively simultaneously develop new structure target task unlike previous work dealt mapping task variables source target grusm knowledge structure source domain even knowledge network came required reused instead using mappings facilitate transfer searches directly mappings solution space connections existing source networks target network approach motivated studies shown naturally occurring complex networks milo artificial neural networks swarup ray certain network structures repeat useful across domains without context exactly structure used work motivated idea neural resources human brain reused countless purposes varying complex ways anderson paper implementation grusm based enforced subpopulations esp neuroevolution framework gomez miikkulainen presented approach validated stochastic atari general game playing platform finding improves learning complex target games improvements may predicted based domain complexity features result demonstrates even without traditional transfer learning assumptions successful knowledge transfer via general reuse existing neural modules possible useful sequential control tasks principle approach scales naturally transfer arbitrary number source tasks suggests future may possible build grusm agents accumulate reuse knowledge throughout lifetimes across variety diverse domains background transfer learning encompasses machine learning techniques involve reusing existing source knowledge different target task domain domain environment learning takes place characterized input output space task particular function input output learned pan yang sequentialdecision domains task characterized values sequences corresponding pursuit given goal taxonomy types knowledge may transferred also enumerated pan yang grusm approach reuses structure existing neural networks falls feature representation transfer transfer learning transfer learning sequential domains studied extensively within reinforcement learning paradigm taylor stone reinforcement learning domains often formulated markov decision processes mdps state space comprises possible observations probability observation depends previous observation action taken learning agent however many real world domains including many atari games example velocity moving object determined looking single frame atari platform also supports wide variety games existing approaches transfer differ types differences allowed source target task approaches general respect kind knowledge transferred restricted require consistent konidaris scheidwasser barto priori specification mappings defining relationships source target state action variables brys existing approaches transfer learning encode policies neural networks require specification taylor whiteson stone verbancsics stanley hand existing modular neuroevolution approaches general respect connectivity reisinger stanley miikkulainen khare applied transfer general existing approaches transfer automatically learn task mappings need provided beforehand approaches general enough apply reinforcement learning domains initial approaches taylor kuhlmann stone talvitie singh intractable high dimensional state action spaces due combinatorial blowup number possible mappings however recent approaches policy gradient ammar tractably learn mappings applied across diverse domains approaches successful continuous control domains unclear would scale domains many features atari also approaches assume mdp environments whereas grusm use recurrent neural networks extend pomdps general neural structure transfer existing algorithms similar grusm make possible reuse existing neural structure apply wide range domains tasks automatically select source knowledge avoid mappings example shultz rivest developed technique build increasingly complex networks inserting source networks chosen much reduce error technique applicable supervised learning source selection depends heavily immediate error calculation also connectivity source target networks limited input output layer source another example swarup ray introduced approach creates sparse networks primitives commonly used mined library source networks subgraph mining approach depends computationally expensive graph mining algorithm tends favor exploitation innovation small primitives rather larger networks sources grusm approach general applied unsupervised reinforcement learning tasks makes priori assumptions kind sources mappings work best able develop memory via recurrent connections although evolutionary approach developed paper grusm extensible neural learning algorithm approach section introduces general idea behind grusm provides overview esp neuroevolution framework describe particular implementation general reuse static modules grusm underlying idea agent learning neural network target task reuse knowledge selectively existing neural modules source networks simultaneously developing new structure unique target task approach attempts balance reuse innovation integrated architecture source networks new hidden nodes termed recruits recruits added target network learning process recruits incorporated adaptively target network learns connection parameters target recruit recruit target internal structure source networks frozen allow learning connection parameters remain consistent across recruits mechanism forces target network transfer learned knowledge rather simply overwrite connections source networks general case connect nodes source target minimizing assumptions knowledge useful grusm network traditional neural network feedforward recurrent containing new nodes connections unique target task input output nodes corresponding inputs outputs defined target domain possibly empty set pointers recruited source networks set weighted transfer connections nodes nodes source networks connection construction strictly extends traditional neural networks traditional neural network grusm network evaluated network induced directed paths inputs outputs including pass via connections evaluated evaluation recruited source network inputs fixed since agent concerned performing current target task parameters learned usual parameters along contents internal parameters frozen rewritten motivation architecture solution source task contains information relevant solving target task neural network constructed source task contain structure subnetwork module useful target network previously observed naturally occurring complex networks milo well artificial neural networks swarup ray unlike approach neural structure transfer swarup ray general formalism makes assumptions subnetworks actually useful one interpretation lifelong learning agent maintains system interconnected neural modules potentially reuse time new task even existing modules unlabeled may still useful due fact contain knowledge agent successfully learn furthermore advances reservoir computing jaeger demonstrated power using large amounts frozen neural structure facilitate learning complex chaotic tasks formalism general enough allow arbitrary number source networks arbitrary connectivity source target paper validate approach simplify analysis one source network used time connections target input source hidden layer source output layer target output permitted allowing target input connect source input restriction avoids transformations sensor substrates intuitively captures goals approach differentiating approach previous methods used direct mappings sensor spaces restriction sufficient show implementation reuse hidden source features successfully possible analyze cases transfer useful future refinements discussed discussion future work section current implementation described neuroevolution approach based esp enforced subpopulations esp enforced esp gomez miikkulainen neuroevolution technique different components neural network evolved separate subpopulations rather evolving whole network single population esp shown perform well variety reinforcement learning domains shown promise extending pomdp environments use recurrent connections memory critical gomez miikkulainen gomez schmidhuber schmidhuber traditional esp single hidden layer neuron evolved subpopulation recombination occurs members subpopulation mutants subpopulation derive members subpopulation genome individual subpopulation vector weights corresponding weights connections neuron including node bias generation networks evaluated randomly constructed inserting one neuron subpopulation individual participated network receives fitness achieved network fitness converges improve several consecutive generations esp makes use burst phases initial burst phases subpopulation repopulated mutations single best neuron ever occuring subpopulation reverts searching around best solution found far second consecutive burst phase reached improvements made since previous burst phase new neuron new subpopulation may added gomez idea enforced extended transfer learning via grusm networks reused source network transfer connections evolve distinct subpopulation time new hidden nodes added evolve within subpopulations manner standard esp way integrated evolutionary process simultaneously searches space reuse potential source network innovate new node grusmesp architecture figure composed following elements pool potential source networks experiments paper target network reuses one source time transfer genomes encoding weights connections source target potential source network pool subpopulation evolving transfer genomes target network connection contained transfer genome experiments transfer connections included target inputs fully connected source hidden layer source outputs fully connected target outputs burst mechanism determines innovation necessary based recent history performance improvement new hidden recruits source networks available single nodes otherwise added burst phase evolve within subpopulations standard esp figure architecture showing balance reused new structure example target network three recruits one source network two single nodes bold edge target network nodes source network recruit indicate connections multiple source nodes genome subpopulation encodes weight information connections corresponding recruit hidden output neurons use hyperbolic tangent activation function networks include single hidden layer include recurrent self loops hidden nodes otherwise feedforward details genetic algorithm implementation used evolve subpopulation mirror described gomez algorithm shown work well within esp framework though suitable evolutionary algorithm could potentially substituted place preliminary experiments using approach discussed braylan experiments evaluated stochastic version atari general video platform using arcade learning environment simulator ale bellemare atari currently popular platform challenges modern approaches contains games entertained generation human video game players would regularly reuse knowledge gained previous games playing new games make simulator closely resemble human experience action approach suggested hausknecht stone used paper make environment stochastic manner like human players algorithm easily find loopholes deterministic nature simulator recommended used note vast majority previously published atari results deterministic setting unaware existing scores published setting agents trained play eight games pong breakout asterix bowling freeway boxing space invaders https seaquest neuroevolution techniques competitive atari platform hausknecht esp particular yielded performance several games braylan three conditions evaluated scratch transfer random scratch condition networks trained scratch game using standard esp transfer condition scratch network reused source network training new grusm networks different target games random control condition random networks initialized reused source networks networks contain average number parameters scratch networks run lasted generations evaluations per generation since environment stochastic evaluation consists five independent trials individual playing game resulting score average scores across trials score evolutionary run given generation highest achieved individual generation total runs run split across possible setups evolutionary parameters selected based success standard esp interface ale output layer network consists substrate representing nine directional movements atari joystick addition single node representing fire button input layer consisted series object representations manually generated previously described hausknecht location object screen represented input substrate corresponding object class numbers object classes varied one four although object representations used experiments vision could also learned scratch neuroevolution process via convolutional networks done schmidhuber gomez domain characterization understanding transfer useful important transfer learning approach many cases attempting transfer impede learning leading negative transfer approach able successfully adapt knowledge source target domain negative transfer serious concern many practitioners taylor stone pan yang help understand applied useful consider diverse array games within unified descriptive framework biological neural reuse generally thought useful transferring knowledge simple behaviors complex vast majority previous computational approaches exactly thus characterization games paper grounded sense relative complexity game characterized generic binary features determine successful game play requires horizontal movement joystick vertical movement joystick shooting fire button delayed rewards planning intuitively complex games include features partial ordering games complexity defined none pong breakout asterix bowling freeway boxing invaders seaquest vertical movement horizontal movement shooting delayed rewards planning pong breakout bowling freeway asterix boxing space invaders seaquest figure left feature representation game right games feature inclusion every path none contains along edges complexity feature exactly showing games related across feature space existence hierarchy motivates use atari transfer features shown figure assignment features completely defined based game interface bellemare freeway seaquest said delayed rewards high score achieved long sequences rewardless behavior space invaders seaquest deemed require planning mnih since dynamics games penalize reflexive strategies agents games perform well low frequency braylan addition intuitive features validated based well characterize games complexity well predict successful transfer analysis methods many possible metrics evaluating success transfer depending kind transfer desired expected learning curves irregular across different games illustrated figure makes difficult choose single metric makes sense across pairs thus analysis focused broad notion transfer effectiveness aggregates metrics jumpstart max overall score weighted approximation area curve taylor stone success setup defined sum average score setup series generations series favors early performance later performance general long run training transfer scratch converge scratch eventually relearns everything effectively transferred setup success minus success control target game difference normalized size range max scores achieved across runs game order draw comparison across games first hypothesis transfer would outperform scratch setups setups could predicted coincidental however outperformance transfer scratch could due figure raw mean score learning curves generation target game aggregated transfer solid random dashed scratch dotted setups diversity learning curves shows difficulty comparing performance across games larger number network parameters therefore second hypothesis random setups used control number parameters test transfer could predictably outperform random postulated tested several useful indicators predicting outperformance transfer feature similarity count features source target source feature complexity feature count source game target feature complexity feature count target game source training complexity source game average time threshold target training complexity target game average time threshold threshold game minimum max score achieved across scratch runs game time threshold average number generations reach threshold predict linear regression model trained analysis possible pair model trained pairs dependent variable five indicators independent variables subsequently trained model used predict correlation actual predicted across test pairs used gauge predictability experiment conducted identically transfer versus scratch transfer versus random conditions results hypotheses model proved statistically significant predictor transfer effectiveness test data correlation pvalue transfer versus scratch correlation transfer versus random figure strongest indicators transfer versus scratch target feature complexity target training complexity transfer versus random strongest indicator target feature complexity fact complex games successful targets surprising noted transfer learning scenarios transfer considered ability predict work important tool applying method boxing breakout bowling asterix space invaders pong figure predicted actual transfer effectiveness respect scratch left random right predictors significant correlation predicated actual transfer effectiveness game pong breakout asterix bowling freeway boxing invad seaquest scratch random transfer human dqn table game average scores scratch random transfer best source subscripted interestingly best source target unique also show human dqn scores mnih note dqn uses deterministic ale apt external comparison may humans deterministically optimize trajectories frame level larger problems encouraging predictive indicator coincides common sense expectations transfer effectiveness current experiments pairs visualized figure also although difficult compare deterministic atari domain table provides comparison recent results domain context mnih discussion future work results show evolutionary algorithm general transfer neural network structure improve learning atari game playing reusing previously developed knowledge also make possible characterize conditions transfer may useful specifically improvement learning performance target domain depends heavily complexity target domain effectiveness transfer complex games aligns notion hierarchical knowledge representation argued previously transfer learning konidaris scheidwasser barto well biology anderson milo interesting investigate whether principles extend general video game playing platforms vgdl perez schaul work help understand subsymbolic knowledge recycled indefinitely across diverse domains freeway seaquest bowling asterix breakout boxing pong space invaders seaquest freeway figure transferability graphs pairs tasks respect scratch top random bottom illustrating clustering successful pairs graph includes directed edge see analysis reusing positive transfer likely inefficient simpler games due effort involved finding necessary connections reusing knowledge given source network effectively case efficient relearn scratch particular games also seen random consistently outperforms scratch transfer pong initial flexibility untrained parameters random condition may explain result unfreezing reused networks allowing change low learning rate may help close gap transfer pairs consistently outperform training scratch random indicating negative transfer highlights importance source target selection transfer learning results taken step towards answering problem kinds games make good targets transfer data across many games required answer problem given game sources used next step involve pooling multiple candidate sources testing ability exploit useful structure available despite negative transfer setups technique training classifier predict transfer success shown useful approach helping decide transfer given space complex disparate domains try transfer subset pairs use results build classifier inform attempt transfer future paper features provided features could learned analysis networks learning process interesting avenue future work another area future work involves increasing flexibility combined architecture relaxing requirement transfer connections allowing deeper architectures source target networks including multiple source networks adaptive connectivity extensions promote reuse subnetworks varying depth along flexible positioning combination modules however networks become large plentiful maintaining full connectivity layers become intractable necessary enforcing sparsity also extended include lstm units schmidhuber deep memory primary concern conclusion paper introduced approach general transfer learning using neural networks approach minimizes priori assumptions task relatedness enables flexible approach adaptive learning across many domains stochastic version atari general video gameplaying platform specific implementation developed paper boost learning reusing neural structure across disparate domains success transfer shown correlate intuitive notions domain complexity results indicate potential general neural reuse predictably assist agents increasingly complex environments acknowledgments would like thank ruohan zhang useful feedback research supported part nsf grant nih grant npsc fellowship sponsored nsa references ammar eaton luna ruvolo autonomous knowledge transfer lifelong policy gradient reinforcement learning proc ijcai ammar eaton ruvolo taylor unsupervised transfer policy gradient reinforcement learning via manifold alignment proc aaai anderson neural reuse fundamental organizational principle brain behavioral brain sciences bellemare naddaf veness bowling arcade learning environment evaluation platform general agents jair braylan hollenbeck meyerson miikkulainen frame skip powerful parameter learning play atari workshops braylan hollenbeck meyerson miikkulainen reusability neural modules general video game playing giga workshop brys harutyunyan taylor policy transfer using reward shaping proc aamas gomez miikkulainen incremental evolution complex general behavior adaptive behavior gomez miikkulainen solving control tasks neuroevolution proc ijcai gomez schmidhuber recurrent neurons learn deep memory pomdps proc gecco gomez robust control neuroevolution technical report austin hausknecht stone impact determinism learning atari games workshops hausknecht lehman miikkulainen stone neuroevolution approach general atari game playing ieee trans comp intelligence games khare yao sendhoff jin wersing modular neural networks automatic problem decomposition proc cec konidaris scheidwasser barto transfer reinforcement learning via shared features jmlr schmidhuber gomez evolving deep unsupervised convolutional networks reinforcement learning proc gecco jaeger reservoir computing approaches recurrent neural network training computer science review milo itzkovitz kashtan chklovskii alon network motifs simple building blocks complex networks science mnih kavukcuoglu silver rusu veness bellemare control deep reinforcement learning nature pan yang survey transfer learning ieee trans knowledge data engineering perez samothrakis togelius schaul general video game playing competition ieee trans comp intel games reisinger stanley miikkulainen evolving reusable neural modules proc gecco schaul video game description language modelbased interactive learning proc cig schmidhuber wierstra gagliolo gomez training recurrent networks evolino neural computation shultz rivest cascadecorrelation using knowledge speed learning connection science swarup ray knowledge transfer using structured representations proc aaai talvitie singh experts algorithm transfer learning proc ijcai taylor stone transfer learning reinforcement learning domains survey jmlr taylor kuhlmann stone autonomous transfer reinforcement learning proc aamas taylor whiteson stone transfer via intertask mappings policy search reinforcement learning proc aamas verbancsics stanley evolving static representations task transfer jmlr
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aug discretization orthogonal group using icosahedral symmetries golden numbers robert moody jun morita august department mathematics statistics university victoria victoria canada department mathematics university tsukuba tsukuba ibaraki japan rvmoody morita abstract vertices form root system whose corresponding symmetry group coxeter group two special coordinate representations root system corresponding coxeter groups involve rational numbers golden ratio two related conjugation paper investigates happens two root systems combined group generated versions allowed operate result new infinite root system turns natural structure unitary group ring called golden numbers acting upon naturally associated infinite reflection group prove index orthogonal group paper makes extensive use quaternions leads highly structured discretized filtration use offer simple effective way approximate element degree accuracy required using repeated actions five fixed reflections process may find application computational methods quantum mechanics keywords icosahedral symmetry root systems infinite reflection groups discretization golden numbers quaternions introduction symmetry group icosahedron dodecahedron icosahedral group denoted elements finite coxeter group say finite group generated reflections coxeter relations simply transitive simplicial cells icosahedron partitioned mirrors reflections apart numerous dihedral groups two finite indecomposable coxeter groups big sister symmetry group regular polytope called dual paper involves groups notably latter vertices faces see fig vertices form root system type interpretation vertices roots set reflections roots opposite roots give reflection entire set reflections group figure projection whose vertices form root system type dots make vertices constitute root system type generate order group notice use symbol adjective signifying type root system involved noun signifying reflection group generated roots thing involve golden ratio algebraic conjugate sorts significant ways instance model vertices equivalently roots root system type set points permutations roots roots even permutations roots first component label refers power appearing denominators components second component used distinguish rational versus irrational nature components one help noticing interesting fact even permutations allowed third type root half potential permutations missing half obtained conjugating roots interchanging throughout course produces another model root system type reflections generate second model coxeter group shall also use notation suggests origin groups reflection groups tempting look group generated reflections one quickly realizes group let call since points set lie sphere radius compact certainly discrete set one seems paid much attention objective paper get idea look like fact attractive features shall see surprisingly dense set points unit see prop also group quaternion multiplication shall show amalgamation group similarly amalgamation two groups especially interesting using allows explicitly approximate elements two groups unitary special orthogonal groups respectively elements matrix coefficients form call dyadic integers field fact matrices used arise reflections ones involving even reflections one gets level root system bases approximation made fine one wishes increasing powers two denominators simple efficient algorithm key interpret standard division ring quaternions use fact unit sphere identified made act subspace pure imaginary quaternions change sign quaternionic conjugation paper primarily concerned picture also require information corresponding essentially parallel situation based icosahedral root system corresponding conjugate system arguments identical sufficiently parallel provide sketches arguments case paper organized around understanding structure root system group reflections roots generate approximation results appear final section though read directly icosahedral symmetry coxeter groups continued intrigue people ancient times present day familiar diverse places carbon molecules buckyballs capsids outer shells viruses penrose tilings aperiodic order quasicrystals remains continuing interest mathematical world instance investigates subgroup structure quaternionic context explores ways making affine extensions root systems following success affinization root systems simple lie algebras affinization accomplished extending coxeter diagram might thought based using translations extend root systems present paper although root systems involved defined intrinsically root systems galois conjugation effectively result extend using contractions indeed contractive aspect something seems worthy fuller study might note considerable advances understanding coxeter groups along associated geometrical meanings using clifford algebras see associated references although used ideas indeed require throughout paper see parallel structures pertaining setting around setting around generally use symbol corresponding objects context make clear shall make distinction etc limited number tools clifford algebra approach may offer new insights setting introducing set roots root system type presented standard form union three types vectors let denote set roots types set roots form crystallographic root system type type course distinctions intrinsic root system coordinatization however distinction plays fundamental role follow along conjugate set conjugates corresponding irrelevant whether dot operators applied sets conjugation reflections roots generate group call coxeter group type similarly generated reflections roots also type primarily interested group generated together reflections given roots common systems generate subgroup type example base either choice sign let denote unit sphere set smallest subset containing closed reflections reflection note important fact start use algebraic consequences reflections applied context later interpret everything real quaternions elements discussing interpreted elements useful keep mind basic facts quotient ring galois field elements let denote natural homomorphism elements extend meaning map version inside space define subspace spanned elements taken modulo space basis define note cardinalities card card card already similarly using corresponding subset mentioned sentence suggests true throughout everything say come two versions interchanged conjugation henceforth usually state give proofs one versions understanding equally true figure coxeter diagram pairwise relationships reflection generators bonds two nodes indicate orders product reflections corresponding convention unmarked bonds indicate order bonds commuting reflections omitted completely proof establishing bond marked found cor shall see fact proper factor coxeter group diagram corresponding coxeter diagram relationships generators obtained deleting first node attached bond obvious dot product find totally isotropic another way addition linearly dependent saying simply follows parallel statements however always proposition particular particular reflection group set vectors define reflections correspond coxeter diagram fig reflections generate indeed contain set generators coxeter groups associated gram matrix entries positive semidefinite null vector certainly factor corresponding coxeter group later shall see proper factor see prop proposition satisfies mod one three following cases prevails congruent modulo congruent modulo congruent modulo permutation components language notation introduced three cases correspond proof modulo squares modulo quick check shows sum four elements zero modulo one three conditions thesis true examples three types occur double roots look corollary version prop reads satisfies mod one two following cases prevails congruent modulo congruent modulo permutation components define spherical sets similarly conjugated versions using call level spherical set define similarly add set roots type two consequences prop proposition proof let satisfies using prop one hunt solutions see elements satisfy restrict filter proposition furthermore proof let let minimal prop see reverse inclusion comes definitions last line clear proposition proof let definition elements respectively cases since however furthermore thus wished prove infinite order corollary proof let denote commutator subgroup let proposition dense dense iii dense proof rotation plane orthogonal two roots cor infinite cyclic group generated infinite order let orbit generated infinite hence dense circle sphere particular rotations angles small please rotation infinite order also arbitrarily small different plane orthogonal get second group dense subgroup group rotations two groups generate subgroup rotations since certainly map see also contains subgroup dense subgroup rotations space contains since generate entire rotation group see dense subgroup space iii clear proposition let exists iii three reflections form proof start slightly weaker assumption let let since elements let long prop either whatever case shown either assume possible either case reasoning sphere level contrary case level case noting prop see actually fact since elements change level application would impossible thus proves part last case level drops reflection particularly interesting wish look carefully happen iii notation part let also resulting however remains need show choose corresponding definiteness cases work way assume write begin though ultimately cycle last three components around question mean hold quivalently mean mod latter equivalent mod finally may happen cycle last three terms around get equation coefficients likewise cycled around adding three equations together using fact get contradiction thus least one three possibilities equation fails version produces iii suppose let required see happens easiest fix one particular form take choose forms even permutations seen loss generality even permutation referring assuming form question becomes looking first coordinate clear must one elements since overall sign change makes difference reflection defined equation trying solve replace simply find exactly one solution three sets instance taking second type mod mod mod mod various possibilities sign gives value looking fact two solutions obtained cyclically permuting last three coordinates solutions case three solutions three produce desired effect consequence saw concludes proof part iii worthwhile noting three vectors found plane orthogonal roots lie angles let ring shall call ring dyadic golden numbers proposition proof definition let proceed induction assume shown prop prop iii prop see thus however prop prop show side closed action reflections generate gives reverse reflections roots inclusion final equality see prop proposition card card similarly proceed induction proof consists roots simply note element reflections arising map images see prop however according prop thus elements produced exactly three times use reflections available card card likewise card completes induction step consequence prop prop deeper structure path leads finer finer need apply reflections alternately discretization unitary group situation described pictorially represented fig figure action reflections different types roots indicated diagramatically intersection common crystallographic part two spherical sets lateral actions apply similarly theory outlined far parallel version three dimensions around coxeter group features described case appear notationally distinguished additional suffix dealing settings together shall also use suffix situation order avoid confusion unit sphere consists points namely vectors permutations coordinates points permutations coordinates second set points restrict allowing even cyclic permutations provide points together original produce set points may viewed root system type vertices icosadodecahedron reflection group generated distinct reflections points called roots reflections mirrors planes origin orthogonal roots icosahedral group order base root system type use vectors instead even permutations conjugate thus taking even permutations equivalently take odd permutations thing end set conjugates elements root system type reflections roots generates copy icosahedral group write roots similarly let denote group generated reflections let orbit shall see dense set points unit put setting discussed considering space subspace clear indeed lies fact shall see later shall put everything setting algebra quaternions point come subspace pure quaternions let let denote natural homomorphism place wish use often convenient identify subspace useful using define analogues namely play role case written elements similarly define similarly dot product defined matches dot product subspace prop proposition particular particular obtain similar results mainly list proofs basically however one significant difference appears longer two spherical sets deal define similarly using call level spherical set sets relevant counterparts since add clear parallel statements consequence cor proposition furthermore also reflection taken restriction reflection taken restricted use symbol context distinguishing necessary corresponding primed notation used denote groups generated reflections respectively may think groups subgroups prop proposition infinite order corollary following arguments prop proposition dense dense iii dense define rather easy use results deduce corresponding results situation state without comment proposition proposition card card proof consists roots similarly proceed induction simply note element reflections arising map images see prop however according prop iii actually set make corresponding three dimensional case rather apparent elements produced exactly three times use reflections available thus card card likewise card completes induction step acts proposition acts transitively acts transitively iii set roots elements orthogonal translate element stabilizer contains subgroup conjugate proof suffices show contains basic roots happens coxeter diagram fig connected four others chains bonds odd namely labelled argument easily explained case bond labelled two roots left side also showing sign wrong use one reflection proof similar iii stabilizing reflections consists elements see prop parts iii follow directly instead thinking simply subsets view subsets quaternion algebra identify subspace using subspace pure quaternions similarly generally use interpretation henceforth equip usual conjugation changes signs last three nents vector usual dot product standard one expressed terms quaternionic multiplication well known unit sphere set vectors satisfying identified explicitly interpreted group usual matrix representation using returning ring define set unitary matrices coefficients using identifies set quaternions coefficients norm subgroup prop tells natural structure group namely furthermore since dense dense recall usually conjugation designated would cause confusion earlier notation particular shows reflection effected multiplication since definition smallest set containing closed reflections follows generated group viewing ordinary space since axe axa showing familiar way elements turn elements maps conjugation rotation way obtain double cover mapping however since thus group rotations sphere induced actually acts group symmetries particular lie proposition group quaternion multiplication generated dense iii acts naturally group isometries via mapping mapping iii maps onto subgroup index proof remains prove let begin making observation since mapping kernel clear last rotations something arise shall see obstruction deal since maps let denote image mapping note reads axa suppose wish make simple possible multiplying left elements consider since transitive exists writing rak shows case assume outset however see alter sign actually assume elements form since orthogonal transformation fixing permute two pairs roots around four possible rotations possible two require interchanging two coordinate positions changing one sign requires thus composing elements reduce one pair indeed proves note dense since unit circle likewise subgroup elements whose coefficients dense basically reason det surjective onto unit circle proposition stabilizer permutation operator last two coordinates obvious form working last three coordinates space quaternions group generated proof first observation rotation last three coordinates exists rotation order also changes signs individual coordinates exist reflections coordinates goes course let stabilizer let let since isometry since transitive element fixes ask must map remaining two elements roots form however general form roots shows possibilities signs adjusted please using element weyl group may still interchange two points words may involution shall show prop contain groups two basic root systems began rational parts sets intersection proposition groups quaternion multiplication orders respectively fact see instance convenience reader prove proof rational root system clear product two elements unit vector rational space still inversion quaternion conjugation preserves thus group order shall prove group proof follows immediately roots given first set consists quaternion notation elements multiplication quaternion left turns sign changes important thing observe performs even permutation coordinates interchanging first two coordinates last two coordinates likewise multiplication performs even permutation coordinates makes two sign changes similar things happen multiplication right particular left right multiplications map apart elements every property fact either even permutation property actually characterizes elements norm similar remarks hold roles interchanged let want prove said assume neither set begin showing assumption know even permutations arbitrary sign changes leave particular know prop first coordinate quaternion know first coordinate look second coefficient see shows second coefficient continuing idea third fourth coefficients end point since know element however say see following way first possible neither reason thing note see prop assume shows hence also similarly shows since group contradicts assumption finishes proof note passing since resp group products subgroup determine basic structure seen generation two groups following set elements set representatives four right cosets similarly right cosets element written alternating product elements using coset representatives always write product form wam alternately since subgroup going prove writing unique element written uniquely form key introduce set defined set integers mod congruent modulo congruent modulo number congruent even number congruent odd straightforward see set elements using conjugation second set easily seen four properties except number congruent odd retaining form two sets remarkable properties lemma iii part clear definitions necessary verify first parts iii present time depended brute force computation prove example looking iii following products one convert form show conjugating see using lemma let show possible zam equal except trivially alternate definiteness assume analysis case uses case uses suppose wam happen odd take side get either case right hand side level assume even show level side leading contradiction levels starting shows level next showing level continue way show level shows indeed level side course argument obtained conjugating equation show uniqueness standard form suppose wam zbp two expressions form dense dense amalgamation root systems amalgamation groups figure appear amalgamations supposing two expressions different assume minimal equality push side assume outset invert elements left hand side get new reduced expression equals know happen say rewrite hand side bring standard form increasing number terms side reduce combined length contrary minimality assumption case happen either proves uniqueness expression putting together shown proposition every element uniquely expressible form group amalgamation two groups free product factored normal subgroup generated identifying elements appearing normalizer know generated two copies coxeter group type namely section shall call contain subgroup generated reflections defined coxeter group type using root bases described see together create coxeter diagram see appears additional relation beyond coxeter relations see shall show origin relation together obvious coxeter relations obtain presentation begin looking group similarly stabilizer since root system type complete group automorphisms product symmetric group acts diagram automorphisms lemma product cyclic group order normalizer iii parallel results hold normalizer proof set root system type weyl group thus subgroup diagram automorphisms comparing orders see index either shall see sylow order fact direct product two cyclic groups order see fact subgroups type inside root pairs bases let associated root system multiplying find three cycle performs diagram automorphism root system together generates although product thus product element serve produce three cycle hence lie normalizer thus element maps stabilizes thus let using write presentation group presentation uses abstract generators relations involve reflections discussed next section comment sylow conjugate elements ones sylow conjugate elements particular one used conjugations must conjugate root systems root systems generates another relation like course ones follow conjugation presentation purpose section prove proposition generated following generators relations generators iii relations relations similar relations given notice similarly proof depends working explicitly group reflections appear terms algebra quaternions showing untangle relation written terms generators using relations corresponding itemized statement proposition begin recalling quaternionic form reflection effect product reflections rak acting odd aek aek even write products reflections products involves alternately conjugating elements quaternion conjugation designating tilde explicit form depending whether even odd number reflections involved possible overall sign change rather messy write quite trivial practice order avoid lots notation simply signify whole conjugation process putting hat symbol capable conjugated thus symbol beneath hat may may conjugated according overall length even odd thus write rak abk abk notice conjugation stabilizes proof prop let rak relation written terms root reflections raj objective show relation consequence relations equivalently wish show word side reduced empty word using relations rewrite side form begin consider product rbp action rbp bbp bbp let bbp bbp since group mapping assumes form seems simple relationship seems quite possible one would belong would however see rbp fact using fact central inversion see rbp see notice fact rbp fact takes place entirely inside group say determined entirely structure coxeter group view structure rewrite product rbp assume possible exception may see deducible coxeter relations begin parse word side utilizing form right left definiteness assume move left reach first roots involved nothing prove since whole word relation consequence coxeter relations assuming rewrite rap using two blocks abp abp case rap rewrite word using coxeter relations involved possible exception pair order case use replace pair corresponding order thus shifting last two letters passing next block using make blocks two blocks elements case rap little trick make pass along terms subsequent blocks write accordingly may assume rewriting passing along pairs done advance means block multiplies either continue parsing process left time producing new set elements whose left right product produce blocks two blocks ras derivable coxeter relations little bit extra attention needed elements appearing conjugations determined parity positions sequence reflections appearing element action ras might either conjugations may interchanged action would either case ras maps lies continue process finally achieve rewriting side form original length word began equation valid blocks multiply blocks lie saw entire effect element entire relation determined block block using coxeter choose relations let suppose least one block either first block appear either left right call blocks lie suppose right choose definiteness suppose type side partial product set possible blocks see every element side level least see prop discussion preceding level increase alternations blocks right hand side level contradiction types argument works way left contradiction shows blocks must lie relations prop suffice reduce side empty word proves relations presentation succinct way saying useful let set coset representatives right cosets mod let set coset representatives right cosets mod disjoint union natural choice would relate using conjugation however better keep freedom choices shall see proposition let resp set coset representatives right cosets mod resp mod element uniquely expressible form alternate amalgamation intersection proposition let choose prop representatives actually resp coset contains elements subgroups every element uniquely expressible form alternate amalgamation intersection proof notice root system type inside stabilizer product weyl group cyclic group order three arising cycling last three components equivalently cycling three elements see stabilizer way stabilizer see subgroup generated according prop every element uniquely form amongst expressions consider elements occurring actually expressions obviously elements every element evidently written form composed precisely products since expression form unique see fact says amalgamation intersection orthogonal group define group orthogonal transformations stabilize root system already know lot transformations namely elements along shall consider orthogonal group orthogonal matrices coefficients since contains standard orthogonal basis notice orthogonal transformation interchanges namely automorphism interchanges last two coordinates call transformation course numerous conjugates evidently contains subgroup generated prove reverse inclusion proposition proof let stabilizes know root system type root base let root base try use elements bring back basis see almost finish job may need reverse last two coordinates hence use transformation let recall see base traditional coxeter diagram one central node three nodes attached using simple transitivity assume one nodes base elements two nodes take one using time bring node remaining node form elements form use assume since sign changes assume element remains central node scalar product three nodes form given vector length last term must choice sign upshot brought base standard basis using since orthogonal transformation determined entirely action base proved rest follows know iii true iii know stabilizes hence normalizes generated since odd permutation reflections elements also interchanges coordinates stabilizing follows conjugation interchanges stabilizing intersection means choosing coset representatives used prop choose assume prove contrary using prop could write uniquely form zyk elements alternately gives second way writing element form described prop accordingly identical obviously true coset terms present since reversed conjugation etc thus end certainly effects diagram automorphism already noted thus contradiction shows corresponding result case involving take subgroup fixes vector define subgroup stabilizing proposition proof argument essentially repeat saw prop since clear stabilizes proves need prove working contains standard basis vectors let let since transitive element stabilizes span well mapping elements however elements form since sign changes individual coordinates elements assume positive signs thus fixes case interchanges case approximation unitary group plays significant role theory quantum computation group admissible transformations quantum qbit section offer algorithm element approximated degree accuracy repeated use simple fixed elements fact number reduced five five related root base passing via similarly approximate rotation degree accuracy elements idea inside quaternion algebra unit sphere group quaternion multiplication group point view matrix representation full set roots generated fact subgroup generated orbit furthermore dense sphere prop hence appropriate sets mirrors reflecting hyperplanes associated reflections elements partition convex regions arbitrarily small diameter call regions chambers natural choice set use associated hyperplanes points define set chambers use note chambers absolute things rather outcome selection finite number root hyperplanes assuming chambers selected one point chosen one chambers say idea element reflected using reflections mirrors whereby close approximation reversing operations take point equally close take set use points define set chambers note situation chambers customary case root systems chambers isometric copies necessarily simplices cellular decomposition comes sets reflecting hyperplanes use never use finite number finite sets invariants reflection quaternionic form see suppose wish approximate using reflections determined elements idea reflect number times ends close possible choose chamber containing actually vertex chamber choose point close interior element pair choose one satisfies let denote subset chosen way reflection hyperplane lie opposite sides see thus moving point closer always choose large possible process carried end allowing every point one reflecting hyperplanes uniquely specified signs makes roots either positive negative way looks similar dedekind cut real number specified rationals less greater equal almost surely process stop finite number steps point ending suppose infinite sequence successive steps applied starting point limit points choose one say closest follows fact every iteration brings closer closer contradiction almost surely sense lebesgue measure boundary mirror neither thus interior means almost surely interior iteration stop soon reaches point view approximation theory always assume boundary end process achieved relation form rak diameter since reflection isometry relative usual norm rak gives desired approximation odd even treated elements matrix entries come ring dyadic golden numbers need write write fully terms basic reflections roots roots appearing terms elements thus rbm odd even example give sample results computer implementation approximation process set roots formed sets evaluated resulting total elements random element computed process determines case six reflections roots bring point close identity element possible using reflections reflections giving rise total fifteen rewritten products reflections original set matrix typical looking example computing resulting matrix product according obtain approximation difference example entirely typical number reflections required degree terms approximation wished could rewrite reflections roots five generators computing roots result accuracy decent estimates diameters chambers arising acknowledgements thank referee careful parsing manuscript helpful historical suggestions rvm supported discovery grant natural sciences engineering research council canada supported scientific research monkasho kakenhi japan references bourbaki groupes lie hermann paris chen moody patera root systems quasicrystals discrete geometry patera fields institute monographs american mathematical society dechant twarock novel affine extensions noncrystallographic coxeter groups phys math theor dechant clifford algebraic framework coxeter group theoretical computations adv appl clifford algebras val homographies quaternions rotations oxford koca maximal subgroups coxeter group quaternions linear algebra applications moody patera quasicrystals icosians phys math
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preprint version improving files availability bittorrent using diffusion model napoli pappalardo tramontana sep published ieee international wetice conference bibitex incollection international wetice conference improving files availability bittorrent using diffusion model ieee napoli christian pappalardo giuseppe tramontana emiliano published version copyright ieee uploaded policies copyright infringement intended improving files availability bittorrent using diffusion model christian napoli giuseppe pappalardo emiliano tramontana dipartimento matematica informatica university catania viale doria catania italy email napoli pappalardo tramontana bittorrent mechanism effectively spreads file fragments copying rarest fragments first propose apply mathematical model diffusion fragments order take account effects peer distances changing availability peers time goes moreover manage provide forecast availability torrent thanks neural network models behaviour peers system combination mathematical model neural network provides solution choosing file fragments need copied first order ensure continuous availability counteracting possible disconnections peers distributed caching neural networks wavelet analysis ntroduction peer peer systems using bittorrent shared content named torrent becomes split fragments rarest fragments automatically chosen sent first users requesting file fragments availability given number peers storing fragment given moment periodically computed server storing peer ids fragments held requested computing priority fragments spread best availability freshly updated however peers often leave system hence file fragments availability quickly changes possibly least available fragments spread occurs frequently fundamental bittorrent mechanism may become ineffective result fragments quickly become unavailable moreover choosing fragments spread communication latencies among peers considered therefore fragments spreading surely occur sooner peers nearby one holding fragment spread result furthest peers could disconnect receiving whole fragment paper proposes model spreading file fragments considers priority fragment spread latencies among nodes priority fragments spread gradually changes according passing time fragments availability priority variation regulated way availability fragments maximised time goes fragments spread selected according results proposed diffusion model developed analogy diffusion model porous medium moreover propose characterise typical availability torrent observed system using appropriate neural network selection fragments spread aims counteracting decreasing availability estimated later time therefore proposed work aims supporting quality service dependability systems attributing priority fragment spread destination peer turn increases availability performances well consistency rest paper organised follow next section provides mathematical representation proposed model section iii develops model diffusion contents system section introduces neural network predicts user behavious section describes experiments based proposed model neural network predictions well preliminary results section compares related work finally conclusions drawn section vii athematical epresentation order put forward analogy bittorrent physical system conventions must chosen extrapolations needed first describe continuum system using continuum metric however later single interesting discrete points continuum due analogy bittorrent use distance metric named assimilated network latency among nodes hosts network holding seeds peers leeches nodes use notations first indicates generic node bittorrent network second indicates node seen node course could different nodes double indexing needed since use something like representing distance node measured node moreover let express pkij probability diffusion file fragment node node finally distinguish time time steps first used continuum measure temporal intervals use latin letter second indicate computational time steps steps iterative cycle use greek letter therefore represent continuous evolution time network latency measured node distance node notation represent latency measured step time taken ping node node specific time step finally suppose node fragment file interested sharing obtaining portions file hence compute probabilitylike function expresses easily shared fragment copied node jesime node certain step call pkij eventually interested analytical computation urgency share fragment time set call following sections distinguish actually measured value value predicted neural network using tilde predicted values iii ragments diffusion network work compare file fragments shared file diffusion mass porous means embrace view mandatory develop mathematical tools explain following spaces metrics users bittorrent network represented elements space metric could given corresponding network communication latency therefore node set nodes possible define function amount time taken bring minimum amount data ping using given definition distance node also possible obtain ordered list way first item list following items ordered according network latency measured using complete ordering peers possible introduce concept content permeability diffusion let consider files shared one user system file consists fragments diffused diffusion file fragment analysed terms fick law fick law use fick second law commonly used physics chemistry describe change concentration per time unit element diffusing another work proposes analogy system physical system key idea model sharing file fragments diffusion substance porous means along one dimension different places porous means would represent different nodes whereas distances along would proportional network latencies entities would accommodated formalism equations using first second fick laws diffusion substance means given solution following vectorial differential equation concentration time permeability means diffusing matter since separable equation use metric based distance assuming constant among nodes equation written scalar differential equation partial differential equation imposed initial boundaries conditions admits least green function particular solution green function lets study diffusion dynamics single substance rewritten solution equation form complementary gaussian error function function computed means taylor expansion however avoid computationally difficult task use approximation proposed pure exponential approximation obtained error order possible following equation every node certain distance time concentration probability equation scaling factor function time hand used formalism developed mainly focus distance managing merely parameter mathematical formalism valid long distances remain common practice considers distance nodes however actual network latencies vary almost continuously time stationary ordered set unlikely approximation network solution make latency embedded model turn makes possible choose different fragments shared time goes system equation states certain file fragment zki node time probability pkij given diffused node distance within time proportional pij pij pkij pij function pij carries diffusion factors temporal dynamics since interested simple proportion direct equation also neglect factor write pij normalised form pij important proper redefinition coefficient let say number users using file fragment whether asking offering number seeders file fragment mean share ratio file fragment among peers leeches possible consider urge share resource osmotic pressure time varies coefficient permeability network order take account mutable state system vary according amounts available nodes file fragments chosen define formal substitution obtain analytical form term pij discrete time evolution node indeed physical nature adopted law works entire variable space however problem hand simplifications needed let suppose given discrete time step node effectively measures network latencies set nodes ordered set equation computed every node computes probability pkij file fragment every node probability corresponds statistical prevision possible file fragments spreading onto nodes suppose measures taken later discrete time step file fragment zki copied first node served chosen according minimum probability diffusion latencies time since last measures taken see following subsection equation moreover file fragment reaching nodes latency nodes less time computed tik index used equation refer file fragment zki indeed since nodes need offer file fragments ordered set nodes referred given node depend resource possible complete mapping probability diffusion reducing time dependence single variable dependence discrete resource pkij stated possible reduce function assuming considering values execution moment computational step assigning priorities corrections pkij computed values stored proper data structure actually simple determine urgent file fragment share resource least probability spread pkij minimum furthermore consider time goes old measured differs actual value hence measure becomes less reliable take account staleness values gradually consider less bound choice fragment behaviour provided negative exponential equation given enough time choice based number available fragments however consider time new value would measured incorporated model choosing fragment generally nodes highest latencies respect given node time needed receive fragment node aim compensating delay incorporating model inescapable latencies network therefore node receive fragment first chosen according distance order model fact distant nodes highest values take time send receive file fragments chosen decay law possible obtain complete analytical form spreading file fragments pkij decay constant chosen heuristically without harming said law tuned according parameters indicates file fragment index urgent file fragment share latter trivially found solution maximum problem max course priorities depend value bidimensional matrix values pkij mark index innovative method reconstruction photometric data step series set coefficients residuals original signal forward reconstruction backward reconstruction midpoint reconstruction topology wrnn corrected flux adu figure intend input set represented matrix time steps level wavelet decomposition row represents time step decomposition mid point forward nnet backward nnet variate within node among row dataset given input value values need compute elements input neurons proposed wrnn figure elements node queue properties network make possible starting resource cases assumed input time step right predict effective number fig neural networks structures left forward backward reconstruction moreover completed transfer requests offers time step way node element indexes set similar wrnn acts like functional fashion peer able identify possible resource ask order maximize diffusion rare ones becauseones wavelets verify basic properties absence instead common local minima provide response graded number time steps forecast future existent rangeand possible transfer functions particular andonly predicted serie classes predictors sers behavior time julian days approximate functional form wavelet work radial basis order make thewere chosen system able properly reb predicted behaviour tionsto radbas transfer functions indeed user particular kind act peaks requests well fast changes even equation possible functions well describes first approximation halfasof described wavelet fragments availability share ratio propose obtain future prediction number requests functions verify properties shown anyway afterofscalan innovative solution based wavelet recurrent neural specific torrent well ing shift chosen activation possible obtainavailability future networks wrnn repetition characteriseofthethe user behaviour function fact considering predicted several mother forecast wavelet filters let ftorrent choosen transfer producing given modeled possible time step function wavelet analysis provides compression denoising take counteracting actions improve probability observed time series amount users prividing diffusion rare torrent achieved practice requesting fragments neural network values using account rnn trained said observed time series provides forecast future time steps modified values estimations future data ensamble computed ourfor wrnn computing node verifies properties wavelet function possible selected said wavelet analysis rnn called wrnn socloud therefore predicted values neural networks simulate wavelet using radbas function defined provides forecasts number users sent node acting peer thea realseveral domain isnetworks indeed possible share fragment neural beento verify time new torrent becomes shared employed find polaritons propagation metal thicknetwork new wrnn created trained ness correspondence predict behaviour cloud system provide users kserver requested requesting resources perform wavelet transform predictions related availability peers novel recursive lifting procedure set shared fragments torrent predictions shown order simulate wavelet function chosen transfer sent peers periodically allow peers update functions wavelet thisupdate thefrequency tuned wrnn setup must symmetrically periodical emulate valuesa reason choosing pair number neurons aim number order correctly match dynamic hosts work initial datasets consists time series positive weights theorreconstruction layer theoretically representing requestsand negative torrent layer coming peers xperiments happens neuron exactly given declaration availability torrentpairs coming fromsecond layer emulating reconstruction filter althoug theoretical schema peers seeds independently specificities shown figure strong initial condition reasons theseries weights experimental havesome sum torrent happen data let call discrete setup system file fragments neuralwith network beyond perform inverse wavelet transform time step data sampled intervals onetohour heterogeneously spread among peers one shares biorthogonal time series fragment nodes fragment must wavelet performdecomposition also signal prediction computed obtain correct input set wrnn required devised architecture shown figure decomposition achieved applying wavelet transform recursive couple conjugate filters way recursion produces time report simulation comprising peers file fragments evolution time steps order step step peer selects file fragment sends peer fragment spread destination peer chosen according equation order initial condition white cells represent fragments normalised certain node different time steps elated ork several studies analysed behaviour bittorrent systems point view fairness users contribute contents uploaded users levelling amount downloads upload fewer works studied problem unavailability contents bittorrent networks authors proposed order peers according later soon possible peer selects another fragment spread fragment could send transfer previous fragment completed concurrently first transmission shown evolution consider file fragment could passed node next time step value would drop zero note highest values indication urgency receiving fragment described model formula allow subsequent sharing activities initial time steps determined terms fragments sent figure shows first time steps becomes urgent node obtain missing fragments possible see highest priority fragment since share ratio relative availability low respect fragment expected behaviour developed model simulation shown figure fragments except since actually unavailable would spread peers low number time steps figure shows decay several computed different peers requiring fragment number long run law benefit nearby nodes short term distant nodes given highest priority figure figure time decay normalised increasing time steps uploading bandwidth hence providing contents selection peers performed accordingly one mechanism proposed increase files availability use ensuring fairness instead forcing users stay longer provide contribution uploaders fragments belonging different files similarly authors show using availability easily increased confirm fast replication rare fragments essential furthermore bundling dissemination number related files together proposed increase availability proposed mechanisms differ proposal since take account several novel factors dynamic data exchange distant peers decay availability peers forecast contents availability factors related proposed model manages select rarest content spread taking account future availability peers provide take content vii onclusions paper proposed improve availability fragments system adopting mathematical model neural network former describes fragments diffusion urgency share fragments whereas latter provides estimation availability peers hence fragments later time using estimate future availability diffusion model select fragments need spread counteract disappearance due users disconnections acknowledgment work supported project prisma funded italian ministry university research within pon framework eferences marletta pappalardo tramontana tackling consistency issues runtime updating distributed systems proceedings international symposium parallel distributed processing workshops phd forum ipdpsw pages ieee doi bokshtein mendelev srolovitz thermodynamics kinetics materials science short course oxford university press oxford bonanno capizzi sciuto napoli pappalardo tramontana cascade neural network architecture investigating surface plasmon polaritons propagation thin metals openmp proceedings artificial intelligence soft computing volume pages springer capizzi napoli bonanno innovative secondgeneration wavelets construction recurrent neural networks solar radiation forecasting ieee transactions neural networks learning systems capizzi napoli innovative hybrid method reconstruction missing data astronomical photometric surveys artificial intelligence soft computing pages springer chiani dardari simon new exponential bounds approximations computation error probability fading channels ieee transactions wireless communications july giunta messina pappalardo tramontana augmenting web server qos means aspectoriented architecture proceedings wetice pages ieee doi giunta messina pappalardo tramontana providing qos strategies web servers means aspects concurrency computation practice experience giunta messina pappalardo tramontana kaqudai dependable web infrastructure made existing components proceedings wetice pages ieee guo chen xiao tan ding zhang measurements analysis modeling systems proceedings acm sigcomm conference internet measurement usenix association kaune rumin tyson mauthe guerrero steinmetz unraveling bittorrent file unavailability measurements analysis proceedings pages ieee menasche rocha towsley venkataramani content availability bundling swarming systems proceedings pages acm napoli bonanno capizzi exploiting solar wind time series correlation magnetospheric response using hybrid approach advances plasma astrophysics number proceedings international astronomical union pages cambridge university press napoli bonanno capizzi hybrid neurowavelet approach prediction solar wind advances plasma astrophysics number proceedings international astronomical union pages cambridge university press napoli pappalardo tramontana hybrid predictor qos control stability advances artificial intelligence volume pages springer novelli pappalardo santoro tramontana infrastructure support multimedia content distribution proceedings pages acm doi qiu srikant modeling performance analysis networks sigcomm computer communication review cohen incentives build robustness bittorrent workshop economics systems volume pages sweldens lifting scheme construction second generation wavelets journal mathematical analysis giunta messina pappalardo toscano tramontana testing replica selection policies paneuropean grid proceedings wetice pages ieee june doi porous medium equation mathematical theory mathematical theory oxford university press
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ieee transactions information theory incidence structures construction capacity analysis jan ardhendu tripathy student member ieee aditya ramamoorthy member ieee instance function computation problem directed acyclic network terminal node wants compute sum finite field information observed source nodes many characteristics multiple unicast network communication problem also hold due known reduction two problems work describe algorithm construct families instances using incidence structures computation capacity several sumnetwork families evaluated unlike coding capacity multiple unicast problem computation capacity sumnetworks depends characteristic finite field sum computed dependence strong show examples solution one characteristic rate close zero different characteristic additionally arbitrarily different computation capacities different alphabets index coding function computation sumnetworks characteristic incidence structures ntroduction applications diverse parallel processing distributed data analytics sensor networks often deal variants problem distributed computation motivated study various problems fields computer science automatic control information theory broadly speaking one model question following manner consider directed acyclic network edges denoting communication links subset nodes observe certain information nodes called sources different subset nodes called terminals wish compute functions observed information certain fidelity computation carried terminals aid information received incoming edges demand functions network topology part problem instance arbitrary framework general encompasses several problems received significant research attention prior work concerning information theoretic issues function computation worked setting correlated information observed sources simple network structures simple sense edges connecting sources terminal without intermediate nodes relays instance characterizes work supported part national science foundation nsf grants material work appeared part allerton conference communication control computing ieee international symposium information theory authors department electrical computer engineering iowa state university ames usa ardhendu adityar amount information source must transmit terminal correlated reliably compute function message observed source reference considered distributed functional compression two messages separately encoded given decoder computes function two messages arbitrarily small probability error advent network coding scope questions considered included setting information observed sources independent network topology complex setting information sent source terminal path edges directed acyclic network one intermediate nodes relay nodes limit memory computational power communication edges abstracted links certain capacity information transfer sometimes referred messages required recovered zero distortion multicast scenario message observed source network demanded terminals network solved sufficient condition solvability multicast scenario terminal source least entropy rate message random process reference established linear network codes sufficiently large alphabet solve problem provided necessary sufficient conditions solving multicast problem instance algebraic framework work also gave simple algorithm construct network code satisfies unlike multicast problem multiple unicast problem admit clean solution scenario multiple pairs directed acyclic network terminal wants recover message sent corresponding source help information transmitted network begin problem instances one use network required solve model network edge viewed carrying vector alphabet symbols message vector alphabet symbols network code specifies relationship vector transmitted edge network message vectors solves network coding problem instance shown linear network codes general sufficient solve problem one define notion coding capacity network supremum ratio network codes allow terminal recover desired message ratio particular network code called ieee transactions information theory rate coding capacity network independent alphabet used network code rational rate less coding capacity exists definition network code rate equal coding capacity exist certain networks even coding capacity rational flow solution multiple unicast problem called routing solution different messages interpreted distinct commodities routed intermediate nodes case multicast network coding provide gain rate traditional routing messages scales size network however evaluating coding capacity arbitrary instance network coding problem known hard general expanding scope demands terminals considered function computation directed acyclic networks one terminal value recovered terminal allowed function messages opposed subset set messages computation performed using information transmitted edges network code analogous coding capacity notion computation capacity defined case network code allows terminal compute demand function interpretation function computed terminal times uses network based upper bounds computation capacity directed acyclic network one terminal given matching lower bound function computation given computation capacity linear network codes different classes demand functions explored different flavor function computation problem often called problem considers directed acyclic networks multiple terminals demands sum messages observed sources reference characterized requirements two three sources terminals must satisfy terminal recover sum unit rate similar network coding scenario whose terminals satisfied network code called solvable reference established deciding whether arbitrary instance problem instance solvable least hard deciding whether suitably defined multiple unicast instance solvable result reduction various characteristics solvability problem network coding instances also true solvability problem establishes broadness class within communication problems directed acyclic networks solvable solvable network coding instances intimately related results paper indicate classes problems diverge focus capacity strictly less one section coding capacity networks shown independent finite field chosen alphabet messages information transmitted edges show analogous statement true demonstrating infinite families sumnetwork problem instances whose computation capacity vary depending finite field alphabet moreover gap computation capacity two different finite fields shown scale network size certain classes sumnetworks two alphabets different cardinality network authors theorem described procedure simulate network code using network code network procedure apply along lines counterexample given regarding minimum maxflow connectivity required solvability three sources terminals provide infinite family counterexamples mandate certain value connectivity allow solvability finite field general three sources terminals problem instances arrived using systematic construction procedure combinatorial objects called incidence structures incidence structures structured set systems include graphs combinatorial designs note combinatorial designs recently used address issues construction distributed storage systems coded caching systems paper organized follows section describes previous work related problem considered summarizes contributions section iii describes problem model formally section describes construction procedure use obtain problem instances considered work section gives upper bound computation capacity section describes method obtain linear network codes achieve upper bound rate several families constructed section vii interprets results paper outlines key conclusions drawn paper section viii concludes paper discusses avenues future work background elated ork ummary ontributions problem setting work information observed sources independent uniformly distributed finite field alphabet network links assumed possibly many terminals wants recover finite field sum messages zero error problem introduced work intuitively reasonable assume network resources capacity network links network structure effect whether sum computed successfully terminals network reference characterized notion class either two sources two terminals class shown source messages one pair enough solve problem shown means ieee transactions information theory counterexample one enough solve three sources terminals however also shown two pair sufficient solve sumnetwork three sources three terminals reference considered computation capacity class sumnetworks three sources three terminals vice versa shown integer exist whose work closely related computation capacity paper gives construction procedure positive rational number returns sumnetwork whose computation capacity assuming relatively prime procedure described constructs sources terminals large large authors asked question exist smaller sumnetworks fewer sources terminals computation capacity work answered affirmative proposed general construction procedure returned prescribed computation capacity could obtained special cases construction procedure smaller instances specific values presented small instances useful determining sufficiency conditions larger networks scope construction procedure proposed widened result shown exist instances whose computation capacity depends rather strongly finite field alphabet work builds contributions particular present systematic algebraic technique encompasses prior results also include proofs results discuss implications results depth summary contributions work define several classes explicitly determine computation capacity networks constructed using appropriately defined incidence structures main contributions work follows demonstrate novel techniques determining upper lower bounds computation capacity constructed cases bounds match thus resulting determination capacity demonstrate strong dependence computation capacity characteristic finite field computation taking place particular network computation capacity changes based characteristic underlying field unlike coding capacity multiple unicast problem known independent network coding alphabet consider class networks every sourceterminal pair minimum cut value least arbitrary positive integer demonstrate exists within class large number sources terminals whose computation capacity made arbitrarily small implies capacity characterized individual minimum cuts iii roblem formulation reliminaries consider communication directed acyclic graph dag set nodes edges denoting communication links edges given additional index model allows multiple edges two distinct nodes instance two edges nodes represented subset denotes source nodes denotes terminal nodes source nodes incoming edges terminal nodes outgoing edges source node observes independent random process sequence random variables indexed time denoted positive integer xij takes values uniformly distributed finite alphabet alphabet assumed finite field characteristic denoted edge represents communication channel unit capacity transmit one symbol per time slot referring communication link edge without third index assume set edges two nodes set denoted define capacity cap number edges use notation represent set incoming edges node edge edge let head tail terminal node demands thepsum individual source messages let xij set natural numbers wants recover sequence information receives incoming edges set network code assignment local encoding functions edge denoted decoding function terminal denoted terminals compute local encoding function edge connected set sources messages observed particular source nodes input arguments likewise input arguments local encoding function edge connected source values received incoming edges inputs decoding function terminal values received incoming edges consider directed acyclic networks seen global encoding function expresses value transmitted edge terms source messages set global encoding function edge denoted following notation describes domain range local encoding decoding functions using two natural numbers general vector network code number source values encoded simultaneously local encoding function edge emanates ieee transactions information theory source node number symbols transmitted across edge network thus edge whose tail local encoding function xsm using row column vectors paper explicitly mentioned defining vector represents transpose local encoding function edge tail tail tail decoding function terminal network code linear finite field local encoding decoding functions linear transformations case local encoding functions represented via matrix products matrix elements example edge tail let tail tail also let denoted column vector size whose elements value transmitted evaluated matrix indicating local encoding function edge consider valid fractional network code solution interpretation sum source symbols communicated terminals time slots definition rate network code defined ratio solvable network coding solution definition computation capacity defined valid network code sup given use different types incidence structures constructing throughout paper formally define present examples incidence structures definition incidence structure let set elements called points set elements called blocks block subset incidence structure defined pair say point incident block general multiset contain repeated elements considering work thus two distinct blocks least one point incident one denote cardinalities sets constants respectively thus set points blocks indexed subscript fig pictorial depiction fano plane point set blocks indicated straight line joining constituent points points lying circle also depict block definition incidence matrix incidence matrix associated incidence structure dimension defined follows otherwise thus incidence matrices viewed general set systems example simple undirected graph viewed incidence structure vertices points edges blocks block size two combinatorial designs form another large class incidence structures work use properties defined next definition incidence structure design points block point set satisfy property present exactly blocks design called simple repeated blocks designs subject much investigation case also called balanced incomplete block designs bibds example famous example fano plane shown figure letting numerals denote points alphabets denote blocks design write corresponding incidence matrix rows columns arranged numerical alphabetical order ieee transactions information theory shown verified every pair points appears exactly one block conditions parameters design satisfy see integer number blocks incident let denote constant note simply total number blocks denoted likewise represents number blocks point incident use symbol represent furthermore follows necessary condition existence design divides counting number ones incidence matrix particular design two different ways arrive equation onstruction family sum networks let construction takes input dimension definition notation row column let denote row vector denote column vector turns constructed interesting properties matrix incidence matrix appropriately chosen incidence structures construction algorithm presented algorithm various steps algorithm construct components described source node set terminal node set contain nodes one row column source nodes denoted line spi sbj correspond row column respectively terminal nodes also denoted similar manner line added vertex set sumnetwork line bottleneck edges add edges indexed line edge set edge corresponds row matrix also add required tail head vertices edges justification notation later use incidence matrix incidence structure construct rows columns incidence matrix correspond points blocks respectively edges bottleneck edges every connect tail source node corresponding row source nodes correspond columns row described line algorithm line describes similar operation used connect head certain terminal nodes direct edges terminal edges directly connect source nodes path particular terminal bottleneck edges using notation rows columns matrix characterized lines algorithm input output initialize head tail spi sbj tpi tbj sbj tail spi tail head tbj head tpi end spj tpi sbj tpi end spi tbj tbj bjt end return incidence structure let represent incidence matrix constructed paper matrix used algorithm either equal ati incidence structure call constructed normal otherwise ati call constructed transpose following definitions useful analysis every denote set blocks contain point hpi every collection blocks intersection denoted set hbi boldface indicates column corresponding block ieee transactions information theory fig two obtained line graph two vertices described example source set terminal set contain three nodes edges point downward edges arrowheads bottleneck edges constructed step construction procedure normal transposed sumnetwork inner product computed reals sequel occasionally need perform operations similar inner product finite field shall explicitly pointed present examples constructed using technique example let unique simple line graph two vertices points corresponding vertices blocks corresponding edges graph denoting points natural numbers get associated incidence matrices follows ati following algorithm two sumnetworks obtained shown figure example example construct using simple let denote design points denoted numbers blocks denoted letters design associated incidence matrix row column permutations fig normal obtained incidence structure described example edges directed downward edges arrowheads bottleneck edges edges denoted dashed lines correspond direct edges introduced step construction procedure case normal transposed sumnetwork identical written follows note ati hence normal transposed identical case following algorithm obtain shown figure remark note edge added algorithm unit capacity proposition section vii modifies algorithm edge cap pper bound computation capacity section describe upper bound computation capacity obtained dimension assume exists fractional network code assignment corresponding global encoding functions decoding functions terminals recover sum independent sources convenience presentation change notation slightly let messages observed source nodes corresponding rows xpi corresponding columns xbj messages column vector length set source messages represented let denote column vector symbols ieee transactions information theory transmitted edge value returned global encoding function edge set source messages denoted apparent encoding functions employed bottleneck edges edges one input brevity denote define following set global encoding functions let entropy function random variable let denote set following lemma demonstrates certain partial sums computed observing subsets bottleneck edges lemma network code allows terminal compute demanded sum value xpi xpi xbj computed similarly value xbj computed set values proof let xbj xbj xbj xbj xbj xbj xbj xbj equalities due fact source messages independent function xpi set xbj since terminal tpi compute get xpi second part lemma argue similarly follows let xpi xpi sum xpi xpi assumption terminal recover demanded sum know evaluated tpi since xbj determine value respectively also determine values transmitted direct edges connect source node tpi get tpi sbj tpi bold lowercase symbols represent specific realizations random variables xpi xbj xbj xbj xbj xbj xbj inequality follows fact tpi function sbj tpi function xbj equality due fact conditionally independent xbj given conditional independence checked follows let assumption sets hbj determine value respectively also values transmitted direct edges connect source node terminal tbj let denote set spi tbj tbj hbj xpi hbj xpi hbj xpi hbj xpi hbj xpi hbj inequality due fact spi tbj function xpi similarly tbj equality follows fact conditionally independent xpi hbj given set random variables verified manner similar done previously gives result ieee transactions information theory next show fact messages observed source nodes independent uniformly distributed imply random variables also uniform introduce notation matrix two index sets define submatrix containing rows indexed columns indexed consider two dimensions respectively indicate multiplicative additive identities finite field respectively row denoted row submatrix column denoted column submatrix define matrix function returns matrix follows product positive otherwise incidence structure incidence matrix let vectors component uniformly distributed collect independent source random variables column vector elements follows xptr recall denotes row denotes column matrix let denote column vector component zero elsewhere defined lemma one check indicates kronecker product two matrices xpi eti xbj identity matrix size stacking values correct order get following matrix equation xptr xbtc matrix defined note first rows linearly independent natural correspondence rows points blocks incidence matrix row corresponds point row corresponds block lemma size let defined eqs matrix defined let rankf integer index set rankf let set blocks correspond rows indexed increasing order proof quantities statement lemma satisfy following system equations xptr xpr xbs xbs vector xptr uniform since matrix full row rank equal equation uniformly distributed giving first statement second statement true marginalization theorem computation capacity constructed algorithm proof construction procedure terminal tpi connected sources spi sbj edge lemmas bits definition maximum amount information transmitted bits implies next show upper bound computation capacity exhibits strong dependence characteristic field denoted computation takes place theorem let dimension suppose construct corresponding using algorithm matrix defined rankf upper bound computation capacity proof let defined lemma lemmas hence log definition get log log proposition rankf rankf furthermore rankf detz detz indicates determinant matrix elements interpreted proof gives rank condition since matrix full rank determinant element base subfield prime order could also interpret elements integers ieee transactions information theory evaluate determinant detz full rank detz example consider normal obtained using fano plane incidence matrix defined verified rankgf mai hence theorem gives upper bound computation capacity fact network code satisfies terminals normal obtained using fano plane described later proposition obtain different upper bound computation capacity considering submatrices necessarily contain initial rows define new index set based index set follows span span indicates subspace spanned vectors set submatrix contains rows indexed numbers theorem let dimension suppose construct corresponding using algorithm network code enables terminals compute sum must min rankf defined proof note choice index set index set defined lemma rankf thus recovering rankf upper bound computation capacity theorem index set rankf rankf collect blocks indexed increasing order set bty one recover following equation set messages xpts xbtt xbtty hence reasoning valid choice gives result example consider transposed corresponding undirected graph shown figure one check matrix matg rows fig simple undirected graph two connected components vertices edges columns incidence matrix atg arranged increasing alphabetical numeric order follows matg choose finite field alphabet example rankgf matg theorem gives computation capacity however theorem gives tighter upper bound case specifically rankgf matg hence theorem states computation capacity transposed graph apply theorems obtain characteristic dependent upper bounds computation capacity infinite families constructed using given procedure corollary let incidence structure obtained simple undirected graph denotes set vertices consists corresponding edges let deg represent degree vertex incidence matrix dimension computation capacity normal constructed using finite field let finite field alphabet operation define deg consider set edges hpi computation capacity transposed proof recall bit row ati inner product two rows mod edges indexed common vertex otherwise ieee transactions information theory observed matrix interest ati ati full rank every finite field transposed obtained applying algorithm matrix ati parameters apply theorem choosing index set defined way obtained using collect points corresponding rows submatrix mati set note depends set edges definitions true consists edges incident least one point indices set correspond points incident edge outside instance example show rankf gives result using theorem recall denotes row corresponds vertex follows inner product mod otherwise together exactly one block thus inner product mod otherwise implies ati ati setting determinant gives result normal argue follows note bit mod since two points determine unique block two blocks either one point none common hence inner product otherwise equation terms matrix ati ati equal zero focus submatrix obtained letting get ati proof definition inner product two rows mod mod otherwise follows ati ati value diagonal elsewhere ati ati ati ati setting determinant gives result corollary let design point present blocks number blocks incident pair points given consider transposed sumnetwork constructed using incidence matrix ati dimension computation capacity transposed ati ati definition points set deg mod diagonal entry corresponding points ati ati matrix zero thus exactly rows equal row vector first third matrices invertible hence get rankf corollary let design normal constructed using incidence matrix computation capacity transposed constructed using ati computation capacity proof first describe case transposed sumnetwork point design incident blocks moreover two points occur mod mod denotes square ones matrix elementary row columns operations det ati ati evaluated equal mod corollary let design incidence matrix define higher incidence matrix dimension row corresponds distinct column corresponds entry points corresponding row incident block zero otherwise computation capacity normal constructed using computation capacity transposed constructed using proof incidence matrix matrix dimension let denote row ieee transactions information theory column respectively row corresponds distinct tdesign criterion set points belongs exactly blocks since columns correspondence blocks row exactly two rows column block corresponding column incident corresponding two rows since block points one block incident two different hence inner product vhave mod union corresponding rows block ptj otherwise describe network code global encoding functions two bottleneck edges shown table using values transmitted three mod gives result transposed normal look columns similar manner column corresponds block since size block column exactly elements also two different blocks points common happens two columns row hence inner product mod blocks points common otherwise terminals recover sum following manner receives value direct edge receives value direct edge recovers sum using first two components recovers sum using first two components additionally receives carry operation thus terminal satisfied mod theorem gives result inear network codes constructed sum networks section propose linear network codes constructed using algorithm recall algorithm takes rows columns input section demonstrated incidence matrix certain incidence structures result whose capacity upper bounded corollaries demonstrate certain conditions obtain network codes whose rate matches corresponding upper bound thus able characterize capacity large family emphasize random linear network codes used widely literature multicast code constructions applicable context particular hard argue random linear network code would result terminal obtaining different linear function subspace thus constructing codes requires newer ideas outline key ideas means following example example consider shown figure matrix used construction dimension described example observed ati ati theorem states computation capacity table function values transmitted across igure network code rate ach message vector components vectors components number within square brackets adjoining vector indicates particular component vector component network code example following structure bottleneck edge first components global encoding vector sum messages incident bottleneck remaining components encoding vectors transmit certain components messages observed source nodes correspond columns matrix example received first component second component thus able recover value used computing demanded sum construction network codes structure first components bottleneck edge used transmit partial sum messages observed sources connected bottleneck edge remaining components transmit portions certain sources uncoded manner given incidence matrix first step identify possible corresponding integral matrix dimensions following properties row sums column sums certain conditions incidence matrix show used construct suitable network codes consideration existence proposed network codes thus intimately related existence integral matrices satisfy certain constraints following theorem corollary special case general theorem gives necessary sufficient conditions existence integral matrices constraints row column sums give proof since use ideas eventual network code assignment ieee transactions information theory theorem let integral vectors satisfying exists nonnegative integral matrix cij theorem let incidence structure let denote corresponding incidence matrix dimension suppose following conditions satisfied diag mod element exists matrix integer elements dimension whose entries satisfy cij proof consider modelled using bipartite graph nodes left part nodes denoted right part nodes denoted directed edge capacity cij two additional nodes source node terminal node directed edges capacity directed edges capacity let set nodes left part whose indices let set nodes right part whose indices consider cut separating nodes complement let value maximum flow network must possible choice subsets cij computation capacity constructed rate using via algorithm achieved linear network code proof note ati ati full rank assumption theorem states computation capacity construct linear network code using matrix since message vector components vector notation two positive integers denotes length vector contains components indices set order need specify global encoding vectors bottleneck edges edges network act repeaters linear network code first components vector transmitted along xpi xbj particular suppose flownetwork substituting get condition possible subsets cij note choosing possible subsets considering every possible cut network theorem set conditions form necessarypbut also sufficient existence flow value network feasible flow value used arrive matrix follows set value element matrix equal value feasible flow edge easy verify matrix satisfies required conditions using existence theorem nonnegative integral matrices obtain network codes constructed certain incidence structures following theorem describes set sufficient conditions satisfied incidence structure allow construct linear network code rate computation capacity sumnetwork proof theorem constructive results explicit network code construction tpi connected source nodes sbj direct edges tpi compute following value information received direct edges xpi xbj adding value enables tpi compute required sum follows focus terminals form tbj since vector components specified yet describe particular assignment components every using matrix enables tbj compute sum recall bipartite flow network constructed proof theorem nodes left part denoted nodes right part denoted edge flow edge denoted value determined constraints row column sums conclude value flow value flow without ieee transactions information theory loss generality assume partition components message vector xbj parts partition contains distinct components xbj partitioning done message vectors xbj flow indicates vector includes uncoded components xbj assigning interpretation every edge possible total number components available also equal flow point construction terminal tbj connected bottleneck edges set assignment based flow tbj receives distinct components xbj since recover components xbj piecewise fashion adding first components transmitted bottleneck edges connected tbj recover xpi xpi bjt xbl assignment tbj obtains value xbj piecewise manner carry following xbj xpi xbj xbl xbj xbl messages present partial sum hbj available tbj direct edges construction hence terminals correspond column also able compute required sum illustrate linear network code proposed means following example example consider normal obtained undirected simple graph shown figure part shown figure incidence matrix satisfies condition theorem therefore associated matrix shown rows columns arranged increasing numeric alphabetical order xbl condition ati ati diag one verify bjt xbl xbj xbl fig undirected graph considered example part corresponding normal constructed undirected graph full normal nine nodes source set terminal set however clarity five sources terminals correspond columns incidence matrix graph shown also direct edges constructed step construction procedure shown edges point downward edges arrowheads bottleneck edges constructed step construction procedure bipartite flow network constructed proof theorem message values corresponding flow solid lines also shown using matrix one construct structured linear network code rate shown table one check enables terminals compute required sum corresponding shown figure messages corresponding flow solid edges shown alongside respective edge also consider transposed graph corollary gives upper bound computation capacity depends ieee transactions information theory subset points upper bound note theorem applicable matrix atg atg diagonal elements proposition gives condition existence network code transposed obtained using irregular graphs apply condition transposed graph considered example following proposition show certain infinite families incidence structures satisfy requirements stated theorem particular incidence structures considered corollaries satisfy conditions hence computation capacity associated calculated proposition following incidence structures transposes satisfy condition theorem incidence matrix dimension denoted exists corresponding integral matrix satisfies conditions equations incidence structures derived regular graph biregular bipartite graph designs higher incidence structure design obtained using procedure described corollary proof existence existence dit rowsums thus suffices argue check validity condition first choose bounds elements matrix set cij choice condition inequality trivially satisfied whenever chosen point incident block exist hence restrict attention choices none points incident block restriction inequality condition equivalent assume show leads contradiction three incidence structures considered corresponds simple graph consider incidence matrix size chosen look submatrix size consists rows indexed points columns let number columns single number columns two counting total number two ways get since number edges incident least one point subset edges incidence point satisfies using get contradiction suppose corresponds biregular bipartite graph vertices degree left part vertices degree right part ldl rdr consider subset vertices let resp set edges incident vertex resp incident vertex resp number edges incident vertex equal suppose choice ldl rdr ldl rdr contradiction leaves case implies also contradiction next consider design blocks repetition degree point employ similar procedure case graph choose submatrix size corresponds rows indexed points let denote number columns exactly count total number two ways yielding klk number incident least one point pblocks equal hence subset blocks incidence point satisfies using get contradiction higher incidence structure obtained design corollary definition ieee transactions information theory table function values transmitted across igure network code rate ach message vector components vectors components number inside square brackets adjoining vector indicates particular component vector component original design points incident exactly blocks also block consists points submatrix whose rows correspond points condition let denote number columns exactly counting total number two ways get ili number blocks incident least one point ptotal number blocks incident point satisfy using get contradiction thus three kinds incidence structures considered shown admit existence associated matrix stated qualifying conditions enables apply theorem obtain lower bound computation capacity undirected graph regular proposition applicable theorem describes sufficient condition existence linear network code achieves upper bound computation capacity normal constructed undirected graphs necessarily regular upper bound capacity transposed constructed using incidence matrix ati however different depending finite field corollary theorem needs modified applicable case following example illustrates example consider transposed irregular graph described example corollary gives upper bound computation capacity case show submatrix atg equation also associated matrix whose support atg whose sum rows columns arranged increasing alphabetical numeric order atg using construct linear network code shown table iii achieves computation capacity transposed constructed using irregular graph shown figure particular terminals need information partial sums obtained respective bottleneck edges compute sum terminals need value respectively transmitted piecewise fashion according matrix bottleneck edges undirected graph regular let set points edges chosen statement corollary describe condition submatrix ati consists rows columns ati corresponding blocks points sets respectively condition allows construct capacityachieving linear network code transposed proposition undirected graph let subsets points blocks defined corollary let ati indicate element submatrix indices suppose matrix dimension ati linear network code rate allows terminal transposed constructed using compute required sum proof describe network code enables terminal compute sum corollary know code since transposed bottleneck edges sumnetwork correspond blocks undirected graph ieee transactions information theory table iii function values transmitted across bottleneck edges transposed sum network corresponding graph shown igure rate network ach message vector components vectors components number inside square brackets adjoining vector indicates particular component vector dash indicates value transmitted component used decoding terminal component first components transmitted bottleneck obtained following equation xbi xpj show partial sum satisfies terminals set tbi tpj terminals tbi recover sum messages present partial sum available tbi direct edges terminals set carry following operation part decoding procedure xpj xbi pptj xpj deg pptj also condition points deg mod hence coefficients partial sum messages set hpi xpj available direct edges hence recover sum remaining components available bottleneck edges used transmit information enable terminals set compute sum specifically construct flow bipartite graph whose one part corresponds points part corresponds blocks incidence determined submatrix ati since exists matrix specified row column sums use construct flow bipartite graph messages set xpi transmitted piecewise fashion bottleneck edges manner similar proof theorem arguing way one show network code based flow solution allows obtain value information transmitted bottleneck edges set hpi terminal computes sum part decoding procedure since deg mod every term rhs except coefficient since knows value subtract multiple recover relevant partial sum messages present partial sum available direct edges hence also compute value sum proposition describes families incidence structures constructed admit linear network codes upper bound computation capacity obtained corollaries describe linear network code corresponding incidence structures satisfy qualifying conditions upper bounds theorem computation capacity obtained using algorithm proposition incidence structure finite field exists linear network code satisfies following listed design normal transpose design normal obtained using higher incidence matrix transpose obtained using higher incidence matrix proof suppose construct using algorithm dimension following linear network code xpi xbj satisfies every terminal following manner terminal tpi receives messages present partial sum transmitted along direct edges hence compute sum terminal carry following operation xpi xbj xpi blt xbl since coefficients sum equal sum messages set xpi hbj messages present set available tbj direct edges linear network code gives proposition following manner let incidence matrix design let higher incidence ieee transactions information theory matrix defined corollary design proofs corollaries ati ati ati ati ait ait ait ait thus whenever matrices zero matrix scalar linear network code achieves computation capacity associated vii iscussion comparison prior work discussion sections establishes computation capacity derived several classes incidence structures discuss broader implications results appealing existence results incidence structures bibds subject much investigation literature combinatorial designs particular following two theorems theorem theorem exists bibd also known steiner triple system mod theorem theorem exists bibd mod particular results show infinite family steiner triple systems bibds block size since steiner triple system demonstrate existence whose computation capacity greatly affected choice finite field used communication proposition consider normal constructed using design computation capacity odd computation capacity normal constructed using computation capacity otherwise proof number blocks design equal corollary odd computation capacity constructed using steiner triple system moreover proposition construct linear network code rate equal upper bound hand computation capacity proposition number blocks design recover result computation capacity normal constructed using manner similar previous case thus result shows network computing sum even characteristic capacity capacity goes zero odd characteristic moreover dichotomy unique prime number similar results hold derived higher incidence structures corollary fig schematic shown represents undirected graph three components denotes star graph vertices one vertex degree rest degree vertices maximum degree three star graphs respectively addition connected connected deg deg deg theorem two integers mod design repeated blocks exists number blocks design consider normal evaluated obtained using higher incidence matrix corollary proposition computation capacity hand divisor theorem proposition computation capacity normal constructed using higher incidence matrix thus computing sum field whose characteristic divides parameter done rate however field characteristic divide computation done rate goes zero theorem describes infinite family bibds existence results bibds particular exist bibds value given table section example exists design whenever mod choice bibd infinite family construct corresponding normal whose computation capacity particular finite field found using corollary proposition even though theorem states existence satisfy qualifying conditions explicit constructions rare transposed obtained undirected graph regular computation capacity show involved dependence finite field alphabet following example demonstrates example consider transposed obtained applying algorithm undirected graph shown figure corollary gives upper bound computation capacity transposed sumnetwork based finite field alphabet upper bound three different choices follows ieee transactions information theory upper bound upper bound upper bound use proposition check construct linear network code case rate respective upper bound focus appropriate submatrix case see satisfies required condition row column sums rows corresponding vertices order shown indicate row vectors size specified subscripts using one verify appropriate submatrix three choices satisfies conditions proposition hence construct linear network code case thus previous example demonstrates computation capacity particular need take one two possible values range different values based finite field chosen generalize example obtain arbitrary different possible values computation capacity constructed unit maximum flow source terminal modify construction edge network capacity specifically following result shown proposition let denote obtained applying algorithm matrix dimension integer let denote obtained modifying algorithm structure edge cap satisfies qualifying conditions theorems computation capacity proof since satisfies conditions theorem exists vector linear network code every edge unitcapacity edges tail head tail every edge apply network code except distinct edges transmit encoded value thus need transmit symbols edges integer one appropriately multiply constant since also satisfies modified network code rate conditions theorem upper bound computation capacity applying argument get upper bound computation capacity matches rate modified vector linear network code described result interpreted follows consider class maximum flow pair least results indicate always demonstrate existence sumnetwork computation capacity strictly smaller indicates crucial role network topology function computation comparison prior work work rai das closest spirit work authors gave construction procedure obtain computation capacity equal two natural numbers procedure involved first constructing whose capacity edge unitcapacity inflating capacity edge sumnetwork modified shown computation capacity work significant generalization work particular capacity obtained applying algorithm incidence matrix complete graph vertices provide systematic procedure constructing sumnetworks much larger classes incidence structures authors also posed question smaller sumnetworks lesser sources terminals capacity existed using procedure described paper answer question affirmative example normal undirected graph figure computation capacity nine sources terminals obtain computation capacity using method described would involve constructing normal complete graph vertices would source nodes terminal nodes shown class maximum flow pair enough guarantee solvability network code rate exists counterexample observed shown figure characterization computation capacity family provides significantly general impossibility results similar vein particular note edge version maximum flow sourceterminal pair least suppose consider class consider complete graph vertices consider obtained applying procedure edge added capacity computation capacity sumnetwork less implies sourceterminal pair necessary condition ensuring sumnetworks solvable written similar argument made finding undirected graph whose incidence matrix satisfies condition theorem minimal ieee transactions information theory viii onclusions uture ork large class function computation problems directed acyclic networks notion computation capacity central function computation problems various counterexamples problem instances used different authors obtain better understanding solvability computation capacity general networks provide algorithm systematically construct sumnetwork instances using combinatorial objects called incidence structures propose novel upper bounds computation capacity cases give matching achievable schemes leverage results existence integer matrices prescribed row column sums demonstrate dependence computation capacity underlying field characteristic rather strong several opportunities future work proposed linear network codes constructed require corresponding incidence structures specific property particular techniques work case diagonal matrix would interesting find capacity achieving network codes cases diagonal generally would interesting obtain achievability schemes upper bounds general topologies eferences korner marton encode sum binary sources ieee trans info vol orlitsky roche coding computing ieee trans info vol mar doshi shah medard effros functional compression graph coloring ieee trans info vol aug ahlswede cai yeung network information flow ieee trans info vol koetter medard algebraic approach network coding transactions networking vol oct yeung cai linear network coding ieee trans info vol feb dougherty freiling zeger insufficiency linear coding network information flow ieee trans info vol cannons dougherty freiling zeger network routing capacity ieee trans info vol mar dougherty freiling zeger unachievability network coding capacity ieee trans info vol june jaggi sanders chou effros egner jain tolhuizen polynomial time algorithms multicast network code construction ieee trans info vol june huang ramamoorthy achievable region double unicast problem based minimum cut analysis ieee trans vol multiple unicast capacity directed acyclic networks trans networking vol lehman lehman complexity classification network information flow problems proceedings fifteenth annual symposium discrete algorithms jan kamath tse wang hard ieee intl symposium info june appuswamy franceschetti karamchandani zeger network coding computing bounds ieee trans info vol feb huang tan yang upper bound function computation directed acyclic networks ieee info workshop april appuswamy franceschetti karamchandani zeger linear codes target function classes network computing capacity ieee trans info vol sept ramamoorthy communicating sum sources network ieee intl symposium info july rai dey network coding ieee trans info vol ramamoorthy langberg communicating sum sources network ieee select areas vol stinson combinatorial designs construction analysis springer olmez ramamoorthy fractional repetition codes flexible repair combinatorial designs ieee trans info vol rouayheb ramchandran fractional repetition codes repair distributed storage systems annual allerton conference control computing sept tang ramamoorthy coded caching low subpacketization levels intl symp network coding netcod dec coded caching networks resolvability property ieee intl symposium info low subpacketization schemes coded caching online available https rai das capacity sumnetworks ieee info workshop rai das capacity annual allerton conference control computing oct tripathy ramamoorthy undirected graphs construction capacity analysis annual allerton conference control computing sept das rai number sources terminals capacity national conference communications ncc feb tripathy ramamoorthy capacity different message alphabets ieee intl symposium info june brualdi combinatorial matrix classes cambridge university press vol mirsky combinatorial theorems integral matrices journal combinatorial theory vol teirlinck without repeated blocks exist discrete mathematics vol colbourn dinitz handbook combinatorial designs crc press ardhendu tripathy student department electrical computer engineering iowa state university obtained degree electrical engineering indian institute technology kanpur research interests areas information theory machine learning signal processing aditya ramamoorthy received degree electrical engineering indian institute technology delhi degrees university california los angeles ucla respectively systems engineer biomorphic vlsi data storage signal processing group marvell semiconductor inc since fall electrical computer engineering department iowa state university ames usa research interests areas network information theory channel coding signal processing bioinformatics nanotechnology ramamoorthy served editor ieee transactions communications currently serving associate editor ieee transactions information theory recipient early career engineering faculty research award iowa state university nsf career award professorship
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isolated subgroups generic permutation representations jan yair glasner daniel kitroser julien melleray abstract let countable group sub compact metric space subgroups chabauty topology sub collection isolated points denote polish group permutations countable set following properties equivalent dense sub admits generic permutation representation namely exists hom collection permutation representations hom permutation isomorphic hom call groups satisfying properties solitary examples solitary groups include finitely generated lerf groups groups countably many subgroups introduction let countable group sub space subgroups endowed chabauty topology makes compact metrizable totally disconnected space easiest way define topology embed sub closed subset induce tychonoff topology group acts sub continuously conjugation one naturally led question structure topological space sub generally topological dynamical system sub reflected algebraic structure structure theory compact spaces leads consider decomposition sub sub collection isolated points isolated subgroups shall refer sub complement isolated subgroups special algebraic point view clearly countably many one think isolated subgroups algorithmic terms subgroups detectable recognizable via finite algorithmic procedure subgroup sub isolated identified making finite number membership tests specific elements sometimes convenient think terms schreier graphs let symmetric generating set subgroup sub isolated one find finite algorithm would recognize schreier graph sch schreier graphs group note might well infinite consequently schreier graphs question may fail locally finite still algorithm allowed look finitely many edges characterizations easy see isolated groups always finitely generated special case finitely generated every finite index subgroup isolated obtain inclusions subfi subfg subfi subfg stand finite index finitely generated subgroups respectively mathematics subject classification primary secondary key words phrases isolated subgroups solitary groups lerf groups amenable groups ample generics amenable actions clear always discrete countable open subset sub main new definition following definition group called solitary isolated points dense sub let countable set group permutations topology pointwise convergence makes polish group separable metrizable complete space hom space permutation representations clearly also polish space natural action hom hom two permutation representations orbit said isomorphic permutation representations interested baire generic properties permutation representations particular existence generic permutation representation sense following definition group said generic permutation representation permutation representation hom whose orbit hom permutation isomorphic hom turns existence generic permutation representation captured structure topological space sub theorem main theorem countable group admits generic permutation representation solitary definition group called subgroup separable locally extended residually finite lerf short every finitely generated subgroup intersection finite index subgroups equivalently finitely generated subgroup closed profinite topology examples lerf groups include finitely generated abelian groups free groups surface groups generally limit groups grigorchuk group many lamplighter groups recently lerf property attracted lot attention agol proof lerf property fundamental group closed hyperbolic central ingredient solution thruston virtual haken conjecture following theorem analogous main theorem shows particular finitely generated lerf groups solitary theorem let finitely generated group following conditions equivalent lerf collection subfi finite index subgroups dense sub generic permutation representation whose orbits finite remark fact first third condition equivalent also follows earlier work rosendal proposition proves lerf generic permutation representation finite orbits theorem shows finitely generated lerf group admits generic permutation representation note theorem formulated groups acting isometries rational urysohn space see remark last paragraph pointed proofs adapt metric spaces notably urysohn space distances infinite countable set paper state prove detailed version theorem holds also countable groups leave realm finitely generated groups isolated subgroups solitary groups longer generalize finite index subgroups lerf groups respectively theories goes way impression settings choice isolated subgroups solitary groups natural one following theorem summarizes examples structural results solitary groups theorem properties solitary groups finitely generated lerf groups solitary sub countable solitary let short exact sequence countable groups finitely generated abstract group solitary free product two countable groups solitary one following two options hold lerf finitely generated solitary trivial situation considered condition identical one appearing famous rips construction naturally leads following question true every finitely generated solitary group placed short exact sequence solitary hyperbolic finitely generated abstract group theorem tightly connected notion ample generics polish groups adopt notation first paper express property term generic orbits presentation varieties definition say polish group ample generics hom admits generic orbit every notion ample generics first introduced order study small index property polish groups namely every subgroup index open indeed additional consequences automatic continuity abstract homomorphisms separable group subsequently established groups ample generics refer readers mentioned papers see also survey references therein terminology place theorem shows following two well known facts ample generics finitely generated free groups lerf fact two different realizations phenomenon cases one seeks generic hom study groups ample generics one fixes precisely lets range finitely generated free groups lets polish group vary whereas study solitary groups fix polish group consider class countable groups give rise generic orbit view natural characterization arises theorem following question seems natural question given polish group describe class finitely generated groups hom generic particular answer contain finitely generated free groups whenever ample generics group probably simplest example polish group ample generics many others turn generalizing theorem notion lerf groups different direction subgroup called mean see also definitions subgroups generalize finite index subgroups much way amenable groups generalize finite groups view theorems generalize notion lerf groups follows definition group amenably separable short set subgroups dense sub view theorem every lerf group another obvious source examples class amenable groups since subgroups chapter work initiate study groups hope notion prove useful generalization different properties lerf amenability situation perhaps reminiscent way sofic groups simultaneously generalize notions residual finiteness amenability terms analogue theorem following theorem countable group generic action countable set action every orbit amenable properties groups theorem following properties hold class groups lerf groups amenable groups class groups closed free products exist groups neither lerf amenable group property lerf higher rank lattices simple lie groups satisfy congruence subgroup property never paper arranged follows section dedicated systematic investigation topological spaces sub hom standard stabilizer map hom sub section prove theorem chapter dedicated solitary groups prove theorems finally chapter prove theorems give examples non amenable non lerf groups results work also appear part dissertation second author dense generic properties actions subgroups space permutation representations hom let countable set full symmetric group bijections onto endow topology pointwise convergence makes polish topological group words topological group separable admits complete metric latter fact important shows baire space though never consider specific metric explicit basis topology given sets finite always put product topology still polish every let countable group given presentation identify hom closed subset via following embedding hom thus hom closed subspace induced topology makes polish space note topology depend choice presentation basis topology hom given hom hom finite happens finite sets form basis hom mentioned introduction see equation group acts left hom orbits action exactly standard isomorphism classes permutation representations well known two permutation representations isomorphic contain transitive components appearing multiplicity transitive components turn isomorphic quasiregular actions form sub sub countable finite collection subgroups denote isomorphism class permutation representation exactly transitive components isomorphic care due notation always possible identify action element hom sum finite finite index underlying set finite case identify action element hom via arbitrary bijection different choices bijection yield different points corresponding orbit make frequent use following definition let let word trace set tracew space subgroups let countable group consider space subsets equipped product topology compact metrizable space let sub denote set subgroups easy verify sub closed compact metrizable space induced topology sub called chabauty topology basis topology given sets sub sub finite sub denote env sub envelope subsets sub env closed sub finitely generated env also open denoting subfg sub collection finitely generated subgroups easy check collection env subfg sub env subfg forms another basis topology sub isolated subgroups let occ denote isolated points sub subgroups open conjugacy classes respectively note subgroup occ open neighborhood consisting conjugates subsets open conjugation invariant proposition basic properties subgroups occ every finitely generated subfg env open neighborhood finitely generated every finite index subgroup isolated subfg isolated particular every finitely generated maximal subgroup isolated proof clear occ sets open opposite inclusion follows baire theorem let occ let gkg conjugacy class open definition since countable countable finite union closed points baire theorem one open since transitive points open particular proves finitely generated find sequence finitely generated subgroups clearly topology sub none subgroups equal finitely generated clear follows directly finally finitely generated every finite index subgroup follows directly thus finitely generated group isolated subgroups form class subgroups sits finitely generated subgroups subgroups finite index namely subfg subfg find useful think isolated subgroups generalizations finite index subgroups generic properties subset polish space called generic alternatively residual contains countable intersection dense open sets baire category theorem generic sets always dense say property generic generic element property set property generic paper interested generic properties permutation representations hom simplest example hom following well known proposition summarizes generic properties space proof exercise baire category theorem leave readers chose mention main theorem proof basically far reaching generalizations fact proposition residual conjugacy class conjugacy class described explicitly terms definition proposition says admits generic permutation representation definition say admits generic permutation representation exists permutation representation hom whose orbit action hom residual hom properties stabilizer map given permutation representation hom point denote stabilizer point fixing gives rise stabilizer map hom sub lemma main lemma every stabilizer map hom sub continuous surjective open proof clear map surjective let hom let finite proves map continuous prove map open let hom basic open neighborhood hom point therein extending may assume set contains identity symmetric exhibit open neighborhood sub let orbit stabilizer identify orbit map set let finite symmetric set claim basic open set satisfies requirements set product fixing group complete proof finding consider finite subset define partial map map injective domain since definition open set every similar reason map partially respects action sides sense define bijection satisfying following conditions extends namely identity let define action acts action let hom defined easy verify also every indeed choice hence concludes proof open corollary subset sub denote hom dense hom dense sub hom sub generic whenever particular proof suppose dense sub let hom let finite let lemma set dense hom exists hom denote note since apply argument get permutation representation agrees stabilizers points belonging get action hom defined belongs every stabilizer point belonging set repeating process described get finitely many steps action set every finally extend action action hom way since stabilizers points belong thus proves part assume write open every every since open get lemma open every since countable means also part proven lemma let sub open generic permutation representation hom proof let hom claim generic baire theorem would enough show open dense fact open follows directly continuity map prove density add new orbits stabilizer far away made explicit following way given basic open set hom want find element fix subgroup consider set endowed diagonal action hom given ggm let identity map tosthe copy contained restriction finite set let extension bijection one easily checks required lemma given hom arbitrarily close action infinitely many fixed points proof given finite sets seek action infinitely many fixed points consider action obtained adding countably many fixed points desired action obtained intertwining action via bijection property identity restricted lerf property section prove theorem fact promised introduction prove following slightly general version theorem arbitrary countable groups necessarily finitely generated theorem let finitely generated group following conditions equivalent lerf collection finite index subgroups dense sub collection permutation representations whose orbits finite dense hom collection permutation representations whose orbits finite generic hom generic permutation representation whose orbits finite countable necessarily finitely generated first three conditions equivalent easy verify theorem implies theorem finitely generated apparent complications fails finitely generated lack thereof proof theorem emphasize one main points isolated subgroups natural finite index subgroups setting mention equivalence first third fourth conditions already present rosendal works see discussion introduction proof theorem countable group every subgroup ascending union finitely generated subgroups hence subfg dense sub lerf property implies subfg subfi every finitely generated subgroup descending intersection finite index subgroups shows follows directly corollary prove assume given finitely generated infinite index subgroup hsi let hom permutation representation isomorphic quasiregular action way identified trivial coset action finite orbits proving lerf property assume finitely generated implications obvious enough prove implication start describing generic permutation representation hom let subfg enumeration finite index subgroups let representation define namely take countably many copies representation list let act naturally disjoint union corresponding sets follows corollary applied open dense set subfi sub collection permutation representations whose orbits finite generic follows lemma applied open set sub collection permutation representations appears countably many times transitive component also generic baire category theorem generic permutation representation finite orbits appears countably many times permutation representation must permutation isomorphic remark examples demonstrate infinitely generated lerf group longer true generic permutation representation finite orbits still true however restriction generic action every finitely generated subgroup finite orbits even true restriction generic permutation admits well defined isomorphism type isomorphism permutation representations details proof quite similar proof leave reader order demonstrate use theorem give basic examples analyze situation free groups providing short proof hall theorem free groups lerf proposition let free group lerf generic permutation representation hom transitive countable lerf group finitely generated sub perfect proof first notice hom let subgroup dense finitely supported permutations clearly hom hom dense set permutation representations whose orbits finite proves establishing theorem prove enough baire theorem show set hom open dense openness obvious density fix free generating set given basic open set hom finite set contains words involve finitely many generators say find setting every defining way finally fails finitely generated finite index subgroups proposition none isolated also lerf finite index subgroups dense particular isolated subgroups proves solitary groups section dedicated proof theorem proof assume dense sub let representatives conjugacy classes isolated subgroups made distinction groups finite index normalizer groups denote corresponding quasiregular actions important note finitely many points whose stabilizer action infinitely many similar points terminology place describe generic permutation representation countably many orbits isomorphic one orbit isomorphic applying corollary open dense subset sub conclude generic permutation representation stabilizers words generic lemma applied open set know generic representation infinitely many points whose stabilizer immediately implies note identical argument tells generic representation countably many points whose stabilizer even one orbit isomorphic enough ensure add information coefficients left prove generic representation one orbit isomorphic every order simplify notation hence fix index denote let denote sub conjugacy class bad event existence two different orbits stabilizers hom continuity stabilizer map lemma sets hence also closed baire theorem suffices prove nowhere dense assume contrary basic open set replacing necessary smaller basic open set may assume let define similarly consider quasiregular action unique invariant isomorphism since assumption infinitely many possible choices points would work similarly let two choices satisfying additional property let bijection define new action following formula otherwise easy verify completes proof first implication assume exists generic permutation representation hom assumption isomorphism class hom collection subgroups appearing point stabilizers given sub arbitrary basepoint since lemma map surjective continuous dense sub particular show thus showing latter dense completing proof let conjugacy class proposition enough show open fails open must empty interior acts transitively since countable follows baire theorem sub dense set corollary hom sub also dense set contradicts fact dense intersection two sets empty conclude section proving theorem proof theorem view fact finitely generated group finite index subgroups isolated follows directly comparison theorems density isolated points statement general fact countable baire spaces indeed set sub nowhere dense since countable union closed nowhere dense points consider short exact sequence statement since finitely generated env clopen easy verify correspondence principle subgroups subgroups containing gives rise homeomorphism env sub claim follows immediately well known free product two lerf groups lerf one groups say trivial hei situation clear thus establish show neither group trivial fails lerf solitary let subfg subfi finitely generated subgroup approximated finite index subgroups let map identity trivial show ker sub subgroup approximated isolated subgroups indeed let finite set neighborhood wsub contain finite index subgroup use define open neighborhood clear every subsub thus theorem proved view following lemma seems useful right lemma let two countable groups infinite proof convenient argue level actions schreier graphs note hom hom hom denote isomorphism namely unique action whose restriction let sub assume let hom action isomorphic quasiregular action identified trivial coset note action transitive need transitive still assumption know orbit infinite argument simple enough obtain approximating actions form carrying small perturbations action since orbit infinite arbitrarily small perturbations action still affecting stabilizer point elaborate basically complete proof let finite sets ascending whole finite sets ascending generating set assume course finitely generated take fixed symmetric set generators set since infinite find bijections additional properties let hom fixed action using data construct sequence actions hom follows let clear definitions particular consequently course hom let hom trivial regular left action latter defined via arbitrarily chosen identification play role discussion finite replace regular left action countably many copies action make sure action infinite set use obtain two convergent sequences actions hence two convergent sequences subgroups sequences different element thus least one sequences eventually constant proving limit point isolated sketch another proof theorem briefly sketch another proof theorem along lines arguments first note countable group action hom topologically transitive given two nonempty open subsets hom always exists equivalently exists elements hom dense conjugacy class true simply two actions infinite countable set embed third one instance obtained considering disjoint union two infinite countable sets acting first copy second copy closure conjugacy class contains proving desired result brings setting following lemma equivalence criterion using due rosendal equivalence conditions appears new seems potentially useful including even though needed lemma assume polish group acting continuously topologically transitively polish space following conditions equivalent exists comeager orbit open identity neighborhood collection points somewhere dense int dense open identity neighborhood nonempty open subset exists nonempty open nonempty open one proof fix open identity neighborhood sequence group elements exists comeager orbit baire category theorem every orbit exists somewhere dense translating deduce somewhere dense proves implies assume holds fix open identity neighborhood nonempty open subset using assumption continuity action may find symmetric open identity neighborhood open closure contains nonempty open contained thus holds finally assume false since exist dense orbits orbit must meager comeager topological law case orbits meager given family closed subsets empty interior must nonempty interior proving exists open neighborhood nowhere dense thus union sets form nowhere dense ranges countable basis neighborhoods one sets must nonmeagre hence since sets borel comeagre nonempty open assume holds pick witnessing assumption amounts saying dense open nonempty open implies dense comeagre contradiction fact must nowhere dense generic element hence also need understand criterion satisfied countable group hom given open set let denote number distinct elements may pick number minimal among elements enlarging shrinking needed reduce situation situation orbits elements interfere enables reduce case singleton working inside polish space transitive consider open set let denote stabilizer action let group permutations fixing finite set enlarging needed assume readily checked two elements belong iff stabilizers let discussion shows criterion lemma satisfied iff exists open set contained stabilizer two elements exist finite sets subgroup sub thus isolated point sub since arbitrary subgroup encode arbitrary open neighborhood established exists generic action hom solitary one try use approach understand exists generic conjugacy classes hom polish groups structure becomes complicated analysis harder carry particular reduction transitive actions longer works one case one aut automorphism group random graph reasoning much way one obtains following criterion probably simplified proposition given countable group exists generic element hom aut iff following condition satisfied subgroups finite exists finitely generated subgroups contains double coset space finite one property hard grasp imply finitely generated finitely generated subgroup intersection finitely generated subgroups finite must group satisfying previous conditions lerf definition action discrete countable group called amenable satisfies one following equivalent conditions every finite admits subset finite set exists finitely additive probability measure action transitive form conditions equivalent following acts continuously compact space admits invariant borel measure transitive case sometimes convenient adopt group theoretic terminology follows definition subgroup group called quasiregular action amenable equivalence three conditions classical definition amenable action always admits sequence finite subsets lim recall following remark chosen increasing respect inclusion defined introduction group set subgroups dense sub prove theorem giving characterization language generic actions proof theorem denote coam set subgroups note denote hom amenable hom coam generic dense hom lemma image set dense subset sub consisting subgroups conversely assume coam dense sub wish prove set generic hom enough show generic hom every density assured hypothesis fact coam lemma show enough show condition specific finite set open finite assume case hom seek open neighborhood hom still contained orbit still every every pick group element let desired neighborhood given mentioned introduction lerf groups amenable groups examples groups order give example aseparable group neither lerf amenable first prove closed taking free products theorem let countable groups proof every element hom hom defined setting expanding definition free product every finite subsets let hom contains set want prove finite generic hom since countable enough show sets open dense every finite subsets argument shows open given proof theorem fix prove dense hom let hom let finite find hom hom assume exist hom hom actions amenable every orbit let case orbits contained finite let finite set containing let define representation hom declaring every every element fixed point every element identifies notice since well defined acts particular agrees finite invariant contains setting finite orbit set case contains either infinite infinite assume loss generality contains infinite denote let increasing since sets finite none stabilize implies particular contains set let minimal length respect canonical presentation denote lemma assume infinitely many fixed points particular exits set acts trivially intersect finite set trace think word corresponding given presentation denote let permutation order takes bijectively onto acts trivially define action hom since acts trivially every element fixed hence thus thus notice minimality length trace acts trivially trace means since contained implies contained required recall group group well known solvable hence amenable proposition every group lerf proof write notice hsit hsn hsi thus element hsi hsit separated hsit homomorphism finite group corollary exist groups proof let amenable hence proposition hand lerf since lerf lerf property passes subgroups also amenable since contains free subgroup two generators order complete proofs statements promised introduction prove following proposition group kazhdan property lerf particular following groups never groups property residually finite particular simple group property irreducible lattices higher rank lie groups compact factors satisfy congruence subgroup property proof follows directly fact transitive action amenable finite argument follows property action amenable set invariant vector taking kazhdan constants deduce existence invariant vector since action transitive must constant function constant function finite lattice statement theorem lerf strong approximation theorem window every zariski dense subgroup finite index closure topology coincides profinite topology assumption kazhdan theorem lattice property statement follows note conjecturally congruence subgroup property automatically holds higher rank lattices indeed proven many different cases particular groups sln good examples generically finite groups conjecture lerf property never occur nontrivial way property groups namely conjecture countable group kazhdan property lerf finite pointed matthew stover similar question already asked long reid question grateful constructive comments previous version paper eli glasner matthew stover wishes thank romain tessera interesting conversations suggestions solitary groups work written first author sabbatical university utah grateful math department hospitality acknowledges support national science foundation grants dms rnms geometric structures representation varieties gear network enabled visit first second authors partially supported israel science foundation grant isf third author partially supported agence nationale recherche grant grupoloco references ian agol virtual haken conjecture doc appendix agol daniel groves jason manning burns finitely generated subgroups free products austral math grigorchuk kravchenko lattice subgroups lamplighter group internat algebra grigorchuk wilson structural property concerning abstract commensurability subgroups london math soc eli glasner benjamin weiss topological groups rokhlin properties colloq marshall hall subgroups finite index free groups canadian wilfrid hodges ian hodkinson daniel lascar saharon shelah small index property structures random graph london math soc daniel kitroser generic actions countable groups phd thesis university negev alexander kechris christian rosendal turbulence amalgamation generic automorphisms homogeneous structures proc lond math soc long reid subgroup separability virtual retractions groups topology alexander lubotzky dan segal subgroup growth volume progress mathematics verlag basel julien melleray todor tsankov generic representations abelian groups extreme amenability israel nikolay nikolov strong approximation methods lectures profinite topics group theory volume london math soc stud texts pages cambridge univ press cambridge residual finiteness free products respect subgroups izv akad nauk sssr ser christian rosendal finitely approximable groups actions part property symbolic logic christian rosendal finitely approximable groups actions part generic representations symbolic logic bhaskara rao bhaskara rao category analogue law proc amer math peter scott correction subgroups surface groups almost geometric london math soc london math soc henry wilton hall theorem limit groups geom funct yair glasner department mathematics university negev sheva israel yairgl daniel kitroser department mathematics university negev sheva israel kitrosar julien melleray claude bernard lyon institut camille jordan cnrs umr boulevard novembre villeurbanne cedex melleray
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feb list heaps andrew february abstract paper presents simple extension binary heap list heap use list heaps demonstrate idea adaptive heaps heaps whose performance function size problem instance disorder problem instance focus presortedness input sequence measure disorder problem instance number practical applications rely heaps deal input random even random input contains presorted subsequences devising heaps exploit structure may provide means improving practical performance present basic empirical tests support claim additionally adaptive heaps may provide interesting direction theoretical investigation introduction heap data structure holds finite set items item associated key drawn totally ordered set heaps support following operations make heap insert find min delete min decrease key delete meld create return new empty heap insert item key heap return reference stored return reference item minimum key stored heap delete item minimum key heap return decrease key item heap delete item heap return heap formed taking union heaps binary heap introduced williams simplicity speed made generalization heap popular choice andrewfrohmader practice supports insert delete min decrease key log time used sort items log matches lower bound comparison sort vuillemin introduction binomial queue added meld list operations supported log fibonacci heaps extension binomial queue achieved amortized time insert decrease key meld decrease key result particularly important improved bounds number graph algorithms recently structures achieved time decrease key meld see work produced interesting important theoretical results failed yield structure consistently outperforms original binary heap variants practice paper return binary heap develop simple extension list heap straightforward extension given adaptive operations operations whose performance depends problem size also level presortedness disorder problem instance bit work gone developing theory adaptive sorting algorithms see knowledge work migrated related work heap data structures believe adaptive heaps may provide interesting angle theoretical investigation additionally may provide means improving empirical performance current heap variants list heap first step direction list heaps support decrease key insert delete min log number lists list heap show number lists list heap function size problem instance disorder problem instance returned binary heap simplicity ubiquity without costs list heaps lose insert decrease key meld sophisticated structures preliminaries present notational conventions definitions used remainder paper let sequence distinct elements totally ordered set monotonically increasing increasing monotonically decreasing decreasing sequence monotonic either increasing decreasing head sequence tail set cardinality sequence length two sequences concatenation sequence sequence contains elements write sequence obtained deleting zero elements called subsequence subsequence hxi consecutive indices consecutive integers let hxi hxk subsequences intersection subsequence obtained deleting similarly union subsequence obtained deleting either disjoint let set disjoint subsequences union subsequences equals partition adaptive sorting section gives brief review adaptive sorting heaps solve generalized sorting problem adaptive sorting provides intuition adaptive heaps might useful detailed survey adaptive sorting see consider sorting problem take input arbitrary sequence elements totally ordered set return permutation sequence increasing sorted order comparison based sorting lower bound log however clear lower bound must always hold input sequence already sorted one element place concatenation two sorted subsequences lower bound refined account disorder input sequence main achievements adaptive sorting literature proposing variety measures disorder proving new lower bounds respect measures developing sorting algorithms whose performance matches new lower bounds developing partial order set measures stop direct reader information outline paper remainder paper organized follows section discusses adaptive heaps might worth developing section presents list heaps structure operations section presents results series brief empirical tests suggesting list heaps may promise practice section summarizes results obtained adaptive heaps section presents reasons developing adaptive heaps use term adaptive heap loosely throughout paper refer heap whose performance function level presortedness disorder input sequence clearly complications glossing largest deal decrease key heap problem instances disorder different types related presortedness input sequence starting point formalize notions adaptive sorting literature fairly easy extend results adaptive sorting arbitrary sequences insert delete min decrease key may prove challenging build heap whose performance adapts presortedness input sequence lower bound sorting restated function presortedness input sequence lower bound arbitrary heap problem restated function disorder problem instance delete min operation performance available disorder problem directly impacts bound delete min thus restate bound delete min log measure disorder less equal number elements heap intuitively bound delete min ranges sorted input log random input bounds algorithms rely heaps dijkstra shortest path algorithm similarly adjusted reflect disorder example restate dijkstra bound log number vertices number edges measure disorder discussion suggests applications input sequences level presortedness could benefit adaptivity less obvious even applications random input sequence benefit adaptivity benefit asymptotic performance constant factors let random sequence elements minimum number increasing subsequences partitioned approximately subsequences show section create extension binary heap adaptive minimum number increasing subsequences thus binary heap constructed performs roughly log comparisons delete min adaptive heap constructed perform delete min log comparisons cutting constant factor half list heaps section introduces list heap list heap structure closely mirrors binary heap hope similarity makes changes required add adaptivity heaps clear additionally believe adaptive variants binary heap greatest potential immediately useful practice however binary heaps flawed choice decrease key insert take log time implementations operations clearly possible thus list heap optimal problem instances tolerate flaw show construct heap whose performance function number measures disorder particular focus developing heaps adaptive runs enc defined later measures disorder partition input monotonic subsequences particularly simple heap use paper focuses insert delete min decrease key assumes keys unique first consider structure heap next outline operations adaptive respect runs finally present operations adaptive respect enc structure list heap array circular doubly linked lists nodes node unique key associated throughout paper refer node key interchangeably use notation refer jth node list list heaps must maintain two invariants list heap nodes increasing sorted order key value lists arranged heap order array define invariants view list heap standard binary heap elements binary heap element list key list key head node structure minimum node head node list refer root list heap figure list heap structure sorted linked lists key feature list heap enable capture existing order input sequence result list heaps given operations adaptive respect measures disorder runs adaptive list heap section present operations list heap adaptive respect runs abbreviated list heap show performance heap operation function number lists heap insert operation partitions sequence inserted elements lists less equal number runs sequence definition run consecutive decreasing subsequence maximum length subsequence hxj run consecutive integers total number runs sequence gives measure disorder example consider following random sequence sequence sixteen elements partitioned eight runs decreasing sequence consist one run increasing sequence elements contain runs one element outline operations list heap insert insert new item heap following less head node recall denote set otherwise set set repeat loop terminates append front list note might equal case new empty list need created appended figure shows list heap generated inserting random sequence empty heap delete min perform delete min heap remove head node root list set return value function empty replace last list heap empty need swap either way point might heap order restore heap order calling heapify identical heapify operation binary heap except manipulates entire lists instead individual nodes heapify list compare two children less children nothing otherwise swap smallest child repeat decrease key perform decrease key node set key new value two cases head node decreasing key maintains sorted order list may destroy heap order array call heapify restore heap order heapify list compare parent swap lists recurse heap order restored head node may longer sorted less left sibling list sorted order remove reinsert list heap using insert routine analysis clear descriptions functions run log time number lists list heap theorem sequence consecutive insert operations partitions inserted elements lists less equal minimum number runs input sequence proof say consecutive sequence elements inserted heap must show appended existing list couple cases consider appended one ancestors also appended one ancestors appended one ancestors appended new list must created compared first since thus appended one ancestors instead appended ancestor created compared first compared ancestors since appended ancestor must less one ancestors appended existing list enc adaptive list heap section present operations list heap adaptive respect runs enc abbreviate list heap due levcopoulos petersson enc proposed skiena definition shuffled upsubsequences minimum number increasing subsequences sequence partitioned differs runs subsequences increasing required consecutive example sequence partitioned following seven increasing subsequences fact optimal partition sequence seven definition encroaching set ordered set increasing sequences head head tail tail thus increasing sequences nest encroach upon one another skiena describes encroaching set algorithm melsort builds given input sequence permutation ordered set build encroaching set follows item fits either end one increasing sequences put otherwise form new sequence place item oldest upon fits note strategy bears lot similarities patience sorting see encroaching set example sequence thus enc sequence five adaptive sorting enc optimal algorithm also optimal runs optimal therefore focus developing heap adaptive enc thereby also achieving adaptivity respect runs emphasize paper treats notation adaptivity informally order achieve level adaptivity respect enc need alter insert function decrease key delete min functions remain insert insert function attempts build encroaching set adapt skiena strategy dynamic context heaps fixed input sequence let new item insert heap context roughly equal encroaching list set three cases try insert front list use binary search lists find list minimizes note lists maintained heap order sorted order result binary search may approximate heuristic moreover binary search may failed consider direct parent appending front without checking parent could destroy heap order thus must also compare ancestors set repeat find done otherwise move case try insert tail list use binary search tail nodes find list tail minimizes tail found case create new list new node element insert list array analysis easy see insert routine described runs log number lists list heap theorem sequence consecutive inserts empty list heap partitions input optimal encroaching set proof simply observe insert function reduces list creation phase melsort heap starts empty algorithmic difference insert routine skiena melsort addition heap order check corollary given empty list heap perform consecutive insert operations elements uniformly random keys grows large expected number comparisons delete minimum item less equal half number comparisons required binary heap proof theorem lists form optimal encroaching set mentioned enc optimal partition also optimal less equal thus approaches something less equal number comparisons delete min standard binary heap enc adaptive list heap number comparisons less equal binary heap bound approaches empirical results implemented list heap list heap testing purposes made attempts optimize code results series brief empirical tests similarly unoptimized binary heap presented tests performed using codebase written workloads dimacs challenge modifications simple sorting routine results presented raw wallclock times divided minimum time attained heap thus minimum wallclock time times minimum list heap list heap binary heap sorted sorting random sorting random dijkstra table normalized wallclock times sorted sorting refers task inserting decreasing sorted sequence heap followed consecutive calls delete min random sorting refers task inserting random sequence heap followed consecutive calls delete min results presented sorting million random dijkstra refers dijkstra workloads dimacs results presented dijkstra strongly connected randomly generated network million nodes million edges test results suggest variant list heap may useful practice stress results presented derived simplistic testing merely suggestive list heaps may potential practice way final word empirical performance list heap conclusion paper introduced adaptive heaps heaps whose performance function size problem instance disorder problem instance introduced list heap generic structure endowed adaptive operations finally presented operations list heap adaptive respect number measures disorder input sequence discussion within paper relatively informal list heap introduced far theoretically optimal interest topic number directions additional empirical testing list heap needed presented two insert functions list heap clearly runs optimal choice likely depends intended application decrease key needed list heaps could implemented entirely arrays theory side list heap discussion adaptivity included paper leaves much desired working formalizing notions adaptivity developed variant fibonacci heap closer optimal theoretical perspective modification existing heap variants may provide even better results references david aldous persi diaconis longest increasing subsequences patience sorting theorem bull amer math soc gerth brodal efficient priority queues proceedings seventh annual symposium discrete algorithms soda pages philadelphia usa society industrial applied mathematics gerth brodal survey priority queues pages springer berlin heidelberg berlin heidelberg gerth brodal george lagogiannis robert tarjan strict fibonacci heaps proceedings annual acm symposium theory computing stoc pages new york usa acm thomas cormen charles leiserson ronald rivest clifford stein introduction algorithms third edition mit press edition dimacs dimacs challenge https priority queues vladmir derick wood survey adaptive sorting algorithms acm comput december michael fredman robert endre tarjan fibonacci heaps uses improved network optimization algorithms acm july daniel larkin siddhartha sen robert tarjan empirical study priority queues proceedings meeting algorithm engineering expermiments pages philadelphia usa society industrial applied mathematics levcopoulos petersson sorting shuffled monotone sequences information computation ola petersson alistair moffat framework adaptive sorting discrete applied mathematics steven skiena encroaching lists measure presortedness bit numerical mathematics dec jean vuillemin data structure manipulating priority queues commun acm april john william joseph williams algorithm heapsort commun acm june edsger wybe dijkstra note two problems connexion graphs
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balancing explicability explanations emergent behaviors planning tathagata chakraborti sarath sreedharan subbarao school computing informatics decision systems engineering arizona state university tempe usa rao feb abstract human aware planning requires agent aware intentions capabilities mental model human loop decision process involve generating plans explicable human observer well ability provide explanations plans generated paper bring two concepts together show agent account needs achieve plan generation process means search method mega effect provides comprehensive perspective means agent bringing together existing principles planning umbrella single plan generation process situate discussion context recent work explicable planning explanation generation illustrate concepts modified versions two planning domains well demonstration robot involved typical search reconnaissance task external supervisor human factor studies latter highlight usefulness proposed approaches ccs concepts computing methodologies artificial intelligence planning scheduling cognitive robotics computing computer systems organization robotics keywords planning explicable planning plan explanations explanation model reconciliation minimal explanations introduction first two authors contributed equally robot instead attempts update human mental model robot original plan intermediate model equivalent respect metric cost similarity optimal hence explicable two processes plan explanations explicability remained separate far role agent deliberative process considered planner either generates explicable plan best ability produces explanations plans required however may situations combination provide much better course action expected human plan costly planner model human might aware safety constraints cost communication overhead explanations high limited communication bandwidth consider example human working robot received software update allowing perform new complex maneuvers instead directly trying conceive sorts new interactions right away might end spooking user robot could instead reveal certain parts new model still using older model even though suboptimal rest interactions slowly reconcile drifted model user focus current paper try attain sweet spot plan explanations explicability often useful planning agent interacting human loop use process deliberation model task also model mhr human thinks refer figure mental model human addition physical model human essence fundamental thesis recent works plan explanations explicable planning summarized umbrella planning addition originally studied planning hap problems actions human hence actual human model robot belief also involved planning process need explicable planning plan explanations fact occur two models mhr diverge means optimal plans respective models may hence optimal behavior robot model inexplicable human loop explicable planning process robot produces plan closer human expected plan explanation process related work agents become pervasive daily lives need agents cognizant beliefs expectations humans environment well documented perspective task planning depending extent involvement human life cycle plan work direction ranged spectrum planning robot passively tries account plans humans cohabiting workspace explicable planning robot generates plans explicable predictable human observer plan explanations agent uses explanations bring human may different understanding agent abilities page planning general humans planners participating plan generation execution process together technical report asu tathagata chakraborti sarath sreedharan subbarao kambhampati evolving scope planning hap subsumption architecture hap figure expanding scope planning hap acknowledging need account mental model human loop deliberative process autonomous agent planner example choose bring human model closer ground truth using explanations via process called model reconciliation mrp otherwise inexplicable plan makes sense human updated model compute explicable plans closer human expectation capabilities stacked realize complex behavior paper concentrate explicability versus explanation form argumentation planning evolving scope planning ongoing efforts make planning illustrated figure initial work topic largely focused incorporating agent understanding human model decision making process since importance considering human understanding mhr agent actual model planning process also acknowledged sometimes implicitly later explicitly considerations engender interesting behaviors space plans models example model space modifications human mental model mhr used explanations reasoning actual model reveal interesting behavior affecting belief state human planning serendipity plan space agent use mhr compute joint plans teamwork generate behavior conforms human preferences expectations point view planner sense asymmetric epistemic setting nested beliefs models indeed existing literature epistemic reasoning also provide interesting insights planning process agent settings subsumption architecture hap different forms behavior composed form sophisticated forms behavior hierarchical composition behaviors viewed form subsumption architecture planning similar motivation illustrated figure basic reasoning engines plan mrp model reconciliation modules former accepts model planning problems produces plan latter accepts produces new model former operates plan space gives rise classical joint explicable planning depending models operating latter operates model space produce explanations belief shaping behavior composed form argumentation modules trading explanations explicability topic current paper planning general perspective design autonomy two important implications agent explain also plan setting compromise optimality possible explanations mind argumentation process known crucial function reasoning capabilities humans extension autonomous agents well result algorithms develop incorporate explanation generation process agent decision making process general argumentation frameworks balancing explicability explanations resolving disputes plans indeed explored work seen specific case argumentation process set constraints prove correctness quality plans considering cost argument specifically relate plan quality cost explaining plan first kind algorithm achieve scope plan explanations explicable presence model differences human planning revisited problem formulation closely follows introduced reproduced clarity methods built definitions classical planning problem tuple domain set fluents define state set actions initial goal states action tuple pre cost pre preconditions effects pre else transition function cumulative transition function solution planning problem sequence actions satisficing plan cost plan otherwise cheapest plan arg cost optimal plan refer cost planning hap problem given tuple mhr planner model planning problem mhr ihr ghr respectively planner estimate human model human understanding model solution planning problem joint plan irh grh robot component plan referred purposes paper ignore robot belief human model mrh effect making human observer passive consumer plan focus instead challenges involves planning human model planner planning human model indeed studied extensively literature noted assumption change way relevance work specifically following concepts built top joint planning problem explicable plan paper would general sense correspond robot component joint plan explicable human loop thus purposes paper without loss generality focus simplified setting model planner human approximation explicable planning explicable planning solution planning problem plan executable may longer technical report asu optimal robot model closer expected plan human model given particular planning problem mhr closeness distance expected plan modeled terms cost optimality general preference metric like plan similarity existing literature usually achieved modifying search process heuristic guides search driven robot knowledge human mental model heuristic either derived directly human model known learned interactions form affinity functions plans purported goals plan explanations approach would compute optimal plans planner model usual also provide explanation form model update human plan also optimal human updated model problem thus solution involves plan explanation note model update indicated operator may include correction belief goals state information well information pertaining action model authors explored various ways generating solutions including methods minimize lengths explanations given result however done fashion optimal plan already generated matter finding best explanation ignores possibility finding better plans equally optimal smaller explanations also avenues compromise manner discussed previously whereby planner sacrifices optimality reduce overhead explanation process mega bring notions explicability explanations together novel planning technique mega explanation generation algorithm trades relative cost explicability providing explanations plan generation process output mega plan explanation executable robot model explanation form model updates optimal updated human model cost length explanations cost deviation optimality model explicable human traded according constant note model planning problem includes action model well initial goal states agent assume human mental model known computation power also suggests possible ways address issues discussions apply well also refer discussion model learning later technical report asu tathagata chakraborti sarath sreedharan subbarao kambhampati arg clearly higher values planner produce plans require explanation lower generate explicable plans thus help hyperparameter autonomous agent deliberate costs incurs explicable human second minimizing term versus explaining decisions first minimizing term note irrespective cognitive burden decisions human loop example robot collapsed building search rescue task rover mars may limited bandwidth communication hence prefer explicable instead instead employ model space search algorithm compute expected plan explanations given value similar define state representation planning problems mapping function represents planning problem transforming every condition predicate set actions contains unit model change actions make single change domain time start initializing min node tuple human mental model empty explanation new possible model come across model space search test objective value new node smaller current min node stop search identify model capable producing plan also optimal robot model different stopping condition used original authors trying identify first node given plan optimal algorithm mega beyond offered model reconciliation search computes smallest explanation given plan optimal planner model property mhr yields optimal plan planner model along minimal explanation possible given planning problem easy see since mhr latter total model difference penalty departure explicable plans high enough planner must choose possible explanations note explicability penalty always positive search hits nodes point onwards penalty exactly zero general works since mce known retrospectively search complete condition suffices since entire model difference known front largest possible explanation worst case mce minimally complete explanation shortest model update given plan optimal robot model also optimal updated human model closed list node minimum objective value optimal plan explained plan expected human nmin mhr priority true nmin update min node nmin emin nmin emin min optimal return else models satisfy condition removes models satisfy condition adds property mega yields smallest possible explanation given planning problem means high enough see algorithm guaranteed compute best possible plan planner well smallest explanation associated construction search process search terminates exhausted nodes allow procedure input hap output plan explanation procedure fringe procedure return property yields explicable plan condition planner minimize cost explanations course point produce plan requires shortest explanation hence explicable plan note distinct computing optimal plan human model since plan may executable planner model explanations required even worst case also welcome additions explicability view plan generation introduced human model also guides plan generation process instead directly though none works provided insight make remainder model reconciliation possible cases done explanations associated generated plans property required per problem independent algorithm terminates nodes containing minimally complete explored means different values agent needs nodes new objective function mind thus large part reasoning process particular problem evaluations provide internal evaluations mega modified versions two ipc domains rover barman balancing explicability explanations domain name problem technical report asu time secs time secs time secs rover barman table computation time plans rover barman domains along length explanations demonstrating cost computation time plans respect varying size model difference follow demonstration mega action robot search reconnaissance domain finally report human factor studies received users code domain models available review process rover meets martian domain empirical results cost value determines much agent willing sacrifice optimality versus cost explaining perceived suboptimal plan human following illustrate modified versions two ipc domains rover meets martian domain mars rover model described ipc domain gone update whereby carry rock soil samples needed mission time means need empty store collecting new rock soil samples anymore new action definitions longer contain precondition empty mission runs across martian unaware robot expanded storage capacity older extremely cautious model rover learned spying cave believes time collect rock sample also need collect soil sample need communicate information lander also believes rover perform action needs send soil data rock data waypoint taking image clearly rover follow model order spook martians end spending lot time performing unnecessary actions like dropping old samples collecting unnecessary samples example rover communicate image objective needs move waypoint visible perform action rover produce plan better represents martian expectations would look like store general store store general barman bar domain figure explicability versus explanation cost plans produced different values rover chose directly use mce could end explaining six different model differences based problem plan execution case may acceptable others may make sense rover bear extra cost rather laboriously walking updates impatient martian mega lets naturally model scenarios use parameter rover would choose execute martian expected optimal plan parameter set zero means rover care extra cost needs incur ensure plan makes sense martian least explaining involved figure shows explicability cost explanation cost varies three typical problem instances domain algorithm starts converging smallest possible mce set one smaller mega chooses save explanation costs choosing expensive explicable plans barman bar domain brand new twohanded barman robot wowing onlookers technical report asu tathagata chakraborti sarath sreedharan subbarao kambhampati skills even admirers may unsure capabilities expect much like original ipc domain required one hand free perform actions like shake etc means make single shot cocktail two shots ingredient three shots one shaker human expects robot execute following plan left right left left right left leave left grasp left robot however directly start picking shot shaker need put either making cocktail similar rover domain illustrate three typical problems barman domain figure lower values robot choose perform plans require less explanation increases algorithm produces plans require larger explanations explanations finally converging smallest mce required problem empirical results computation time contrary classical notions planning occurs state plan space planning model space every node search tree new planning problem seen table becomes quite time consuming increasing number model differences human robot even significant gains terms minimality explanations reduction cost explicable plans result motivates need developing approximations heuristics search explanations demonstration usar domain first demonstrate mega robot performing urban search reconnaissance usar task remote robot put disaster response operation often controlled partly fully external human commander typical usar setup robot job infiltrate areas may otherwise harmful humans report surroundings required instructed external external usually map environment map longer accurate disaster setting new paths may opened older paths may longer available due rubble collapsed structures like walls doors robot internal however may need inform external changes cause information overload commander may otherwise engaged orchestrating entire operation calls instantiation mega algorithm model differences contributed changes map initial state planning problem human model original unaffected model world figure shows relevant section map environment whole scenario plays orange marks indicate rubble blocked passage green marks indicate collapsed walls robot fetch currently located position marked blue tasked taking picture location marked orange figure external commander figure typical search reconnaissance scenario internal agent robot external supervisor human video demonstration accessed https expects robot take path shown red longer possible robot armed mega two choices either follow green path explain revealed passageway due collapse compromise optimal path clear rubble proceed along blue path video demonstration scenario viewed https watch first part video demonstrates plan generated mega low values expected chooses blue path requires least amount explanation thus explicable plan fact robot needs explain single initial state change make plan optimal namely explanation also instance plan closest human expectation explicable plan still requires explanation balancing explicability explanations technical report asu previous approaches literature provide moreover order follow plan robot must perform costly action traverse corridor could avoided optimal plan shown green map indeed mega switches robot optimal plan higher values along following explanation explanation explanation explanation providing explanation robot able convey human optimality current plan well infeasibility human expected plan shown red human factors evaluations finally use search reconnaissance domain analyze humans respond explicability versus explanations done exposing external commander interface participants get analyze plans mock usar scenario participants incentivized make sure explanation indeed help understand optimality plans question formulating interaction form game make sure participants sufficiently invested outcome well mimic nature usar settings accurately evaluate explanations figure shows screenshot interface displays participant initial map told may differ robot actual map starting point goal plan illustrated form series paths various waypoints highlighted map participant identify plan shown optimal player unsure ask explanation explanation provided participant form set model changes player map scoring scheme game follows player awarded points correctly identifying plan either optimal satisficing incorrectly identification costs points every request explanation costs points skipping map result penalty participants additionally told selecting inexecutable plan either feasible optimal would result penalty points even though actual incorrect plans dataset information provided deter participants taking chances plans understand well participant paid dollars received additional bonuses based following payment scheme scores higher equal paid scores higher paid scores higher paid scores higher paid scores received bonuses scoring systems game designed make sure participants ask explanation unsure quality plan due small negative points explanations figure interface external commander mock search reconnaissance study figure responses explicable plans versus balanced robot optimal plans explanations figure rates explanations participants incentivized identify feasibility optimality given plan correctly large reward penalty wrongly participant shown total maps maps participant assigned optimal robot plan asked explanation randomly shown different types explanations introduced rest maps place robot optimal plan participants could potentially assigned plan optimal human model explicable plan explanation somewhere balanced plan shorter explanation note maps balanced plans well explicable plans either balanced plan optimal human plan total participants study including female male participants age range participant reveal demographic technical report asu optimal plan balanced plan tathagata chakraborti sarath sreedharan subbarao kambhampati explicable plan table statistics explicability versus explanation tradeoff respect explanation length plan cost figure shows people responded different kinds explanations plans results problem instances required explanations instances contained balanced explicable plans respectively perspective human balanced plan robot optimal plan make difference since appear suboptimal evident fact rate explanations two conditions similar however rate explanations significantly less case explicable plans desired table shows statistics explanations plans results problem instances required minimal explanations per instances contained balanced explicable plans respectively desired robot gains length explanations loses cost plans produced progresses along spectrum optimal explicable plans thus table demonstrates cost explanation versus explicability robot point view figure shows perceived human perspective interesting see figure third time participants still asked explanations even plan explicable thus optimal map artifact behavior incentivized gamification explanation process indicative cognitive burden humans cost optimal planners thus going forward objective function incorporate cost difficulty analyzing plans explanations point view human addition current costs equation mega table modeled perspective robot model finally figure show participants responded inexplicable plans terms rate explanation request button information used model parameter situate explicability versus explanation according preferences individual users interesting see distribution participants right inset seem bimodal indicating people particularly skewed towards behavior others rather normal distribution response motivates need learning interactively particular human loop discussion future work following section elaborate exciting avenues future research borne work model learning picking right assumed set designer determining much costs explicability versus explanations part autonomous agent however design adaptive sense parameter learned course interactions human loop determine kind plans preferred seen figure much information transmitted also relevant cases human mental model known precisely uncertainty towards new model update explanation topic future work existing literature iterative model learning provide useful guidance towards authors discuss useful representations learning models purposes task planning various levels granularity note search uncertainty learned human mental model often times compiled planning process described using annotated models techniques introduced paper still apply cost explanations cognitive load currently considered cost explanations explicability point view robot however might additional cognitive burden human measured terms complexity interpreting explanation far away final plan optimal plan human mental model ties back assumptions cognitive abilities optimality human loop needs calibration based repeated interactions seen figure conclusion saw agent achieve behavior time keeping mind cost departure optimality could otherwise explained away given opportunity raises several intriguing challenges plan generation process notably finding better heuristics speeding model space search process well dealing model uncertainty identifying sweet spot algorithm indeed revised planning paradigm opens exciting new avenues research learning human mental models providing explanations different levels abstractions references mitchell john bresina len charest adam chase hsu ari jonsson bob kanefsky paul morris kanna rajan jeffrey yglesias mapgen planning scheduling mars exploration rover mission ieee intelligent systems rachid alami clodic vincent montreuil emrah akin sisbot raja chatila toward robot task planning aaai spring symposium boldly team gone rachid alami mamoun gharbi benjamin vadant lallement adolfo suarez task motion planning abilities teammate robot collaboration industrial manufacturing workshop rss james allen mixed initiative planning position paper labs planning initiative workshop cade earl bartlett communication teammates urban search rescue thesis alexandros belesiotis michael rovatsos iyad rahwan agreeing plans iterated disputes proceedings international conference autonomous agents multiagent systems volume international foundation autonomous agents multiagent systems rodney brooks robust layered control system mobile robot ieee journal robotics automation chakraborti briggs talamadupula zhang scheutz smith kambhampati planning serendipity iros balancing explicability explanations tathagata chakraborti subbarao kambhampati matthias scheutz zhang challenges cognitive teaming arxiv preprint tathagata chakraborti sarath sreedharan zhang subbarao kambhampati plan explanations model reconciliation moving beyond explanation soliloquy ijcai tathagata chakraborti zhang david smith subbarao kambhampati planning resource conflicts cohabitation aamas marcello cirillo planning inhabited environments task planning activity recognition dissertation university marcello cirillo lars karlsson alessandro saffiotti task planning application mobile robots acm transactions intelligent systems technology anca dragan kenton lee siddhartha srinivasa legibility predictability robot motion interaction chukwuemeka emele timothy norman simon parsons argumentation strategies plan resourcing international conference autonomous agents multiagent international foundation autonomous agents multiagent systems george ferguson james allen bradford miller towards planning assistant aips maria fox derek long daniele magazzeni explainable planning ijcai xai workshop dylan stuart russell pieter abbeel anca dragan cooperative inverse reinforcement learning nips marc hanheide moritz graham horn andrzej pronobis kristoffer alper aydemir patric jensfelt charles gretton richard dearden miroslav janicek robot task planning explanation open uncertain worlds artificial intelligence international planning competition ipc competition domains https subbarao kambhampati kartik talamadupula planning decision support http uwe koeckemann federico pecora lars karlsson grandpa hates robots interaction constraints planning inhabited environments aaai anagha kulkarni tathagata chakraborti yantian zha satya gautam vadlamudi zhang subbarao kambhampati explicable robot planning minimizing distance expected behavior corr pat langley ben meadows mohan sridharan dongkyu choi explainable agency intelligent autonomous systems lydia manikonda tathagata chakraborti kartik talamadupula subbarao kambhampati herding crowd using automated planning better crowdsourced planning journal human computation hugo mercier dan sperber humans reason arguments argumentative theory behavioral brain sciences tim miller jens pfau liz sonenberg yoshihisa kashima logics common ground journal artificial intelligence research christian muise paolo felli tim miller adrian pearce liz sonenberg planning single agent environment using fond ijcai stefanos nikolaidis przemyslaw lasota ramya ramakrishnan julie shah improved team performance approach inspired human team training practices international journal robotics research sailik sengupta tathagata chakraborti sarath sreedharan subbarao kambhampati radar proactive decision support system planning aaai fall symposium groups sreedharan chakraborti kambhampati explanations model reconciliation perspective aaai fall symposium humanagent groups kartik talamadupula gordon briggs tathagata chakraborti matthias scheutz subbarao kambhampati coordination teams using mental modeling plan recognition intelligent robots systems iros international conference ieee stevan tomic federico pecora alessandro saffiotti cool school adding social constraints human aware planning workshop cognitive robotics cogrob zhang sarath sreedharan anagha kulkarni tathagata chakraborti hankz hankui zhuo subbarao kambhampati plan explicability robot task planning rss workshop planning interaction zhang sarath sreedharan anagha kulkarni tathagata chakraborti hankz hankui zhuo subbarao kambhampati plan explicability predictability robot task planning icra technical report asu
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action perception spatiotemporal patterns martin daniel polani araya tokyo university hertfordshire hatfield jun martin abstract contribution formalization concept agents multivariate markov chains agents commonly defined entities act perceive multivariate markov chain cellular automaton transition matrix completely determines dynamics seems contradict possibility acting entities within system present definitions actions perceptions within multivariate markov chains based entitysets represent largely independent choice set spatiotemporal patterns considered entities within markov chain example entityset chosen according operational closure conditions complete specific integration importantly perceptionaction loop also induces multivariate markov chain show definition actions leads perceptions specialize usual concept perception loop introduction loop used formalize mostly information theoretic terms various properties associated agents include empowerment klyubin autonomy bertschinger decisions tishby polani embodiment zahedi literature agents usually seen entities act perceive way cmp barandiaran assumes entities make agents well environments captured interacting stochastic processes convenient assumption since actions perceptions easily identified interactions see section requirements autonomy introduced distinguish stochastic processes actually constitute agents formally established whether assumption set entities contains agents represented stochastic processes justified argued previous work naively using stochastic processes capture entities within given multivariate markov chain example cellular automata like game life fails account essential proporties agents irrespective chosen additional conditions instead argued use spatiotemporal patterns stps previously employed beer represent entities immediate advantage stps superset structures like gliders game life spots reaction diffusion systems virgo froese bartlett bullock particle based systems exhibiting individuation cells schmickl formally capturing structures becomes matter selecting according subsets stps see disadvantage stp based entities lack formal actions perceptions provides far know formal definitions actions perception exists stp based entities first contribution paper proposals formal definitions called entity action section entity perception section second contribution formal connection entity actions perceptions section connection achieved via notion set stps multivariate markov chain considered entities according independently specified criterion organizational closure beer complete local integration biehl importantly quite naturally identify use definitions entity action entity perception result entity perception coincides standard notion perceptions entity actions necessary sufficient condition nonheteronomy determined environment proposed bertschinger part information theoretic measure autonomy closely related work beer apart generalization stochastic systems set entity perceptions seems straightforward surprisingingly tedious formalization cognitive domain stps described glider game life use perceptions instead cognitive domain motivation came unlike beer directly autopoiesis concerning entity actions deviate beer requiring action continuation entity note ikegami taiji propose use counterfactual trajectories game players signs autonomy construct capability act counterfactual trajectories find imply related autonomy figure first timesteps processes represent environment tet utpt agent memory tmt utpt notation restrict finite multivariate markov chains unrolled time formally described bayesian networks bns index random variables via index set set timesteps set spatial degrees freedom index also write convenient set txi uipv random variables together set edges determining parents papiq papj node associated mechanisms ppj pxj assume parents node subset nodes previous timestep papj tpj write pxi qipa joint random variable consisting random variables indexed elements also sometimes write tpj elements correspond indices timestep refer state space random variable denoted specific values denoted lower case letters joint random variables write ipa spatiotemporal pattern stp value joint random variable since arbitrary subset stp specify values random variables multiple timesteps multiple spatial locations set stps txa important envision difference set stps set subsets random variables txi uipv latter isomorphic power set written txa txa set random variables set values random variables trajectory stp occupies random variables txi uipv say stp occurs within trajectory eptxi uipv subset stps one choice would use entire set stps entity set choices include using organizational closure conditions like beer complete specific integration criterion biehl following definitions theorems assume txi uipv multivariate markov chain given loop given two interacting stochastic processes fig always extract random stochastic process explicitly represent interactions see one processes agent memory process tmt utpt environment process tet utpt extracted random variables seen perceptions actions agent perceptions tst utpt capture exactly influence environment agent actions tat utpt capture influence agent environment means introduce another containing two processes action process tat utpt sensor process tst utpt result extraction extended fig identical joint probability distribution two initial stochastic processes tmt utpt tet utpt perception action loop used example bertschinger idea behind extraction perceptions tst utpt conversely actions tat utpt partition state space environment blocks identical influence next memory state blocks possible perceptions states formally definition time let pmt pmt pmt pmt sensor defined set equivalence classes equivalence relation set sensor values defined element also block called perception sensor value symmetrical way define actions via partition arrive extended fig straightforward prove following theorem construction sensor partition new also used example balduzzi obtain coarser states alphabet joint random variables authors thank benjamin heuer originally pointing construction figure first extended processes tat utpt tst utpt mediate interactions tmt utpt tet utpt without changing probability distributions latter see theorem theorem invariant extension theorem given perception action loop txi uipv tmt utpt extended txi uipw tmt utpn let pmt probability distribution entire perception action loop txi uipv let marginal probability distribution memory environment process obtained probability distribution entire extended pmt proof probably fairly well known see biehl explicit proof shows introduction action sensor process way makes interactions agent environment processes explicit introduce additional dynamics theorem also shows sensor process conversely actions captures influences environment agent else dynamics original processes could remain identical section want capture influences environment set stps entities instead stochastic process like tmt utpt require generalization perception extraction procedure definition entity action define concept actions given multivariate markov chain first briefly sketch main ideas behind definition due setting given multivariate markov chain actions occur within concept actions differs approaches paraphrasing wilson shpall slightly distinguishes actions among events merely happen individuals rather made happen individuals problematic setting stps entities take role individuals happens multivariate markov chain trajectories stps occurring within markov chain dynamics determined mechanisms turn determine possibly stochastically going happen times anywhere within chain therefore impossible multivariate markov chain contains stp entity make something happen beyond happens anyway due mechanisms means explain define actions different way also note unlike accounts actions wilson shpall require actions necessarily purposeful way way conceive agency entity actions considered actions sense get definition actions note events called actions usually attributed limited bounded region part universe body living organism sometimes brain one parts usually contain mechanisms configurations matter either directly observable human observer hidden opaque container well understood human observer factors inevitably lead unpredictability events words events attributed well understood therefore predictable mechanisms sunrises considered actions point view actions beyond possibly complex unobserved origin special events may appear special observers lack sensory computational capacity resolve understand approach construct actions events way fundamentally unpredictable observer within system see done without need definition observer note approach remains compatible notions apparent actions cmp mcgregor fundamental actions apparent actions every possible observer events actions observers plain predictable events others ignored randomness true randomness sense stochastic independence event event universe exists system never explained predicted understood combined reasoning suggests random events actions fundamentally intuition random events seen actions agents however place burden ruling random events interpreted actions agents expect useful notions entities consist completely independent events would prevent events becoming actions approach furthermore independent events seem useful order achieve particular goal however usefulness random number generators might seen counterexample formal definition answering questions future work want define actions entities first issue run entities already fixed stps therefore entity already consists consequences whatever actions took sense result actions freedom left order investigate actions therefore deconstruct entity see actions could taken look counterfactual entities whose occur exactly environment xvt zat different immediate future requiring environment makes different futures unpredictable anything part environment observer therefore predict futures either even observer distinguished two entities past state independent entity faced therefore must forgotten difference way get definition actions unpredictable events observers without needing formally define observers formal definition first define definition environment stp let stp environment time spatial pattern xvt zat state definition action entity time particular trajectory formally definition action entity let pxv also let entity nonempty performs action xat time trajectory exists entity occurs pyv entities occupy random variables iii trajectories otherwise identical xvt zat yvt zat entities different xat ybt also call entity trajectory ybt note requirements symmetric therefore performs action also performs action also notion entities easily extended one entity make sure entities set entities mutually different furthermore easy generalise definition actions situations must occupy variables interval time action case environment must identical interval finally note condition two acting entities differ time fulfilled two ways call actions extent actions else actions differ value xat yat call actions value actions difference value actions extent actions made possible due definition entities stps intriguing question future whether capabilities agents act value extent truly superior agents act value modelled see section probabilistic information theoretic expressions easy formulate actions value however actions extent done yet entity perception section formally define perception entities make distinction perception experience sensory input tradition modelling systems using dynamical systems probabilistic generalisations stochastic processes define perception effects environment beer run similar problem actions entity already fixed stp contains influence may subjected sense result influence influences surroundings order investigate influences therefore deconstruct entity see formed external influences perceptions idea fix past entity use set counterfactual entities past alternative futures partition possible environments counterfactual entities according influence probability distribution entities futures done basically way defined perception definition however technical issues overcome set entities identical pasts time interpreted set entities like different entities differ future futures including next therefore close analogue next states agent memories make sure however entities next also require next requirements together define notion entities entity time entities also perceive something maybe thing trajectories perceives something definition entities entity let entity set entities spxa entity set entities identical spxa tyb next want define conditional probability distribution entities similar pmt pmt need random variable ranges possible futures entities naivest way would use xkak spxa pxkak zat pak zat pxkak xvt zat pvt zat pxvt zat however two problems first general denominator may vanish environments xvt zat xvt zat second conditional probability since sum indices necessarily one problems solved restricting set environments introducing condition futures coperception entities restrict environments therefore define coperception environments following way definition environments let entity spxa entities define associated environments xvst zat xvt zat xvst zat zat dyb spxa zat pyb zat environments set spxa spatial patterns xvt zat least one environment definition environments denominator vanish anymore require construction allows environments however order sum entities equal one environments need pak xvt zat pxkak xvt zat pvt zat pxkak xvt zat case general xvt zat xvst bsat entities futures mutually exclusive xkak xlal spxa prpxkak xlal xvt zat condition guaranteed require form condition states two different entities identical point time single trajectory positive probability point reveal difference entities identical pasts ever reveal difference must different trajectories must mutually exclusive definition satisfies implies entities mutually exclusive theorem let entity timeslices spxa entities satisfies spxa mutually exclusive proof let spxa identical pasts get prpxb stronger means well defined conditional probability distribution however conditional probability distribution still quite different pmt pmt since ranges entire futures xkak entities next timesteps transition coperception entities spxa split sets entities identical call sets branches one sets set spxa example entity spxa past different ybt xat part different branch case branch spyb spyb spxa summary dynamics system split entities disjoint sets branches entities identical pasts interpret branches time distinctions among entities revealed time distinctions among entities revealed later times also means differences could possibly due influence environment show effect later way perceptions also defined respect branches call partition defined via identification entities spxa identical branching partition definition branching partition let entity spxa entities define branching partition spxa partition induced equivalence classes equivalence relation ybt zct spxa note definition branching partition easily generalised one future instead requiring equality require equality next given branching partition noninterpenetrating entity set define conditional probability distribution branches summing probabilities entities branch remember mutually exclusive get probability branch gives defined definition let txi uipv multivariate markov chain index set entity set let entity spxa entities branching partition furthermore let xvst zat xvt zat associated environments also write every block zat zat zat zat xvst zat define branchmorph probability distribution zat zat zat zat define expected perceptions equivalence classes environments respect associated first define partition environments called environment partition perceptions blocks partition definition let txi uipv multivariate markov chain index set entity set let entity spxa entities branching partition furthermore let xvst zat xvt zat associated environments define environment partition pxa xvst zat partition induced equivalence classes equivalence relation zat zat zat zat means associated environments block pxa words lead branch entity futures future branch probabilities elements environment blocks identical effects future branches branches distinguish environments within blocks definition perceptions let txi uipv multivariate markov chain index set entity set let entity spxa entities furthermore let xvst zat xvt zat associated environments pxa environment partition blocks pxa called perceptions entity action perception show systems modelled multivariate markov chains containing specific choice entity sets trajectory considered consist agent environment agent therefore occurs every trajectory occupies degree freedom every trajectory stp since entities stps define entityset set agent process tmt tpt similarly define entities environments add note exhibit interpenetration still define subsets entities since choice subsets arbitrary however lead uniquely defined notion perception details see biehl entity actions write every trajectory pair pmt entity entity performs entity action time trajectory pmt pmt pmt entity occurs pmt pmt entities occupy random variables case entities environments identical entities different since entities occupy random variables value actions assume conditions fulfilled time two entities derive conditional entropy hpmt next agent state given current environment state greater zero see note pmt pmt pmt directly follows pmt pmt pmt pet pet plugging definition get hpmt also seen different entities time higher conditional entropy hpmt get final value hpmt depends actual probabilities maximum value log also note actions entity trajectory hpmt entity actions entities therefore necessary sufficient hpmt conditional entropy hpmt measures uncertainty next agent state current environment state known proposed part autonomy measure measure bertschinger means agent determined history environment terminology entity actions necessary sufficient condition entity perception look entity perception defined section specialises case argument effect constitutes proof definition generalisation conditional probability distributions pmt arbitrary sets entities result surprising since set also instructive work recovery original expression conditional probability distribution starting general pick entity entity set consider perceptions arbitrary order get perceptions need entities spmt branching partition branches environments environment environment partition pxa blocks perceptions identified following way entities spmt entities identical entities time slices times spmt first note entity set satisfies noninterpenetration since occupy set tmt utpt random variables branching partition composed blocks branches entities identical therefore identify blocks future branches values entities take define branch associated via spmt branching partition spmt environments stps xvt zat compatible least one entity entity xvt zat therefore xvst zat ets ets ets tet spmt pmt marginalize see equivalent ets tet pmt probability distributions branches environment ets defined using becomes rewrite sum right hand side using spmt pmt definition ppb rewritten work pmt pmt used environment partition pmt induced using becomes pmt pmt pmt pmt equivalence relation used extract section seen definitions section specialise case concept perception section conclusion defined actions perceptions therefore sets spatiotemporal patterns provides formally defined way associate gliders similar spatiotemporal patterns systems actions perceptions step towards foramization agency patterns notion goaldirectedness still missing future work also shown definitions specialize necessary sufficient condition standard notion perceptions agent process paloop future research interesting note generalisations conditional probability distribution pmt conditional probability distributions play role various information theoretic concepts formulated suggests might able translate bakc forth concepts spatiotemporal patterns future noted unique definition entity perception dependent condition beer dynamical systems perspective agentenvironment interaction artificial intelligence beer characterizing autopoiesis game life artificial life beer cognitive domain glider game life artificial life bertschinger olbrich jost autonomy information theoretic perspective biosystems biehl ikegami polani towards information based spatiotemporal patterns foundation agent representation dynamical systems proceedings artificial life conference pages mit press biehl ikegami polani specific complete local integration patterns bayesian networks entropy biehl formal approaches definition agents phd thesis university hertfordshire hatfield froese virgo ikegami motility origin life characterization model artificial life ikegami taiji uncertainty possible worlds coupled dynamical recognizers http klyubin polani nehaniv empowerment universal measure control ieee congress evolutionary computation volume pages mcgregor bayesian stance equations sensorimotor agency adaptive behavior page schmickl stefanec crailsheim lifelike system emerges simplistic particle motion law scientific reports tishby polani information theory decisions actions cutsuridis hussain taylor editors cycle springer series cognitive neural systems pages springer new york doi references virgo thermodynamics structure living systems university sussex unpublished phd thesis balduzzi detecting emergent processes cellular automata excess information advances artificial life ecal wilson shpall action zalta editor stanford encyclopedia philosophy summer edition barandiaran paolo rohde defining agency individuality normativity asymmetry spatiotemporality action adaptive behavior zahedi quantifying morphological computation entropy math bartlett bullock emergence competition different dissipative structures free energy source proceedings european conference artificial life pages mit press
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gated recurrent networks seizure detection golmohammadi ziyabari shah von weltin campbell obeid picone neural engineering data consortium temple university philadelphia pennsylvania usa meysam saeedeh vinitshah obeid picone recurrent neural networks rnns sophisticated units implement gating mechanism emerged powerful technique modeling sequential signals speech electroencephalography eeg latter focus paper significant big data resource known tuh eeg corpus tueeg recently become available eeg research creating unique opportunity evaluate recurrent units task seizure detection study compare two types recurrent units long memory units lstm gated recurrent units gru evaluated using state art hybrid architecture integrates convolutional neural networks cnns rnns also investigate variety initialization methods show initialization crucial since poorly initialized networks trained furthermore explore regularization convolutional gated recurrent networks address problem overfitting experiments revealed convolutional lstm networks achieve significantly better performance convolutional gru networks convolutional lstm architecture proper initialization regularization delivers sensitivity false alarms per hours introduction diagnosis clinical conditions epilepsy dependent electroencephalography eeg recording brain electrical activity electrodes placed scalp delivering conclusive diagnosis illness without eeg often unfeasible large amounts time required specialized neurologists interpret records created workflow bottleneck neurologists overwhelmed amount data needs manually reviewed great need partial complete automation eeg analysis process automated technology slowly emerging fill void automatic analysis eeg scans reduces time diagnosis reduces error enhances neurologist ability administer medications ability search eeg records symbolically greatly accelerates review process paper focus specifically problem seizure detection many algorithms applied problem including analysis methods nonlinear statistical models modern machine learning approaches neural networks support vector machines despite much progress current eeg analysis methodologies far perfect many considered impractical due high false detection rates machine learning challenges application described extensively significant big data resource known tuh eeg corpus tueeg become available eeg interpretation creating unique opportunity advance technology using subset data manually annotated seizure events novel deep structure recently introduced achieves low false alarm rate eeg signals system integrates convolutional neural networks cnns recurrent neural networks rnns deliver state art performance paper goal investigate use rnns using architecture described also explore improved initialization methods regularization approaches recurrent neural networks recurrent neural network rnn extension conventional feedforward neural network handle input rnn handles sequence recurrent hidden state whose activation time dependent previous time standard rnns hard train due vanishing exploding gradient problems address vanishing gradient problem gated recurrent network architectures long memory lstm unit gated recurrent unit gru proposed lstm presented commonly used architecture described formulated input gate forget gate cell state output gate block output time instance respectively input time weight matrices applied input recurrent hidden units respectively sigmoid tangent activation functions respectively connections biases respectively means product alternative lstm gated recurrent unit gru architecture proposed gru architecture found achieve better performance lstm tasks gru formulated one see gru architecture similar lstm without separate memory cell unlike lstm gru include output activation functions connections also integrates input forget gates update gate balance previous activation candidate activation reset gate allows forget previous state iii experimental design basic architecture employs convolutional recurrent neural network presented figure architecture integrate cnns cnns lstm networks better exploit dependencies structure currently uses lstms however easily replace lstms grus feature extraction performed using fairly standard linear frequency cepstral feature extraction approach lfccs popularized applications speech recognition also use first second derivatives features since provide small improvement performance drawing video classification analogy input data first layer cnns composed frames distributed time frame image width equal length feature vector height equals number eeg channels number image channels equals one input data consists frames equal window length multiplied number samples per second optimized system window duration seconds first convolutional layer filters frames eegs distributed time size using kernels size stride first max pooling layer takes input vector frames distributed time size applies pooling size process repeated two times two convolutional layers kernels size respectively two max pooling layers pooling size output third max pooling layer flattened frames size convolutional layer filters output flattening layer using kernels size decreases dimensionality space apply maxpooling layer size decrease dimensionality input deep bidirectional lstm network dimensionality output space output last bidirectional lstm layer fed sigmoid function produces final classification epoch epochs typically sec duration figure deep recurrent convolutional architecture decoding eeg signals integrates cnns cnns lstm networks shown structure lstms easily replaced grus overcome problem overfitting force system learn robust features regularizations used first two layers cnns increase nonlinearity exponential linear units elu used adam used optimization process along mean squared error loss function system sensitivity specificity hrs table recognition results convolutional recurrent neural networks using gru lstm architectures sensitivity range results lack big data resources used train sophisticated statistical models compounds major problem automatic seizure detection agreement task low especially considering short seizure events manual annotation large amount data team certified neurologists extremely expensive time consuming difficult employ large numbers neurologists perform task study reporting results tuh eeg seizure corpus tusz dataset publicly available subset tuh eeg corpus focuses problem seizure detection summary corpus shown table comparison performance convolutional recurrent neural networks using gru lstm architectures sensitivity range shown table related det curve illustrated figure systems evaluated using method scoring popular eeg research community known overlap method true positives defined number epochs identified seizure reference annotations correctly labeled seizure system true negatives defined number epochs correctly identified false positives defined number epochs incorrectly labeled seizure false negatives defined number epochs incorrectly labeled sensitivity shown table computed specificity computed false alarm rate number fps per hours comparing results demonstrated figure find lower false positive rates significantly better performance due fact gru unit controls flow information like lstm unit memory unit lstms description patients sessions files seizure secs secs total secs train eval table overview tuh eeg seizure corpus figure det curves convolutional recurrent neural networks using gru lstm architectures remember longer sequences better grus outperform task since seizure detection requires modeling long distance relationships additionally training time less hence training time two systems comparable since cycles used training convolutional layers neural networks determining proper initialization strategy parameters model part difficulty training hence investigated variety initialization methods using structure introduced figure results presented table related det curve illustrated figure experiments observed proper initialization weights convolutional recurrent neural network critical convergence example initialization zeros ones methods completely stalled convergence process also see table performance system sensitivity change different initialization methods decrease performance deceleration convergence arises initializations result deeper layers receiving inputs small variances turn slows back propagation retards overall convergence process best performance achieved using orthogonal initialization method simple yet effective way combatting exploding vanishing gradients orthogonal matrices preserve norm vector eigenvalues absolute value one means matter many times perform repeated matrix multiplication resulting matrix explode vanish also orthogonal matrices columns rows orthonormal one another helps weights learn different input features initialization orthogonal lecun uniform glorot uniform glorot normal variance scaling lecun normal normal random uniform truncated normal uniform sensitivity specificity fas initialization dropout gaussian sensitivity specificity fas table recognition results convolutional lstm architecture sensitivity range using different regularizations table results sensitivity range using different initialization methods figure det curves architecture using different regularizations summary figure det curves architecture using different initialization methods overfitting serious problem deep neural nets many parameters study used five popular regularization methods address problem using techniques apply penalties layer parameters optimization penalties incorporated loss function network optimizes alternative approach used dropout prevents units much randomly dropping units connections neural network training also studied impact introducing gaussian noise network results experiments presented table along det curve figure generally best performance move towards low rate dropout delivers lower rate additionally found primary error modalities observed false alarms generated brief delta range slowing patterns intermittent rhythmic delta activity experiments showed regularizing methods presented table playing important role increasing false alarms slowing patterns even though dropout effective cnns dropout placed kernels leads diminished results solve problem future work efficient bayesian convolutional neural network explored places probability distribution cnn kernels approach offers better robustness overfitting small data show improve robustness training process paper investigated two deep learning architectures lstm gru automatic classification eegs using cnns lstms outperformed grus also studied initialization regularizations networks future research designing powerful architecture based reinforcement learning concepts also optimizing regularization initialization algorithms approaches goal approach human performance range sensitivity false alarm rate per hours robust training procedures needed make technology relevant wide range healthcare applications acknowledgements research reported publication recently supported national human genome research institute national institutes health award number content solely responsibility authors necessarily represent official views national institutes health material also based part upon work supported national science foundation grant opinions findings conclusions recommendations expressed material author necessarily reflect views national science foundation tuh eeg corpus work funded defense advanced research projects agency darpa mto auspices doug weber contract temple university college engineering temple university office senior research references obeid picone machine learning approaches automatic interpretation eegs biomedical signal processing big data sejdik falk eds boca raton florida usa crc press lopez suarez jungries obeid picone automated identification abnormal eegs ieee signal processing medicine biology symposium philadelphia pennsylvania usa direito teixeira ribeiro sales dourado modeling epileptic brain states using eeg spectral analysis topographic mapping neurosci methods vol temko thomas marnane lightbody boylan neonatal seizure detection support vector machines clin vol gotman automatic recognition epileptic seizures eeg electroencephalogr clin vol gotman automatic detection seizures clin vol stam nonlinear dynamical analysis eeg meg review emerging field clinical neurophysiology vol alotaiby alshebeili alshawi ahmad abd elsamie eeg seizure detection prediction algorithms survey eurasip adv signal vol ramgopal seizure detection seizure prediction closedloop warning systems epilepsy epilepsy vol varsavsky mareels patient detection epileptic seizures changes variance proceedings annual international conference ieee engineering medicine biology society swisher white mace dombrowski diagnostic accuracy electrographic seizure detection neurophysiologists adult icu using panel quantitative eeg trends clin vol obeid picone temple university hospital eeg data corpus front neurosci sect neural vol golmohammadi tuh eeg seizure corpus proceedings american clinical neurophysiology society annual meeting golmohammadi ziyabari shah obeid picone deep architectures automated seizure detection scalp eegs proceedings aaai conference artifical intelligence bengio simard frasconi learning dependencies gradient descent difficult ieee trans neural networks vol pascanu mikolov bengio difficulty training recurrent neural networks international conference machine learning hochreiter hochreiter schmidhuber schmidhuber long neural vol chung gulcehre cho bengio empirical evaluation gated recurrent neural networks sequence modeling arxiv prepr graves schmidhuber framewise phoneme classification bidirectional lstm networks proceedings international joint conference neural networks vol cho learning phrase representations using rnn encoderdecoder statistical machine translation arxiv prepr king investigating gated recurrent networks speech synthesis international conference acoustics speech signal processing harati golmohammadi lopez obeid picone improved eeg event classification using differential energy proceedings ieee signal processing medicine biology symposium lopez golammadi obeid picone analysis two common reference points eegs proceedings ieee signal processing medicine biology symposium clevert unterthiner hochreiter fast accurate deep network learning exponential linear units elus arxiv prepr wilson scheuer plummer young pacia seizure detection correlation human experts clin vol saxe mcclelland ganguli exact solutions nonlinear dynamics learning deep linear neural networks international conference learning representations lecun bottou orr mulller efficient backprop lect notes comput sci glorot bengio understanding difficulty training deep feedforward neural networks proc int conf artif intell zhang ren sun delving deep rectifiers surpassing performance imagenet classification proceedings ieee international conference computer vision srivastava hinton krizhevsky sutskever salakhutdinov dropout simple way prevent neural networks overfitting mach learn vol gal ghahramani bayesian convolutional neural networks bernoulli approximate variational inference arxiv prepr
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nodetrix planarity testing small clusters emilio giuseppe maurizio alessandra sep degli studi perugia italy roma tre university italy patrigna abstract study nodetrix planarity testing problem flat clustered graphs maximum size cluster bounded constant consider case sides matrices edges incident fixed case arbitrarily chosen show nodetrix planarity testing fixed sides solved time every flat clustered graph reduced partial collapsing clusters single vertices general case nodetrix planarity testing fixed sides solved time nodetrix planarity testing remains also free side model introduction motivated need visually exploring graphs hybrid planarity one emerging topics graph drawing see hybrid planar drawing graph suitably represents restricted geometric regions dense subgraphs classical representation paradigm would visually effective regions connected edges cross different representation paradigms dense subgraphs give rise different types hybrid planar drawings angelini consider hybrid planar drawings dense portions graph represented intersection graphs sets rectangles study complexity testing whether graph admits representation context social network analysis henry introduce nodetrix representations dense subgraphs represented adjacency matrices batagelj study question minimizing size matrices nodetrix representation graph guaranteeing planarity edges connect different matrices batagelj choose subgraphs represented matrices lozzo consider problem testing whether flat clustered graph graph clusters subclusters admits nodetrix planar representation paper lozzo cluster must represented different adjacency matrix edges represented simple jordan arcs prove nodetrix planarity testing flat clustered graphs even constrained case matrix specified edges must incident top left bottom right side motivated hardness results paper study whether nodetrix planarity testing efficiently solved size clusters big precisely consider flat clustered graphs whose clusters size bounded fixed parameter want understand whether nodetrix planarity testing problem fixed parameter tractable solved time polynomial function depends main results listed follows describe algorithm test nodetrix planarity fixed sides flat clustered graphs partial informally flat clustered graph partial graph obtained collapsing every cluster single vertex partial flat clustered graph partial nodetrix planarity testing fixed sides still solved time becomes larger value finally extend hardness result free sides model show nodetrix planarity testing remains maximum cluster dimension larger four done proving even triconnected boolean formulas may result independent interest solution solves special type planarity testing problem order edges around vertex suitably constrained take account fact vertex matrix four copies along four sides may worth recalling gutwenger considered apparently similar problem namely studied planarity testing order edges around vertices may arbitrarily permuted unfortunately problem fall cases addressed gutwenger seem solvable introducing gadget polynomial size models embedding constraints vertex characteristic associates nodetrix planarity testing known variants planarity testing including clustered planarity use gadgets polynomial size far elusive goal rest paper organized follows preliminary definitions section sections describe polynomial time algorithm clustered bounded section show general flat clustered graphs fixed sides nodetrix planarity testing solved polynomial time section extend completeness result nodetrix planarity testing flat clustered graphs free sides finally open problems found section proofs found appendix preliminaries assume familiarity basic definitions graph theory graph drawing particular notions tree see flat clustered graph simple graph vertex set edge set partition sets called clusters edge edge edge nodetrix representation flat clustered graph cluster called trivial cluster represented distinct point plane cluster called cluster represented symmetric adjacency matrix rows columns drawn plane boundary square sides parallel coordinate axes iii intersection two distinct matrices point representing vertex matrix intracluster edge cluster represented adjacency matrix edge represented simple jordan arc connecting point boundary matrix point boudary matrix point belongs column row resp associated resp nodetrix representation flat clustered graph planar intersection two edges except possibly common intersection edge matrix flat clustered graph nodetrix planar admits planar nodetrix representation fig example nodetrix planar representation formal definition problem investigated paper follows let flat clustered graph vertices let maximum cardinality cluster clustered graph nodetrix planar fixed sides nodetrix planar representation edge sides matrices attaches specified part input nodetrix planar free sides sides matrices edges attach arbitrarily chosen let matrix representing cluster nodetrix representation let vertex let edge edge intersect boundary four points since row column represent intersect four sides boundary call points top copy bottom copy left copy right copy respectively side assignment specifies edge whether edge must attach matrix representing top left right bottom side precisely side assignment mapping set edges clusters adjacent side assignment set side assignments denote flat clustered graph given side assignment let nodetrix representation every edge incidence points matrices representing exactly points respectively call nodetrix representation consistent say nodetrix planar admits nodetrix planar representation consistent edge heavy belong clusters light otherwise flat clustered graph light every intercluster edge light heavy edge flat clustered graph replaces path defines new flat clustered graph light reduction flat clustered graph obtained performing every heavy edge consequence theorem edge density nodetrix planar graphs light reduction nodetrix planar flat clustered graph vertices edges property flat clustered graph nodetrix planar light reduction nodetrix planar based property remainder shall assume flat clustered graphs always light call clustered graphs short frame clustered graph graph obtained collapsing cluster single vertex called representative vertex let two representative vertices respectively every edge connecting vertex vertex edge connecting observe frame graph general multigraph however simple light since nodetrix planarity clustered graph implies planarity frame graph test nodetrix planarity clustered graphs planar frame graph recursively defined follows edge graph obtained adding vertex connecting two adjacent vertices planar graph partial subgraph planar biconnected partial graph clustered graph partial frame partial sometimes talk clustered graphs frames series parallel nodetrix representations wheel reductions algorithms described sections based decomposing planar frame clustered graph biconnected components storing tree process block using spqr decomposition tree rooted reference edge visited leaves root visited node decomposition tree block test whether subgraph whose frame pertinent graph satisfies planar constraints imposed side assignment edges key ingredient efficiently perform test notion wheel replacement let clustered graph side assignment let cluster vertices admits permutations vertices associate suitable graph permutation let permutation vertices wheel consistent wheel graph consisting vertex degree adjacent vertices oriented cycle edge cycle oriented forward intuitively oriented cycle embedded clockwise encode constraints induced matrix representing order columns precisely wheel replacement cluster consistent clustered graph obtained follows remove edges incident insert wheel consistent iii edge insert edge incident call edge image edge let clustered graph side assignment set permutations one cluster call permutation assignment say nodetrix planar side assignment permutation assignment admits nodetrix planar representation side assignment matrix permutation columns wheel reduction consistent graph obtained performing wheel replacement consistent theorem let clustered graph side assignment permutation assignment nodetrix planar planar wheel reduction admits planar embedding external oriented cycle wheel embedded clockwise fig fig show respectively nodetrix planar representation clustered graph corresponding wheel reduction planar embedding based theorem test graph nodetrix planarity exploring space possible permutation sets corresponding wheel reductions search nodetrix planar note maximum size cluster given parameter every cluster replaced wheel graphs one possible permutation vertices order test planarity wheel replacement cyclic order edges incident vertex arbitrarily permuted wheel reduction yields instance constrained planarity testing solved algorithm void void void void ext fig nodetrix planar representation clustered graph planar embedding corresponding wheel reduction labeling vertices complete internal external sequences highlighted described approach repeats algorithm possible wheel reduction may lead testing planarity different instances instead visited node decomposition tree compute succinct description possible nodetrix planar representations subgraph represented subtree rooted done storing poles pairs wheel graphs compatible nodetrix planar representation efficiently compute succinct description subject next sections testing nodetrix planarity partial section prove nodetrix planarity testing fixed sides solved polynomial time clustered graph maximum size cluster bounded constant frame graph partial contrasts nodetrix planarity testing fixed sides proved case size clusters unbounded first study case clustered graph whose frame graph seriesparallel graph biconnected spqr decomposition tree consider case partial graphs whose biconnected components frame graphs section prove nodetrix planarity testing fixed sides solved time clustered graphs whose frame graphs cluster size let clustered graph side assignment let frame graph let spq decomposition tree rooted simplify description without loss generality assume every exactly two children let node let poles consider pertinent graph represented subtree rooted let pole pole frame graph may correspond cluster case call pole cluster pertinent cluster edges incident edges edges incident edges intracomponent edge corresponds edge incident one vertex pertinent cluster call intracomponent edge edge associate wheel graphs pole wheel replacement pertinent cluster consistent one permutations vertices let pole let pertinent cluster let permutation vertices let wheel replacement consistent every edge incident image edge labeled either int ext depending whether edge vertex external cycle assigned one label set void int ext follows vertex labeled void edge incident image edge vertex labeled int resp ext label int resp ext every edge incident image edge otherwise vertex labeled see fig example concerning wheel fig dashed curve fig shows subgraph wheel reduction corresponding clockwise sequence vertices external cycle external sequence pole consistent labeled either ext vertices sequence labeled either void ext external clockwise sequence pole complete contains vertices labeled ext note complete external sequence may contain many void vertices int vertex internal complete internal sequences pole defined analogously observe complete internal sequence complete external sequence may exist vertices labeled int vertices labeled ext alternate twice traversing clockwise external cycle three vertices labeled special case exactly two vertices labeled vertices void case clockwise sequence clockwise sequence complete internal complete external sequences order test nodetrix planarity implicitly take account possible permutation assignments considering pole node possible wheels computing complete internal complete external sequences visit decomposition tree leaves root equip node information regarding complete internal complete external sequences poles let internal node let pole let permutation pertinent cluster let wheel consistent denote iseq complete internal sequence consistent pole eseq complete external sequence consistent pole distinguish different types nodes node since light one poles let edge pertinent graph one representative vertex pertinent cluster pole fact edge corresponds edge trivial cluster side assignment defines whether incident top bottom left right copy wheel possible permutation iseq labeled eseq external cycle starting ending otherwise traverse external cycle starting following direction edges eseq consists encountered vertices first labeled ext last labeled ext node let children observe pole also children consider every permutation complete internal sequence complete external sequence compatible complete internal sequence consistent union complete internal sequences children iseq iseq determine complete external sequence consistent consider intersection complete external sequences children intersection consists exactly one sequence consecutive vertices eseq eseq otherwise intersection empty consists one sequence consecutive vertices complete external sequence consistent node let child shares pole consider every permutation iseq eseq complete internal external sequence consistent iseq iseq eseq eseq test nodetrix planarity execute traversal node poles check whether possible pair induces complete internal external sequences compatible planar embedding wheel reduction case theorem nodetrix planar otherwise reject formally let respectively permutation respectively complete internal sequence complete external sequence compatible respectively say compatible pair permutations either one poles trivial pole one following cases applies node case possible pairs permutations recall one compatible node let children consider pair permutations recall poles first condition pair compatible pair also compatible pair second condition asks pair defines opposite orders poles namely let wheel consistent traversing clockwise external cycle starting first vertex eseq let iseq iseq iseq order internal sequences encountered pair defines opposite orders poles traversing clockwise external cycle starting first vertex eseq order encounter internal sequences opposite one order iseq iseq iseq node let children pair compatible pair exists permutation pair compatible pair compatible fig suggests nodetrix planar representation clustered graph defines permutation assignment every node pair compatible pair lemma let clustered graph side assignment let spq decomposition tree frame graph graph nodetrix planar exists permutation assignment every node poles permutation permutation form compatible pair permutations lemma let clustered graph side assignment let maximum size cluster let cardinality exists algorithm tests whether nodetrix planar side assignment computes nodetrix planar representation consistent proof let frame graph possible choice edge repeat following procedure construct spq decomposition tree rooted whose pertinent graph visit leaves root test whether permutation assignment nodetrix planar first equip pole every node possible complete internal complete external sequences maximum number complete internal sequences true complete external sequences complete internal fig nodetrix planar representation induces permutation assignment planar embedding wheel reduction complete internal external sequences pair poles also highlighted external sequence pole encoded means first last vertex clockwise order around complete internal external sequence needs constant space follows intersection union two complete internal external sequences computed constant time therefore complete internal external sequences pole computed time hence whole traversal equip poles every possible complete sequence executed time test whether exists permutation assignment node compatible pair permutations aim look complete internal external sequences pair poles children pair permutations poles equip information whether pair compatible requires space every pair permutations compatible follows compatible pairs computed time recall one hence time children one permutations equip one permutations equip testing whether pair compatible pair executed time follows compatible pairs computed time hence time children one permutations equip one permutations equip testing whether pair compatible pair executed time corresponding choosing possible permutations pole shared follows compatible pairs computed time hence time conclusion time complexity visit rooted rooting possible overall time complexity stirling approximation thus clustered graph vertices side assignment mum cluster size tested nodetrix planarity time note compatible pair permutations stored node implicitly define planar embedding wheel reduction shown possible construct nodetrix planar representation time proportional number edges statement lemma follows partial consider clustered graphs whose cluster size frame graph partial planar graph whose biconnected components handle case decomposing frame graph blocks store tree following theorem generalizes result lemma theorem let partial clustered graph side assignment let maximum size cluster let cardinality exists algorithm tests whether nodetrix planar side assignment computes nodetrix planar representation consistent general planar frame graphs section study problem extending theorem planar frame graphs may partial prove nodetrix planarity testing fixed sides solved polynomial time maximum cluster size however problem becomes fixed sides remains even free sides scenario every block frame graph decomposed triconnected components means spqr decomposition tree block adopt approach graphs look permutation assignment every pair poles forms compatible pair either extend definition compatible pairs permutations follows let clustered graph side assignment let frame graph let spqr decomposition tree let poles pair permutations forms compatible pair exists planar embedding skeleton skel following conditions hold vertex skel let virtual edges skel incident clockwise order around edge associated child exists permutation complete internal sequences iseq iseq iseq appear clockwise order around every vertex skel assigned permutation virtual edge skel corresponds child permutation pair compatible observe case maximum cluster size possible permutations induced cluster vertex skel exactly two denoted order test whether forms compatible pair perform traversal skel starting permutation clockwise order edges incident impose choose one two permutations available vertex adjacent corresponding cluster incident edges turn propagate constraints possible permutations neighbors till reached therefore testing whether form compatible pair reduced suitable problem labeling edges vertices skel verifying end labeled theorem let clustered graph side assignment maximum size cluster two exists algorithm tests whether nodetrix planar given side assignment computes nodetrix planar representation consistent proof following theorem based reduction theorem nodetrix planarity testing fixed sides cluster size extend hardness result free sides model show nodetrix planarity testing remains maximum cluster dimension larger four done proving even triconnected boolean formulas may result independent interest theorem triconnected boolean formulas theorem nodetrix planarity testing free sides cluster size open problems conclude paper listing open problems opinion worth investigating study complexity nodetrix planarity testing free sides scenario values study families clustered graphs nodetrix planarity testing fixed parameter tractable free sides scenario iii determine whether time complexity algorithms theorems improved references angelini lozzo battista frati patrignani rutter representations graphs giacomo lubiw eds proceedings international symposium graph drawing network visualization lecture notes computer science vol batagelj brandenburg didimo liotta palladino patrignani visual analysis large graphs using hybrid visualizations ieee trans vis comput graph lozzo battista frati patrignani computing nodetrix representations clustered graphs eds graph drawing network visualization lecture notes computer science vol battista eades tamassia tollis graph drawing prentice hall upper saddle river giacomo liotta patrignani tappini planar graphs new family beyond planar graphs frati eds graph drawing network visualization lncs springer appear gutwenger klein mutzel planarity testing optimal edge insertion embedding constraints graph algorithms appl harary graph theory series mathematics addison wesley henry fekete mcguffin nodetrix hybrid visualization social networks ieee trans vis comput graph special planar satisfiability problem consequence discrete applied mathematics moret planar sigact news schaefer complexity satisfiability problems proceedings annual acm symposium theory computing omitted proofs proof theorem theorem let clustered graph side assignment permutation assignment nodetrix planar planar wheel reduction admits planar embedding external oriented cycle wheel embedded clockwise proof nodetrix planar construct planar embedding wheel reduction external oriented cycle wheel embedded clockwise follows let nodetrix planar representation replace matrix representing cluster wheel consistent permutation also embedded way forward traversal external cycle clockwise traversal cycle every edge incident vertex wheel also cyclic order edges incident cyclic order edges incident pvj immediate see since two edges cross two edges cross constructed embedding wheel reduction conversely suppose given planar embedding wheel reduction external oriented cycle wheel embedded clockwise show construct nodetrix planar representation wheel remove center vertex wheel insert matrix inside created face morph every vertex external cycle point pvj maintain around pvj cyclic order edges incident planar embedding wheel reduction proof lemma lemma let clustered graph side assignment let spq decomposition tree frame graph graph nodetrix planar exists permutation assignment every node poles permutation permutation form compatible pair permutations proof prove first nodetrix planar exists permutation assignment every node poles pair compatible let nodetrix planar representation side assignment let matrices representing clusters matrix let left right order columns replace wheel consisting vertex degree adjacent vertices cycle vertex drawn point pvj represents attachment edges incident vertex side matrix edges external cycle drawn along external boundary every edge incident vertex wheel also cyclic order edges incident cyclic order edges incident pvj straightforward verify computed drawing defines planar embedding wheel reduction consistent planarity wheel reduction follows pole frame graph complete internal complete external sequence consistent every node spq decomposition tree poles pair compatible example described procedure illustrated fig show exists permutation assignment every node poles permutation pair compatible nodetrix planar side assignment construct planar embedding wheel reduction consistent external cycles wheels embedded clockwise theorem implies nodetrix planar let two wheels consistent respectively visit leaves root incrementally construct desired planar embedding wheel reduction visited node one poles light assume without loss generality pole let cluster represented frame graph embed wheel consistent traversing edges external cycle forward direction cycle traversed clockwise embed external face planarly connect top bottom left right copy specified suppose let children planar embedding wheel reduction node obtained composing planar embedding wheel reduction node planar embedding wheel reduction done identifying planar embedding wheel consistent planar embedding wheel consistent note possible compatible pair compatible pair finally assume let children similar case planar embedding wheel reduction node obtained composing planar embeddings wheel reductions nodes since pair compatible defines opposite orders poles opposite circular orders correspond planar embedding wheel reduction obtained combining planarly embedded wheel reductions children follows nodetrix planar permutation assignment proof theorem theorem let partial clustered graph side assignment let maximum size cluster let cardinality exists algorithm tests whether nodetrix planar side assignment computes nodetrix planar representation consistent proof compute tree tbcv frame graph root block broot visit tbcv let currently visited block let parent tbcv execute testing algorithm lemma rooting spq decomposition tree one poles test fails block tbcv conclude nodetrix planar given side assignment chosen root broot tbcv repeat test rooting tbcv different block otherwise test whether among permutation assignments computed blocks children exists set complete internal sequence block permutation assignment overlap complete internal sequence block permutation assignment equip permutations pass test let block parent tbcv testing nodetrix planarity consider permutations computed processing blocks check complete internal sequences intersect complete internal sequences let number vertices block using lemma procedure described executed time per block therefore visit tbcv computed time root block since possible roots tbcv follows overall time complexity proof theorem lemmas discussion section prove theorem completeness report statement theorem theorem let clustered graph side assignment maximum size cluster two exists algorithm tests whether nodetrix planar given side assignment computes nodetrix planar representation consistent let clustered graph side assignment let frame graph assume first frame graph biconnected let spqr decomposition tree analogously case frame graph described section traverse leaves root compute node poles pairs permutations compatible formal definitions pair permutations compatible found section formal definition pair permutations compatible section differently algorithm frame graphs handle case skeleton skel triconnected graph fact cases handled exactly described section overall time complexity lemma let poles children given vertex skel since maximum cluster size two possible permutations exactly two denote sake simplicity assume vertex skel labeled one two labels set corresponding two possible permutations pertinent cluster also assume edge skel corresponding child assigned one labels set initialized based pairs permutations compatible child first phase algorithm node discard edge vertex labels give rise planar embedding skel particular consider one two possible planar embeddings one let chosen embedding perform rotation coherence check purpose enforcing compatibility respect embedding two permutations vertex skel namely consider vertex skel one possible permutations let virtual edges skel appear clockwise order around embedding skel let corresponding nodes respectively complete internal sequences iseq appear different clockwise order around respect corresponding virtual edges nodetrix planar permutation assignment hence label discarded correspondingly permutation discarded cluster end phase vertex edge remains without label nodetrix planar given embedding want decide whether pairs permutations compatible affirmative case want compute problem deciding whether permutations pairs compatible equivalent following given plane graph vertex least label edge least one label label selected vertex way edge label call plane graph described labeling vertices edges instance solution coherent labeling observe vertex edge labels filtered based following obvious properties selected vertex vertex coherence coherent labeling label exists virtual edge skel label edge coherence coherent labeling label selected virtual edge admit label skel admit label hence discard vertex edge labels satisfy properties instance vertex label satisfies property vertex coherence edge label satisfies property edge coherence said succinct observe time vertex label discarded property vertex coherence vertex labels virtual edge incident checked property edge coherence also time edge label discarded property edge coherence property vertex coherence checked labels one incident vertices due fact one label per vertex three labels per edge discarded otherwise instance nodetrix planar instance reduced succinct instance time linear number edges skel succinct instance called reduced vertex exactly two labels edge exactly three labels use following lemma efficiently solve problem lemma let instance exists reduced instance admits coherent labeling admits coherent labeling proof graph found sequence transformations starting original instance first transformation described proof following claim claim let succinct instance exists succinct instance whose vertices labels admits coherent labeling admits coherent labeling proof claim let vertex one label let graph obtained removing incident edges show admits coherent labeling admits one one direction easy prove every solution also solution since vertices edges also vertices edges prove opposite direction suppose vertex label case analogous consider edge incident property edge coherence follows edge may label label labels labels turn property vertex coherence labels vertex following three cases respectively vertex label vertex label vertex labels suppose admits solution easily obtain solution original instance reinserting incident edges selecting labels follows select label select edge respectively depending incident depending label label three cases prove admits coherent labeling since solution selects label vertex selecting label vertex label edge yields solution since solution selects label vertex selecting label vertex label edge yields solution solution may select either label case label case vertex selecting label vertex label case case edge yields solution repeating procedure every vertex one label equivalent instance vertices labels obtained finally observe since removal edges instance disrupt properties vertex coherence edge coherence labels applying transformation described obtain succinct instance whenever original instance succinct claim several important consequences order explicit need notation say edge label set following sets labels label set immediate consequence claim following property claim vertex instance labels edges exclusively label sets proof claim proof based observation remaining label sets either miss label sets miss among others label sets miss among others label set among others label set however property vertex coherence labels edge allow claim let succinct instance exists succinct instance edge label set admits coherent labeling admits one coherent labeling obtained coherent labeling constant time proof claim edge four labels one could remove compute coherent labeling obtained instance insert back suitable label depending pair labels selected claim let succinct instance exists succinct instance edge label set admits coherent labeling admits one coherent labeling obtained coherent labeling constant time proof claim suppose edge labels contract edge merge vertices new vertex suppose obtained graph admits coherent labeling expand vertex back label select label label label edge otherwise select label label label easy verify obtained selection labels coherent labeling claim let succinct instance exists succinct instance edge label set admits coherent labeling admits one coherent labeling obtained coherent labeling constant time proof claim suppose edge labels replace labels labels edges incident observe instance labels renamed admits coherent labeling instance admits one particular edges label set respectively label set respectively hence edge claim applies edge contracted reinserted coherent labeling resulting instance found claims imply assume every vertex labels every edge label set hence obtained instance reduced instance equivalent instance concludes proof lemma every based lemma assume every vertex exactly two labels edge exactly three labels consider edge whose missing label implies instance coherent labeling label selected vertex label selected vertex solution must consideration generalized follows property let reduced instance let edge missing label coherent labeling label selected label selected label different analogously label selected label selected label different property shows selecting label vertex may consequences label selected neighbor vertices local choice may propagate graph however property also implies following property property missing label edge selection label vertex implication selection label selection label vertex compatible selection label vertex gives rise following procedure compute pairs permutations compatible since skel two possible planar embeddings procedure repeated twice embedding let current embedding skel build instance vertex edge zero labels admit compatible pair embedding chosen skel obtain reduced instance admits coherent labeling admits one let two vertices corresponding vertices original instance consider one one four possible permutation pairs decide whether permutation pair compatible select label corresponding propagate selection according property vertex reached two times imposing selection label imposing selection label pair compatible suppose first propagation phase stops without contradiction label vertex selected yet start second propagation selecting label corresponding vertices label selected start new propagation phase selecting arbitrary label one property guarantees vertex traversed propagation phase traversed successive propagation phase hence labels selected one propagation phase cause contradiction labels selected successive propagation phases contradiction found selected label permutation pair compatible proof theorem concluded observing transformations described proof lemma requires constant time therefore overall time complexity compute pairs permutations linear number edges skel observation together lemma implies nodetrix planarity fixed sides tested time clustered graph whose maximum cluster size two frame graph biconnected connected case construct tree tbcv frame graph apply strategy described proof theorem namely every node either test whether exists compatible pair means procedure lemma use strategy described gives rise procedure possible choice root tbcv since possible choices overall time complexity frame graph biconnected statement theorem follows proof theorem theorem nodetrix planarity testing fixed sides cluster size proof nodetrix planarity testing fixed sides trivially fact given clustered graph possible permutations assignments could computed problem deciding whether clustered graph admits nodetrix planar representation side assignment permutation assignment solvable linear time order prove reduce instance consists collection clauses set boolean variables clause consists exactly three literals problem asks whether exists truth assignment variables clause least one true literal least one false literal problem shown thomas schaefer however known polynomial graph adjacencies variables clauses planar starting instance variables clauses construct instance nodetrix planarity testing fixed sides follows first obtain drawing graph variables clauses like one fig clause vertices vertically aligned variable vertices horizontally aligned edges drawn clearly drawing computed polynomial time polynomial number crossings fig drawing instance small circled plus signs circled minus signs represent direct negated occurrences variables clauses respectively replace vertex representing variable variable gadget variable gadget variable degree composed clusters connected together depicted fig namely let clusters composing variable gadget variable let nodes may encode truth value one two possible representations cluster matrix representing column corresponding vertex precedes column corresponding vertex say true otherwise say false correspondingly say true permutation cluster false permutation order connect clusters composing variable gadget add edges set immediate nodetrix planar representation either simultaneously true simultaneously false correspondingly say variable gadget true false fig show example true false drawing variable gadget false true true false fig true configuration variable gadget variable degree four false configuration gadget transforming encoded true value false value gadget transforming encoded false value true value edge attaching variable drawing refer fig corresponds two parallel edges attached one clusters composing variable gadget let cluster set observe order exit depends truth value encoded hence truth value encoded variable gadget fig depicts gadget use replace crossings consisting cluster size three figure apparent representation order edges entering bottom side order edges exiting top side analogous consideration holds order edges entering right side order edges exiting left side implies truth value encoded edges entering truth value encoded edges exiting false true true false true false false true false true false false true true false true true true false false true false true false fig six possible configurations crossing gadget describe clause gadget assume three pairs edges encoding truth value variables occurring clause arrive clause gadget let three variables whose literals occur clause entering clause gadget literal negated literal variable attach edges coming gadget depicted fig use edges exiting gadget instead edge coming directly variable effect three pairs edges entering clause gadget encode truth value true literal true false literal false following therefore consider truth values literals rather truth values variables clause gadget depicted fig composed three clusters size three vertices three clusters connected together way nodetrix planar representation permutations always present sequence labels example also edges encoding truth value literal attach cluster prescribed side right side matrix two edges encoding truth value literal attach respectively two edges encoding truth value literal attach respectively finally two edges encoding truth value literal attach respectively hence literal true matrix must permutation columns column precedes column literal false matrix must permutation columns column follows column analogously truth value literal determines whether matrix column precedes follows column truth value literal determines whether matrix column precedes follows column true literal true true literal true true literal true fig clause gadget follows three literals true induce unsatisfiable constraints ordering columns matrices since column precede column precede column precede holds three literals false easily checked combination truth values literals exists ordering columns matrices makes planar drawing edges possible therefore constructed instance nodetrix planarity fixed sides admits planar nodetrix representation original instance admits solution proof theorem theorem triconnected boolean formulas proof use reduction strategy similar one used prove hardness planar triconnected however worry planarity since searching triconnected instance let instance graph represents occurrences variables clauses triconnected possibly biconnected connected show construct instance graph triconnected strategy consists adding suitable number variables clauses obtained instance triconnected solution one consider edge graph refer fig suppose variable occurs clause direct literal negated literal respectively remove remove occurrence clause replace occurrence direct literal negated literal respectively new variable also add new variable two clauses see fig shown assigment truth values variables clauses satisfied sense truth value truth value fact whatever truth fig variable occurring clause instance gadget replaces edge pair edges graph gadget replaces edges value variable literals occur clause negative literal positive literal hence positive negative implies follows obtained instance admits solution original instance admits one consider two edges see fig replacing edges described gadget identifying variables two gadgets one obtains bridge gadget instance replaced bridge gadget equivalent original instance path going passing hence suppose graph let two edges belonging two different connected components replacing bridge gadget number connected components decreased assume connected let tbcv tree let two edges belonging two different blocks tbcv replacing bridge gadget number blocks decreased see example fig holds replace two edges belonging two triconnected components fig tree tbcv graph green vertices leaves tbcv bridge gadget added link two leaf blocks tbcv therefore obtain instance graph represents occurrences variables clauses triconnected admits solution admits one furthermore size gadgets used replacement constant therefore size polynomial size concludes proof proof theorem theorem nodetrix planarity testing free sides cluster size proof proof based reduction triconnected starting triconnected instance first build instance gfix nodetrix planarity fixed sides analogously proof theorem using drawing strategy shown fig instead shown fig observe since triconnected special planarization strategy used frame graph gfix also triconnected fact inserting crossing gadgets equivalent planarizing drawing fig fig drawing instance replacing crossings dummy nodes would obtain planar triconnected graph fig cluster size three instance nodetrix planarity fixed sides gadget used replace instance nodetrix planarity free sides second obtain instance gfree nodetrix planarity free sides replacing cluster maximum size three gadget depicted fig uses exclusively clusters size gadget consists nine clusters corresponding nodes frame graph gfree form wheel graph external cycle eight nodes connected central one since wheel one embedding flip gadget admits nodetrix planar representation hub wheel drawn inside cycle formed eight clusters also edges instance gfix constrained attach specific side matrix due side assignment incident cluster wheel since gfree triconnected frame graph embedding frame graph fixed nodetrix planar representation fixed sides gfix immediately obtained nodetrix planar representation free sides gfree
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circuits algorithmic problems nilpotent groups alexei armin stevens institute technology hoboken usa stuttgart germany jul abstract recently macdonald showed many algorithmic problems finitely generated nilpotent groups including computation normal forms subgroup membership problem conjugacy problem computation subgroup presentations done logspace follow approach show problems complete uniform circuit class uniformly nilpotent groups class fixed order solve problems show unary version extended gcd problem compute greatest common divisors express linear combinations moreover allow certain binary representation inputs word problem computation normal forms still uniform problems examine shown reducible binary extended gcd problem keywords phrases nilpotent groups abelian groups word problem conjugacy problem subgroup membership problem greatest common divisors contents introduction preliminaries complexity nilpotent groups mal cev coordinates presentation subgroups quotient presentations word problem computation mal cev coordinates extended gcd problem matrix reduction subgroup membership problem subgroup membership problem subgroup presentations algorithmic problems homorphisms kernels centralizers conjugacy problem computing quotient presentations power problem conjugacy wreath products nilpotent groups conclusion open problem alexei myasnikov armin licensed creative commons license leibniz international proceedings informatics schloss dagstuhl informatik dagstuhl publishing germany myasnikov introduction word problem given word generators represent identity one fundamental algorithmic problems group theory introduced dehn general finitely presented groups problems undecidable many particular classes groups decidability results established word problem also wide range problems finitely generated nilpotent groups class many algorithmic problems efficiently decidable exceptions like problem solving equations see mal cev established decidability word subgroup membership problem investigating finite approximations nilpotent groups blackburn showed decidability conjugacy problem however methods allow efficient polynomial time algorithms nevertheless mostowski provided practical algorithms word problem several problems terms complexity major step result lipton zalcstein word problem linear groups logspace together fact finitely generated nilpotent groups linear see gives logspace solution word problem nilpotent groups later improved uniform robinson typical algorithmic approach nilpotent groups using mal cev mal cev bases see allow carry group operations evaluating polynomials see lemma approach systematically used general setting polycyclic presentations solving among others subgroup membership conjugacy problem polycyclic groups recently polynomial time bounds equalizer subgroup membership problems nilpotent groups given finally following problems shown logspace using mal cev basis approach denotes class nilpotent groups nilpotency class generated elements word problem given given compute mal cev normal form subgroup membership problem given decide whether express word subgroup generators decision version shown logspace expressing word original subgroup generators polynomial time bound given given together homomorphism specified compute generating set ker find given compute presentation given compute generating set centralizer conjugacy problem given decide whether exists find element problems interest also might serve building blocks solving problems polycyclic groups particular interest possible application cryptography work follow extend results several ways give complexity bound uniform problems order derive bound show extended gcd problem given compute gcd input output unary uniform circuits algorithmic problems nilpotent groups description circuits uniform setting part input uniform setting also considered however short remarks since nilpotent groups polynomial growth natural allow compressed inputs give uniform solution word problem allowing words binary exponents input contrasts situation programs contextfree grammars produces precisely one word another method exponential compression input word problem hard thus difficulty word problem programs due compression rather due difficulty evaluating program show problems binary extended gcd problem inputs ambient group subgroup etc given words binary exponents show solve power problem nilpotent groups allows apply result order show iterated wreath products nilpotent groups conjugacy problem uniform thus unary case settle complexity problems completely moreover also seems rather difficult solve subgroup membership problem without computing gcds case results binary inputs would also optimal altogether results mean many algorithmic problems complicated nilpotent groups abelian groups notice explicit length bounds outputs problems proven obtain polynomial length bounds simply fact everything computed uniform following write throughout paper follow outline concise presentation copy many definitions theorems involve two statements one unary encoded inputs one binary encoded inputs order concise presentation always put one result consider finitely generated nilpotent groups without mentioning outline start basic definitions complexity well nilpotent groups section describe subgroups nilpotent groups represented develop nice presentation groups section deals word problem computation normal forms solve unary extended gcd problem introduce matrix reduction order solve subgroup membership problem section present result remaining problems section explain compute nice presentations section apply results order show conjugacy problem iterated wreath products nilpotent groups finally conclude open questions preliminaries complexity finite alphabet set words denoted computation decision problems given functions finite alphabets decision problem formal language identified characteristic function particular word conjugacy problems seen functions use circuit complexity described circuit classes class defined class functions computed families circuits constant depth polynomial size unbounded boolean gates myasnikov majority gates majority gate denoted maj returns number input greater equal number following always assume alphabets encoded binary alphabet letter uses number bits say function following consider circuit families simply write shorthand means deterministic turing machine decides time log input two gate numbers given binary string whether wire two gates circuit also computes type gates note binary encoding gate numbers requires log bits thus turing machine allowed use time linear length encodings gates details definitions refer following inclusions note even known strict logspace reductions function function dlogtimeuniform family circuits computing addition boolean majority gates also may use oracle gates gates input output expressed note particular functions also composition extensively make use observation also guarantee polynomial size bound outputs circuits without additional calculations also use another fact frequently without giving reference input two alphabets coded binary alphabet list pairs occurs precisely one pair word image homomorphism defined computed encoding numbers unary binary essentially two ways representing integer numbers usual way binary number string represents unary number represented respectively number input bits state results paper representations unary representation corresponds group elements given words generators whereas binary encoding used inputs given compressed form example following problem count given length number assume given unary decide whether number ones exactly thus count maj particular word problem encoded simply question whether even arithmetic iterated addition resp iterated multiplication following computation problems input binary integers bits input length compute binary representation sum resp product integer division input two binary integers binary representation integer computed first statement theorem standard fact see statements due hesse circuits algorithmic problems nilpotent groups theorem problems iterated addition iterated multiplication integer division matter whether inputs given unary binary note numbers encoded unary strings division seen easily try whether representing groups algorithmic problems consider finitely generated groups together finite generating sets group elements represented words generators inverses elements make distinction words group elements represent whenever might unclear whether mean equality words group elements write equality words generators correspond unary representation integers generalization binary encoded integers introduce following notion word binary exponents sequence fixed generating set group together sequence exponents encoded binary word binary exponents represents word group element wnxn note fixed nilpotent group every word length rewritten word binary exponents using log bits fact also consequence theorem thus words binary exponents natural way representing inputs algorithmic problems nilpotent groups nilpotent groups mal cev coordinates let group write conjugated commutator subgroups group called nilpotent central series finitely generated abelian quotients let aimi basis generating set presentation aimi aijij aik stands torsion eij aware explicitly allow eij necessary definition quotient presentations section formally put eij note acmc polycyclic generating sequence call mal cev basis associated central series sometimes use interchangeably also set acmc convenience also use simplified notation generators aij exponents eij renumbered replacing subscript generating sequence written allow expression stand notations well also denote choice every element may written uniquely form myasnikov whenever called coordinate vector mal cev coordinates denoted coord expression called mal cev normal form also denote coordi mal cev basis associate presentation follows let gni gni aei gni hence relation aiei holds gni let since series central relations form ijm hold gnj group generators subject relation form presentation relations form resp called nilpotent presentation indeed presentation form define nilpotent group called consistent order modulo precisely presentations form need general consistent derived central series group consistent given consistent nilpotent presentation easy way solve word problem simply apply rules form move occurrences input word left apply power relations reduce number modulo finally continue multiplication functions crucial feature coordinate vectors nilpotent groups coordinates product may computed nice function polynomial integers lemma let nilpotent group mal cev basis exist coord coord coordi coordi iii notice explicit algorithm construct polynomials given background nilpotent groups refer presentation subgroups start algorithmic problems introduce canonical way represent subgroups nilpotent groups important two reasons first course need solve subgroup membership problem second uniform setting allows represent nilpotent groups free nilpotent group modulo kernel represented subgroup let elements given normal form let associate matrix coordinates circuits algorithmic problems nilpotent groups tuple conversely integer matrix associate elements whose mal cev coordinates given rows matrix subgroup generated tuple row let column first entry pivot row sequence said standard form matrix coordinates form pivot columns maximally reduced similar hermite normal form specifically satisfies following properties rows trivial number pivots iii divides sequence resp matrix called full addition hai generated note consists elements first coordinates easy exercise see also show holds given elements use full sequences associated matrices full form interchangeably without mentioning explicitly simplicity assume inputs algorithms given matrices importance full sequences described following lemma proof found propositions lemma lem let unique full sequence generates thus computing full sequence essential tool solving subgroup membership problem focus subgroup membership first solve word problem introduce nilpotent group part input quotient presentations let fixed free nilpotent group class rank defined group subject relations weight commutators trivial throughout fix mal cev basis call standard mal cev basis associated lower central series associated nilpotent presentation consists relations form presentation exists since generates mal cev generators iterated commutators denote set nilpotent groups class every group quotient free nilpotent group normal subgroup assume full sequence generating adding set relators free nilpotent group yields new nilpotent presentation presentation called quotient presentation inputs algorithms assume quotient presentation always given matrix coordinates full form depending whether entries matrix encoded unary binary call quotient presentation given unary binary myasnikov lemma prop let fixed integers let standard mal cev basis moreover denote set relators respect let let sequence subgroup consistent nilpotent presentation proof clearly since nilpotent presentation elements add relators form presentation nilpotent prove consistent suppose order modulo since order infinite must element form lemma must contain element amm divides hence smaller presentation consistent following always assume quotient presentation part input fixed later show compute quotient presentations arbitrary presentation remark lemma ensures group element unique normal form respect quotient presentation thus guarantees manipulations mal cev coordinates word problem computation mal cev coordinates section deal word problem nilpotent groups generalize result allowing words binary exponents recall word binary exponents sequence wnxn using words binary exponents input compressed exponentially making word problem priori harder solve nevertheless turns word problem still solved allowing input given word binary exponents note contrasts situation input given program like words binary exponents allow exponential compression word problem complete counting class theorem let fixed let standard mal cev basis following problem input given binary encoded quotient presentation word binary exponents wnxn compute integers binary aymm moreover input given unary output unary note statement unary inputs essentially one aware formulation theorem depend input group parameters read full matrix coordinates representing recall denotes column index pivot number rows matrix columns pivot immediate consequence theorem obtain corollary let fixed uniform binary version word problem groups input given theorem circuits algorithmic problems nilpotent groups proof theorem follows outline given section however apply rules one one instead make two steps generator first apply possible rules one step apply rules one step proof theorem hardness part clear since already word problem describing circuit proceed induction along standard mal cev basis free nilpotent group contain letter compute induction otherwise rewrite words binary exponents containing completed rewritten aymm induction computing proceed two steps first rewrite possible lemma iii exponent computed iterated addition theorem unary case written unary consists remains eliminated every position compute using iterated addition let lemma fixed polynomials free nilpotent group holds axk axk amk hence order obtain remains replace every wixi empty word every wixi axki amk word binary exponents resp word polynomial length unary case exponents computed theorem since bounded polynomials unary case axki amk written word without exponents second step applied explained decided read directly quotient presentation checking whether pivot first column otherwise empty word rewrite mod word binary exponents containing computed theorem let power relation read quotient presentation row pivot first column write equal use fixed polynomials lemma yielding aqmm binary setting word binary exponents unary setting word without exponents polynomial length desired extended gcd problem computing greatest common divisors expressing linear combination essential step solving subgroup membership problem indeed consider nilpotent group let gcd binary gcds binary extended gcd problem extgcd follows input binary encoded numbers compute gcd myasnikov clearly done using euclidean algorithm known whether actually since need compute greatest common divisors reduce subgroup membership problem computation gcds unary gcds computing gcd numbers encoded unary straightforward exhaustive search yet obvious express gcd max computed logspace however algorithm uses logarithmic number rounds depending outcome previous one work note problem easy example let max gcd easy see assume cases similar given gcd replace change sum iterating step assure hence given unary coefficients computed simply checking polynomially many values max however want express gcd unboundedly many numbers linear combination check possible values max exponentially many expressing gcd linear combination viewed linear equation integral coefficients recently thm shown coefficients given unary decided whether equation system fixed number equations solution since proof thm obvious find actual solution prove following result theorem following problem given integers unary numbers compute either unary binary gcd max proof let max clearly computed assume positive assume numbers appear intermediate results encoded binary indeed numbers grow fast encode unary first observe gcd computed reason simply linearly many numbers less fact computing gcd circuit checks whether every dci found common divisor gcd simply largest one thus remains compute coefficients since compute gcd divide numbers gcd henceforth assume gcd note change coefficients first step computing compute gcd note assumption gcd gcd gcd gcd circuits algorithmic problems nilpotent groups using observation next step computes integers done parallel simply trying possible values example set computed using iterated multiplication see theorem moreover easy induction shows gcd one problem numbers general meet bounds next step modify way meet desired bound idea apply sequence operations example make coefficients small difficulty find exactly multiple let note assume set max max obviously correspond indices large positive indices small negative assumed positive moreover decreased resp increased resp order make reasonably small able reach aim completely sufficiently small error next set computed using iterated addition division see theorem lemma proof definition likewise since obtain meaning argument yields thus let set otherwise otherwise myasnikov lemma clearly computed work numbers also immediate consequence define otherwise note cases overlap however different definitions agree set set lemma proof show statement follows symmetry first assume hence lemma holds let define min max particular notice exist since also implies thus moreover since obtain set notice since means circuits algorithmic problems nilpotent groups finally define new coefficients follows otherwise remains show following numbers computed iii first point straightforward already remarked computed hence also computed simple boolean combination resp addition previous numbers computed using division finally computation simply another application iterated addition second point observe last equality due fact third point let lemma lemma case completely symmetric concludes proof theorem notice straightforward improve bounds theorem getting rid factor however since need order perform matrix reduction additional effort also could find circuit yields bound achievable logspace myasnikov matrix reduction subgroup membership problem matrix reduction procedure converts arbitrary matrix coordinates full form thus essential step solving subgroup membership problem several problems first described however without precise complexity estimate section repeat presentation show fixed actually computed uniformly groups case inputs given unary words inputs represented words binary exponents still show extgcd section defined matrix representation subgroups nilpotent groups adopt notation section let fixed let standard mal cev basis let given quotient presentation matrix full form either unary binary coefficients define following operations tuples subgroup generators elements corresponding operations associated matrix goal converting sequence full form generating subgroup swap corresponds swapping row row replace hlj corresponds replacing row coord hlj add remove trivial element tuple corresponds adding removing row zeros row form replace corresponds replacing row coord append arbitrary product hlikk tuple add new row coord hlikk clearly operations preserve lemma input quotient presentation unary resp binary matrix coordinates given unary resp binary operations done output matrix also encoded unary resp binary operations require exponents given unary resp binary moreover long rows matrix changed pairwise distinct polynomial number steps done parallel proof operations clearly done notice operation means simply row quotient presentation appended matrix unary case follows directly theorem operations since given unary respective group elements written words case binary inputs works follows analogously lemma functions every coord anda coordi functions used compute coord hijj hlikk written word binary exponents theorem applied using row operations defined shown reduce coordinate matrix unique full form let repeat steps let matrix coordinates section recall denotes column index pivot full form produce matrices number pivots full form every circuits algorithmic problems nilpotent groups first columns form matrix satisfying conditions full sequence condition satisfied full form formally denote set assume constructed steps construct let denote number rows columns respectively times computation denotes group element corresponding row denotes coordj may change every operation step locate column next pivot minimum integer least one integer exists already constructed otherwise set copy denote compute linear expression gcd let hlkk hlnn note coordinates form coord occurring position perform operation append row step perform operation replace row coord use replace row coord swap row row using point properties hold first columns step additionally ensure condition follows perform row operation respect append trivial element coord let gcd compute linear expression max let hnk append row row note coord position replace row coord row coord producing zeros column rows swap row row point iii hold first columns need since pivot entry instead replace row coord ensuring step identify next pivot like step last pivot set ensure condition observe steps preserve hence holds since holds range consider range suffices establish obtain notice note subgroup generated appears times last commutator closed commutation since appears times commutator trivial inductive argument shows subgroup hsj coincides similar observations made conjugation therefore appending via operation rows coord delivers note remains true obtain case add row coord commutes therefore preserved note element myasnikov step using operation eliminate zero rows matrix constructed show step performed given mal cev coordinates encoded unary resp extgcd mal cev coordinates encoded binary since total number steps constant depending nilpotency class number generators gives resp extgcd circuit computing full form given subgroup step next pivot found since simply next column matrix entry found simple boolean combination test whether entries zero unary case theorem gcd computed together encoded unary lemma step done binary case computed using extgcd hence lemma step done extgcd step numbers either unary binary computed parallel theorem one operation applied row matrix lemma done parallel rows finally swapping rows done step explained section read directly quotient presentation thus decided whether step executed appending new row computing gcd example unary case extgcd binary case one operation followed two operations one operation finally times operation done lemma step next pivot found outlined step step consists application constant number depending nilpotency class number generators operations thus lemma step clearly thus completed proof main result theorem let fixed following problem given unary encoded quotient presentation compute full form associated matrix coordinates encoded unary hence unique sequence generating moreover given binary sequence binary coefficients computed extgcd subgroup membership problem apply matrix reduction algorithm solve subgroup membership problem theorem let fixed following problem resp extgcd binary inputs given quotient presentation elements decide whether element subgroup moreover circuit computes unique expression sequence encoded unary resp binary alternatively unary inputs output given word note know whether analog second type output binary inputs possible way expressing output would word binary exponents circuits algorithmic problems nilpotent groups however simply applying procedure unary inputs lead word binary exponents proof circuit works follows first full form coordinate matrix corresponding sequence computed resp extgcd using theorem denote pivots lemma element written show find exponents denote coord defined following otherwise check whether divides yes let continue otherwise since bounded constant constant number steps step done theorem division theorem computation mal cev coordinates second type output unary case performing matrix reduction store every row matrix also row expressed word subgroup generators need unary inputs otherwise group elements expressed words polynomial space every operation matrix words updated correspondingly clearly done end writing every substituted respective word since abelian groups nilpotent obtain corollary let fixed following problem given list words generators decide whether moreover case positive answer compute unary words fixed given unary encoded system linear equations unary encoded solution computed subgroup presentations sequence associated subgroup forms mal cev basis allows compute consistent nilpotent presentation note however resulting presentation quotient presentation although transformed one see proposition partly due fact general following extended version thm theorem let fixed following unary inputs extgcd binary inputs input quotient presentation elements output consistent nilpotent presentation given list generators numbers encoded unary resp binary representing relations myasnikov proof first full sequence computed resp extgcd according theorem let hgi proof thm shown mal cev basis hence remains compute relators order give consistent nilpotent presentation order modulo simply read quotient presentation relation computed using resp extgcd circuit theorem input since unique full sequence membership algorithm returns expression right side relations established using method note constant number relations establish everything done resp extgcd algorithmic problems homorphisms kernels given nilpotent groups subgroup generating set homomorphism specified list elements homomorphism consider problem finding generating set kernel given finding following problems solved using matrix reduction group theorem kernels preimages let fixed following unary inputs extgcd binary inputs input given quotient presentations subgroup list elements defining homomorphism via optionally element guaranteed image compute generating set kernel element case unary inputs returned words binary inputs words binary exponents proof let standard mal cev basis standard mal cev basis two embeddings assume mal cev basis chosen way embeddings send mal cev generators mal cev generators note thus read relators via embeddings respectively obtain quotient presentation simply need add relations commute need introduce additional relations mal cev generators image new quotient presentation basically copy computed work direct product identify images let hhi let sequence full form subgroup let greatest integer set thm circuits algorithmic problems nilpotent groups shown sequence kernel sequence image solve suffices compute using theorem return corresponding defined apply theorem express return centralizers focus conjugacy problem need one preliminary result problem computing centralizers theorem centralizers let fixed following unary inputs extgcd binary inputs input given quotient presentation element compute generating set centralizer case binary inputs generating set given set words binary exponents proof let lower central series clearly central series projects onto central series simply write projection denote standard mal cev basis associated lower central series particular generating set proceed induction abelian output assume theorem holds groups particular obtain quotient presentation simply forgetting mal cev generators generating set centralizer computed resp extgcd induction let preimage hki homomorphism define since commutes modulo hence moreover homomorphism therefore commute elements abelian group commutes hki thus centralizer precisely kernel generating set computed resp extgcd using theorem conjugacy problem combine previous theorems solve conjugacy problem following thm theorem conjugacy problem let fixed following unary inputs extgcd binary inputs input given quotient presentation elements either myasnikov produce determine element exists case unary inputs returned word binary inputs word binary exponents proof proceed induction abelian conjugate return let assume theorem holds nilpotent group class particular use notation proof theorem first step circuit check conjugacy done induction elements conjugate conjugate overall answer otherwise obtain let canonical homomorphism denotes centralizer define proof theorem image indeed homomorphism claim conjugate indeed exists hvw hence well required converse immediate suffices express possible case conjugator circuit computes generating set using theorem generated coord computed theorem used determine whether theorem applied find finally returned case previous tests succeed since concatenate fixed constant number resp extgcd computations whole computation resp extgcd remark want outline briefly unary case bounds thm used directly solve conjugacy problem nilpotent groups since thm setting fix nilpotent group generating set let words inputs conjugacy problem total length thm length conjugators polynomial using binary exponents conjugators written respect mal cev basis using log bits constant depends fact see thm particular possible conjugators log checked parallel uniform family circuits whether using circuits word problem note purpose written unary since length computing quotient presentations results previous sections always required group given quotient presentation however use theorem transform arbitrary presentation generators group quotient presentation circuits algorithmic problems nilpotent groups proposition let fixed integers following given arbitrary finite presentation generators group list relators given words compute quotient presentation encoded unary explicit isomorphism moreover relators given words binary exponents binary encoded quotient presentation computed extgcd proof let let set relators presented let free nilpotent group class generators let standard mal cev basis let denote set relations consistent nilpotent presentation consider natural surjection let ker normal closure denoting generated iterated commutators total length generators linear since constant using theorem group produce sequence resp extgcd binary inputs lemma consistent quotient presentation remark proposition theorems input quotient presentation also take arbitrary presentation group input however aware word problem theorem corollary complexity changes extgcd binary case power problem conjugacy wreath products nilpotent groups conjugacy problem iterated wreath products abelian shown definition iterated wreath products refer crucial step transfer result conjugacy problem wreath product conjugacy problems power problem power problem defined follows input words generators decide whether power whether yes case compute binary representation finite order computed smallest also power problem power problems given torsion elements uniformly bounded order latter condition also preserved wreath products thus light remains show power problem nilpotent groups order torsion elements uniformly bounded order establish following theorem note fixed groups therefore formulate also following results setting theorem let finitely generated nilpotent groups let conjugacy problem iterated wreath products well proof following two lemmas together repeated application theorem lemma theorem lemma every finitely generated nilpotent group uniform bound order torsion elements myasnikov proof proceed induction along mal cev basis infinite order done induction otherwise let order torsion elements consider torsion element thus hkm therefore upper bound order torsion elements lemma every finitely generated nilpotent group power problem uniform proof show slightly general statement induction along mal cev basis every fixed arithmetic progression power problem restricted given decided whether computed consider input words quotient let quotient remains solve power problem subgroup done induction next distinguish two cases infinite order finite order case infinite order possible value computed theorem integer contained arithmetic progression mod power otherwise one simply checks whether solving word problem bounded input length lemma done theorem case finite order let denote order checked parallel whether mod case answer power problem answer power problem subgroup restricted arithmetic progression intersection since finitely many possibilities fixed group since modulo bounded least common multiple orders finite order elements mal cev basis answer conclusion open problem seen problems shown logspace indeed even uniform setting number generators nilpotency class fixed moreover binary versions extgcd meaning nilpotent groups complicated abelian groups many algorithmic aspects contrasts slightly larger class polycyclic groups word problem still conjugacy problem even known conclude possible generalizations results question uniform version theorem hold uniform word problem still fixed nilpotency class arbitrary number generators happens complexity also nilpotency class part input note case even clear whether word problem still polynomial time question way solve conjugacy problem nilpotent groups binary exponents notice needed compute greatest common divisors solving subgroup membership problem however might way solving conjugacy problem using another method question complexity uniform conjugacy problem nilpotency class fixed circuits algorithmic problems nilpotent groups way proving subgroup membership problem nilpotent groups established extended gcd problem unary inputs outputs however computed solution small one computed logspace algorithm question following problem given unary encoded numbers compute max gcd references blackburn conjugacy nilpotent groups proceedings american mathematical society boone word problem ann dehn unendliche diskontinuierliche gruppen math eick kahrobaei polycyclic groups new platform cryptology arxiv mathematics elberfeld jakoby tantau algorithmic meta theorems circuit classes constant logarithmic depth electronic colloquium computational complexity eccc garreta miasnikov ovchinnikov properties random nilpotent groups arxiv hall edmonton notes nilpotent groups queen mary college mathematics notes mathematics department queen mary college london hesse division uniform orejas spirakis van leeuwen editors icalp volume lecture notes computer science pages springer hesse allender barrington uniform threshold circuits division iterated multiplication comput syst kargapolov merzljakov fundamentals theory groups volume graduate texts mathematics new york translated second russian edition robert burns kargapolov remeslennikov romanovskii roman kov algorithmic questions groups algebra logika lohrey evaluating matrix circuits computing combinatorics volume lecture notes comput pages springer cham lange mckenzie complexity free monoid morphisms chwa ibarra editors algorithms computation international symposium isaac taejon korea december proceedings volume lecture notes computer science pages springer soicher symbolic collection using deep thought lms comput electronic lipton zalcstein word problems solvable logspace acm july macdonald myasnikov nikolaev vassileva logspace compressedword computations nilpotent groups corr majewski havas complexity greatest common divisor computations algorithmic number theory ithaca volume lecture notes comput pages springer berlin mal cev homomorphisms onto finite groups ser math translation uch zap ivanov gos pedagog inst myasnikov miasnikov vassileva conjugacy problem free solvable groups wreath product abelian groups weil editor computer science theory applications international computer science symposium russia csr kazan russia june proceedings volume lecture notes computer science pages springer mostowski computational algorithms deciding problems nilpotent groups fundamenta mathematicae myasnikov nikolaev ushakov post correspondence problem groups group theory myasnikov nikolaev ushakov lattice problems group theory novikov algorithmic unsolvability word problem group theory trudy mat inst steklov pages russian robinson parallel algorithms group word problems phd thesis university california san diego sims computation finitely presented groups volume encyclopedia mathematics applications cambridge university press cambridge vollmer introduction circuit complexity springer berlin
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feb local wealth redistribution promotes cooperation multiagent systems pinheiro fernando santos collective learning group mit media lab massachusetts institute technology ames street cambridge massachusetts flaviopp instituto superior universidade lisboa porto salvo portugal abstract designing mechanisms leverage cooperation agents goal multiagent systems task especially challenging agents selfish lack common goals face social dilemmas situations individual interest conflicts social welfare past works explored mechanisms explain cooperation biological social systems providing important clues aim designing cooperative artificial societies particular several works show cooperation able emerge specific network structures underlie agents interactions notwithstanding social dilemmas defection highly tempting still pose challenges concerning effective sustainability cooperation propose new redistribution mechanism applied structured populations agents importantly show implemented locally agents share fraction wealth surplus nearest neighbors redistribution excels promoting cooperation regimes defection prevailed ccs concepts computing methodologies systems cooperation coordination keywords emergent behaviour social networks social simulation simulation complex systems cooperation introduction explaining cooperation among selfish unrelated individuals central topic evolutionary biology social sciences simultaneously challenge designing cooperative multiagent systems mas long standing goal researchers artificial intelligence thirty years ago already clear intelligent agents inevitably need interact flexibly entities existence conflicting goals proc international conference autonomous agents multiagent systems aamas dastani sukthankar andre koenig eds july stockholm sweden international foundation autonomous agents multiagent systems rights reserved author version work posted personal use redistribution definitive version record published proceedings proc international conference autonomous agents multiagent systems aamas dastani sukthankar andre koenig eds july https need handled automated agents routinely handled cooperative multiagent interactions agents need collaborate towards common goals introduces challenges associated coordination communication teamwork modeling interactions contrast require design indirect incentive schemes motivate selfish agents cooperate sustainable way cooperation often framed altruistic act requires agent pay cost order generate benefit another refusing incur cost associated act defection results benefits generated whenever benefit exceeds cost plays occur simultaneously agents face prisoner dilemma decisionmaking challenge embodies fundamental social dilemma within mas rational agents pursuing expected defect optimal collective outcome requires cooperation defection likely decision rational agents however justify ubiquity cooperation real world evolutionary biology pursued fundamental question searching additional evolutionary mechanisms might help explain emergence cooperative behavior mechanisms allowed develop solutions found applications computer science informing ways incentivizing cooperation networks wireless sensor networks robotics resource allocation distributed work systems name network reciprocity one popular mechanisms explain evolution cooperation social biological systems context populations structured interactions among agents constrained constraints often modelled means complex network interactions applications mechanism explored design mas reach high levels cooperation despite advances cooperation structured populations still hard achieve considering social dilemmas high levels temptation defect additional complementary mechanisms required consider agents contribute percentage surplus defined later divided among beneficiary set agents context aim answering following questions redistribution wealth promote evolution cooperation beneficiary sets selected potential disadvantages mechanism aamas july stockholm sweden using methods evolutionary game theory egt resorting computer simulations explore wealth redistribution impacts evolution cooperation population agents without memory unable recall past interactions rationally bounded lacking full information payoff structure game engaging assume agents resort social learning peer imitation proves predominant adaptation scheme employed humans also consider strategies binary cooperate defect opting focus attention complexity provided heterogeneous populations redistribution mechanism process agents adapting time role larger strategy spaces lies outside scope present work redistribution show cooperation emerges parameter region previously absent moreover show optimal choice redistributing groups consists picking nearest neighbors local redistribution result fits local polycentric view incentive mechanisms mas may easier implement show establish optimal scale interaction terms eliciting cooperation related work problem cooperation broad intrinsically multidisciplinary topic part mas research agenda long time realm evolutionary biology several mechanisms proposed explain evolution cooperation kin selection direct reciprocity indirect reciprocity network reciprocity constitute important mechanisms proposed remarkably mechanisms applied order design mas cooperation emerges example waibel associated kin selection evolutionary robotics griffiths employed indirect reciprocity promote cooperation networks investigated social norms system reputations indirect reciprocity promote cooperation crowdsourcing markets similarly peleteiro combined indirect reciprocity complex networks design mas cooperation able emerge top han applied egt performed study order investigate role punishment commitments multiagent cooperation pairwise group interactions regarding alternative approaches sustain cooperation mas shall underline role electronic institutions whereby agents actions explicitly constrained desirable collective behaviors engineered role population structure network reciprocity context prolific area research shown complex networks able fundamentally change dilemma stake depending particular topology considered applied tools control theory order study role complex networks evolution cooperation importantly role dynamic networks agents able rewire links also shown significantly improve levels cooperation especially networks high average pinheiro fernando santos degree connectivity survey topic complex networks emergence cooperation mas accessed previous works found cooperation structured population substantially decreases temptation defect increases see model proper definition temptation thereby contribute additional mechanism cooperation structured populations consider mechanism redistribution inspired wealth redistribution mechanisms prevail modern systems mainly taxation particularly interested understanding sample redistribution groups effective way context shall underline works salazar system taxes coalitions shown promote cooperation complex networks regular grids excellent job showing coalitions leaded single agent emerge consider model leaders considered taxes redistributed rather centralized single entity focus analysis showing local redistribution sets optimal approach require additional means reciprocity memory leadership punishment knowledge features network cover wide range dilemma strengths explicitly show local redistribution promotes cooperation notwithstanding analysis performed surely provides important insights address future works explicitly model adherence beneficiary sets guarantee stability also assume egalitarian redistribution individual beneficiary set shall note different redistribution heuristics may imply different levels allocation fairness context recent work introduces concept distributed distributed justice shows local interactions may provide reliable basis build trust reputation agents used regulate decentralized way levels justice agents actions way rewarding note local interactions constitute optimal scale form cooperative beneficiary sets show see also provide convenient interaction environment allow justice contributions sustained model three stage redistribution game propose sequential game dynamics made three stages focusing arbitrary agent stages described follows agent participates game prisoner dilemma neighbors interaction obtains payoff interactions agent accumulates total payoff next agent contributes fraction payoff surplus redistributed group benefits agent contribution called beneficiary set finally agent receives share beneficiary set part refer level taxation defines fraction surplus agents contribute threshold level local wealth redistribution promotes cooperation multiagent systems critical level taxation aamas july stockholm sweden homogeneous network temptation parameter figure solutions game wealth redistribution curve indicates critical taxation levels nature social dilemma changes different payoff thresholds function temptation parameter payoff defines surplus definition agents negative payoff contribute might however receive benefits beneficiary sets agent contributes one beneficiary set part agents receive beneficiary set contribute central question work select show decision profound impact overall cooperation levels system single peak hzi fraction nodes fraction nodes heterogeneous network degree degree figure graphical depiction specific structures used work homogeneous networks correspond structure nodes degree heterogeneous networks characterized high variance among degree nodes color node indicates degree blue tones represent lower degree red tones higher degree panel show respectively degree distributions homogeneous heterogeneous networks analysis particular use networks representatives heterogeneous structures degree distribution decays power law prisoner dilemma game general possible outcomes game two agents engage interaction requires decide independently simultaneously whether wish cooperate defect summarized payoff matrix reads payoff obtained playing row strategy facing opponent column strategy represents reward payoff mutual cooperation punishment mutual defection one individuals defects cooperates first receives temptation payoff second obtains sucker payoff manuscript consider agents interact according prisoner dilemma agents said face whenever relationship payoffs scenario rational agents seeking optimize expected always defect however since best aggregated outcome would players cooperating agents said face social dilemma optimizing clashes optimizing social outcome sense mutual cooperation pareto optimal contributes increase average payoff mutual defection egalitarian social welfare unilateral cooperation noteworthy mention situations different optimal rational responses arise parameters take different relationship stag hunt game snowdrift game harmony game deadlock game name notwithstanding far popular metaphor social dilemmas one presents biggest challenge cooperation emerge reasons shall main focus study manuscript simplify parameter space considering game fully determined temptation value sense higher temptation creates stringent conditions emergence cooperation prisoner dilemma wealth redistribution introductory example let start analyzing particular case two interacting agents event case beneficiary sets agent composed opponent redistribution thus analyzed considering slightly modified payoff matrix takes account second third stages resulting payoff matrix becomes aamas july stockholm sweden pinheiro fernando santos hence depending choice given minimum level taxation required observe change nature game faced agents straightforward notice nature game changes prisoner dilemma harmony game relationship moves figure shows different values clearly populations simple scenario mas composed two agents redistribution mechanism simple effect reshaping payoff matrix trivially changing nature dilemma trivial conclusion drawn large populations playing networks show different ways assigning beneficiary sets profound impact ensuing levels cooperation structured populations let consider population agents agents correspond complex network links dictate interacts structure reflects existence constraints limit interactions agents constraints arise spatial communication limitations number interactions agent participates defines degree distribution degrees describes fraction agents degree work consider two structures homogeneous random graphs scalefree networks homogeneous random graphs generated successively randomizing ends pairs links initially regular graph lattice ring resulting structure random interaction structure nodes network degree figure depicts graphically example structures figure corresponding degree distribution networks generated algorithm growth preferential attachment algorithm follows start three fully connected nodes add sequentially remaining nodes time new node added connects nodes selecting preferentially nodes level cooperation temptation parameter heterogeneous level cooperation results critical values homogeneous level taxation level taxation payoff threshold level taxation rationale arrive payoff structure following whenever players choose act way payoff remains contributions taxes benefits receiving contributions opponent cancel defector playing cooperator sees payoff subtracted amount receiving benefit since cooperator negative payoff contribute likewise cooperator exempt contributing receives additional contribution represents amount taxed defector inspect whether wealth redistribution changes nature social dilemma prisoner dilemma another type game inspect whether difference relationship payoffs sums solving single inequality temptation parameter figure level cooperation homogeneous random networks heterogeneous networks plot shows level cooperation different combination taxation level temptation cases fitness threshold fixed blue indicates regions cooperation dominates red delimits regions dominated defectors top bars panel indicate level cooperation absence wealth redistribution function temptation payoff parameter level cooperation computed estimating expected fraction cooperators population reaches stationary state end run independent simulations start cooperators defectors population size intensity selection higher degree used resulting network characterized heterogeneous degree distribution one decays power law majority nodes connections many figure shows graphical example structure figure degree distribution following explore case networks nodes average degree local wealth redistribution promotes cooperation multiagent systems heterogeneous aamas july stockholm sweden level cooperation level cooperation heterogeneous wealth redistribution nearest neighbors random group homogeneous temptation parameter temptation parameter homogeneous nearest neighbors level cooperation level cooperation legend random group temptation parameter temptation parameter figure level cooperation heterogeneous homogeneous populations different values payoff threshold function temptation payoff parameter gray dashed line shows results obtained absence wealth redistribution scheme population size intensity selection figure comparison effects assigning nearest neighbors agent corresponding beneficiary set dark blue line agents assigned random light blue level cooperation domain temptation payoff parameter panel shows results heterogeneous populations panel impact homogeneous populations population size intensity selection simulations make use independently generated networks type games networks study expected level cooperation attained population estimate quantity computer simulations level cooperation corresponds expected fraction cooperators population evolved iterations estimate quantity averaging observed fraction cooperators final simulation independent simulations simulation starts population equal composition cooperators defectors randomly placed along nodes network update round agent plays nearest neighbors agents directly connected accumulated payoff interactions agent participates computed nid number neighbors defect cooperate equal cooperator otherwise accumulated payoff agents contribute pool fraction surplus fitness agent results subtracting accumulated payoff contributions plus share obtains beneficiary sets participates shall underline agents dilemma everyone population heterogeneous populations introduce additional complexity layer implying different agents may vary maximum values accumulated payoff able earn formalized equal one part beneficiary set towards contributes zero otherwise denotes size set evolution frequency strategies adopted population happens process imitation social learning iteration random agent say compares fitness aamas july stockholm sweden pinheiro fernando santos fitness neighbor say depending fitness difference adopts strategy probability heterogeneous exp level cooperation meaning sigmoid function understood follows performing much better updates strategy adopting strategy conversely performing much worse update strategy parameter often called intensity selection akin learning rate dictates sharp transition two regimes approaches zero large means individuals act deterministic way updating strategies minimum difference small means individuals prone make imitation mistakes nearest neighbors legend temptation parameter homogeneous level cooperation results wealth redistribution level cooperation structured populations section start analyzing scenario beneficiary set agent corresponds nearest neighbors hence size beneficiary set also agents interacts obtains payoff figure shows achieved levels cooperation payoff threshold set function temptation payoff level taxation figure shows results homogeneous networks figure heterogeneous find fixed payoff threshold increasing level taxation results increase levels cooperation effect diminishes increase temptation increasing minimum value necessary promote cooperation increases well behavior observed structures however larger degree cooperation heterogeneous networks always level taxation given temptation guarantees level cooperation hence order cooperation evolutionary viable homogeneous networks stringent conditions necessary higher tax levels figure shows level cooperation depends variations fitness threshold intervals keeping fixed level taxation different levels temptation payoff figure shows results obtained heterogeneous networks panel results homogeneous structures constant level taxation decreasing payoff threshold increases range temptation cooperation possibly evolve case types structures however effect limited homogeneous populations figure highlight positive impact local wealth redistribution mechanism enhancement cooperation also puts evidence success mechanism depends volume payoff redistributed ultimately done either increasing level taxation decreasing payoff threshold defines taxable payoff nearest neighbors legend temptation parameter figure panel compares extending beneficiary sets nearest neighbors nodes distance links away impacts level cooperation heterogeneous networks panel shows extended beneficiary sets impact level cooperation homogeneous networks cases extending set beneficiaries negative negative impact levels cooperation population size intensity selection randomized beneficiary set next explore extent results obtained depend way agents assigned beneficiary set end compare two cases nearest set assignment beneficiary set agent corresponds nearest neighbors random set assignment agents assigned random beneficiary set number agents assigned set equal degree contributing agent cases guarantees collected payoffs agent distributed among number individuals figure show results obtained respectively heterogeneous homogeneous networks consider explore domain dark blue curves show results obtained nearest set assignment light blue curves results obtained random set assignment results show ability wealth redistribution mechanism lies redistribution taxed payoff among agents spatially related random assignment agents local wealth redistribution promotes cooperation multiagent systems homogeneous number generations level taxation temptation parameter heterogeneous level taxation number generations temptation parameter figure panel shows fixation times generations homogeneous networks panel shows fixation times generations heterogeneous networks generation corresponds iteration steps fixation times indicate expected time population takes arrive state dominated cooperators defectors starting state equal abundance strategies population size intensity selection drastically decreases levels cooperation obtained networks extent beneficiary sets need constrained spatially extended beneficiary set answer previous question explore case nodes distance links assigned beneficiary set focal agent previous results thereby obtained figure show results heterogeneous homogeneous networks respectively cases see expansion size beneficiary set leads decrease levels cooperation result reinforces aamas july stockholm sweden conclusion wealth redistribution efficient agents return form taxes share accumulated payoffs agents engaged shall underline distance size play role obtained results previous section size kept constant across different treatments thus disambiguating effect size distance resulting cooperation levels cost wealth redistribution figure shows fixation times populations along domain bounded fixation times correspond expected number generations sets potential imitation steps population reach state one strategy present population plots map directly figure allowing compare relative fixation times regions levels cooperation observe evolution cooperation associated increase fixation times increase situations order magnitude higher regions exhibit larger fixation times lie critical boundary divides areas defectors cooperators dominance figure hence promoting cooperation redistributing wealth also requires longer waiting time population reach state full cooperation however setting higher taxation values bare minimum necessary emergence cooperation allows populations reach fixation quicker multiple contribution brackets real world taxes unlikely defined single threshold separates agents contribute reality taxes progressive sense taxation levels increase increasing level income case accumulated payoff section implement similar approach inspect impact increasing number taxation brackets let consider instead single threshold taxation brackets divided threshold levels bracket define effective tax bottom threshold bracket definition corresponds case taxes collected redistribution wealth absent moreover implies existence single bracket individuals would contribute case explore manuscript corresponds case two brackets scenario explored consider case taxation increases linearly increasing brackets let define individuals bracket payoff surplus taxed accumulated payoff falls tax level affects individuals example bracket would characterized following tax levels individuals individuals individuals individuals aamas july stockholm sweden pinheiro fernando santos sum show wealth redistribution embodies effective mechanism significantly helps cooperation evolve works fundamentally changing nature dilemma stake appropriately choosing level taxation payoff threshold possible shift defector dominance cooperator dominance dynamics moreover find heterogeneous populations allow ease redistribution mechanism imposing lower taxation rates lower taxable surplus values compared homogeneously structured populations additionally show first time different assignments beneficiary sets significantly impact ensuing levels cooperation local beneficiary sets agents receive contributions direct neighbors constitute judicious choice compared beneficiary sets formed agents randomly picked population including agents higher distances naturally local wealth redistribution scheme may prove optimal terms achieved cooperation levels also reveal much simpler implement exempting need central redistribution entities minimizing number peers agents need interact shall highlight however promoting cooperation wealth redistribution mechanism bears longer fixation times terms number iterations required achieve overall cooperation assume redistribution mechanism externally imposed agents able opt taxation scheme given mechanism increases overall cooperation average payoff system argument acceptance rational agents formulated based infamous veil ignorance proposed john rawls agents decide kind society would like live without knowing social position agents would way prefer cooperative society redistribution exists provided average payoff maximized notwithstanding future research shall analyze role complex strategies give opportunity agents voluntarily engage proposed redistribution scheme alongside effective mechanisms discourage second order free riding problem free riding contributing redistribution pot expecting others shall examined future works shall also evaluate whether alternative taxation schemes prone efficient one proposed cases evolutionary game theoretic framework one developed constitutes promising toolkit employ varf varp lit payoff threshold level taxation figure relative wealth inequality redistribution step heterogeneous population dominated cooperators different combinations taxation level threshold quantify relative wealth inequality redistribution step ratio variance fitness distribution variance gains across population redistribution variance accumulated payoff distribution arp variance gains redistribution population size intensity selection way use upper level bound parameters condition find variations number taxation brackets marginal impact overall levels cooperation observed compared scenarios studied far wealth inequality finally discuss effect wealth redistribution fitness inequality first important highlight observed levels inequality depend default distribution strategies network degree homogeneous structures every agent adopts strategy either defectors cooperators everyone obtains fitness heterogeneous structures cooperation dominance scenario bounds feasible equality levels given degree distribution population fact agents engage interactions others beneficiary sets different sizes depending particular connectivity agents shall focus scenario compare variance fitness gains redistribution step variance accumulated payoff gains redistribution step order quantify relative inequality apply proposed redistribution mechanism particular use ratio variance fitness variance accumulated payoff metric resulting wealth inequality figure shows higher levels reduce resulting inequality fact increasing payoff threshold limits taxation richer agents increasing level taxation increases flow fitness rich agents beneficiary sets strict case high variance fitness distribution reduced low accumulated payoff distribution conclusion acknowledgments authors acknowledge useful discussions francisco santos jorge pacheco aamena alshamsi thankful media lab consortium financial support acknowledges financial support para tecnologia fct phd scholarship funding grants local wealth redistribution promotes cooperation multiagent systems references airiau sandip sen daniel villatoro emergence conventions social learning autonomous agents systems ian akyildiz weilian yogesh sankarasubramaniam erdal cayirci wireless sensor networks survey computer networks albert statistical mechanics complex networks reviews modern physics josep arcos marc esteva pablo noriega juan carles sierra engineering open environments electronic institutions engineering applications artificial intelligence juan memetic framework describing simulating spatial prisoner dilemma coalition formation proceedings aaai aaai press ulle endriss nicolas maudet welfare engineering multiagent systems international workshop engineering societies agents world springer eithan ephrati jeffrey rosenschein deriving consensus multiagent systems artificial intelligence marc esteva bruno rosell juan josep arcos ameli middleware electronic institutions proceedings third international joint conference autonomous agents multiagent systems ieee computer society michal feldman john chuang overcoming behavior systems acm sigecom exchanges michael genesereth matthew ginsberg jeffrey rosenschein cooperation without communication proceedings aaai aaai press philippe golle kevin ilya mironov mark lillibridge incentives sharing networks electronic commerce springer nathan griffiths tags image scoring robust cooperation proceedings international conference autonomous agents systems international foundation autonomous agents multiagent systems william hamilton genetical evolution social behaviour journal theoretical biology han emergence social punishment cooperation prior commitments proceedings aaai aaai press han moniz pereira luis tom lenaerts centralized personalized commitments influence cooperation group interactions proceedings aaai aaai press zhang jennifer vaughan mihaela van der schaar towards social norm design crowdsourcing markets aaai technical report aaai press hofmann nilanjan chakraborty katia sycara evolution cooperation agent societies critical study proceedings international conference autonomous agents systems international foundation autonomous agents multiagent systems genki ichinose yoshiki satotani hiroki sayama mutation alters fitness cooperation networked evolutionary games arxiv preprint nicholas jennings katia sycara michael wooldridge roadmap agent research development autonomous agents systems david burth kurka jeremy pitt distributed distributive justice systems saso ieee international conference ieee michael macy andreas flache learning dynamics social dilemmas proceedings national academy sciences martin nowak five rules evolution cooperation science martin nowak evolving cooperation journal theoretical biology martin nowak robert may evolutionary games spatial chaos nature martin nowak karl sigmund evolution indirect reciprocity nature hisashi ohtsuki christoph hauert erez lieberman martin nowak simple rule evolution cooperation graphs nature elinor ostrom governing commons cambridge university press liviu panait sean luke cooperative learning state art autonomous agents systems ana peleteiro juan burguillo siang yew chong exploring indirect reciprocity complex networks using coalitions rewiring proceedings international conference autonomous agents systems aamas july stockholm sweden international foundation autonomous agents multiagent systems pinheiro dominik hartmann intermediate levels network heterogeneity provide best evolutionary outcomes scientific reports flavio pinheiro jorge pacheco francisco santos local global dilemmas social networks plos one pinheiro francisco santos jorge pacheco linking individual collective behavior adaptive social networks physical review letters jeremy pitt julia schaumeier didac busquets sam macbeth selforganising resource allocation canons distributive justice systems saso ieee sixth international conference ieee bijan haitham bou ammar daan bloembergen karl tuyls gerhard weiss theory cooperation complex social networks proceedings aaai aaai press john rawls theory justice harvard university press luke rendell robert boyd daniel cownden marquist enquist kimmo eriksson marc feldman laurel fogarty stefano ghirlanda timothy lillicrap kevin laland copy others insights social learning strategies tournament science norman salazar juan josep arcos ana peleteiro juan emerging cooperation complex networks proceedings international conference autonomous agents multiagent systems international foundation autonomous agents multiagent systems francisco santos jorge pacheco networks provide unifying framework emergence cooperation physical review letters francisco santos jorge pacheco tom lenaerts cooperation prevails individuals adjust social ties plos computational biology francisco santos rodrigues jorge pacheco epidemic spreading cooperation dynamics homogeneous networks physical review fernando santos jorge pacheco ana paiva francisco santos structural power evolution collective fairness social networks plos one fernando santos jorge pacheco francisco santos social norms cooperation costly reputation building aaai aaai press sven seuken jie tang david parkes accounting mechanisms distributed work systems aaai aaai press karl sigmund calculus selfishness princeton university press robert trivers evolution reciprocal altruism quarterly review biology vasconcelos francisco santos jorge pacheco cooperation dynamics polycentric climate governance mathematical models methods applied sciences markus waibel dario floreano laurent keller quantitative test hamilton rule evolution altruism plos biology
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generation massive graphs following lfr michael ulrich manuel hung dorothea institute theoretical informatics karlsruhe institute technology fasanengarten karlsruhe germany jun institute computer science frankfurt frankfurt main germany umeyer htran mpenschuck june abstract lfr popular benchmark graph generator used evaluate community detection algorithms present first external memory algorithm able generate massive complex networks following lfr benchmark expensive component generation random graphs prescribed degree sequences divided two steps graphs first materialized deterministically using algorithm randomized main contributions two external memory algorithms two steps also propose alternative sampling scheme using configuration model rewiring steps obtain random simple graph experimental evaluation demonstrate performance implementation able handle graphs billion edges single machine competitive massive parallel distributed algorithm faster internal memory implementation even instances fitting main memory implementation capable generating large graph instances orders magnitude faster original implementation give evidence implementations yield graphs matching properties applying clustering algorithms generated instances similarly analyse evolution graph properties executed networks obtained find alternative approach accelerate sampling process work partially supported dfg grants parts paper published introduction complex networks web graphs social networks usually composed communities also called clusters internally dense externally sparsely connected finding clusters disjoint overlapping common task network analysis large number algorithms trying find meaningful clusters proposed see overview commonly synthetic benchmarks used evaluate compare clustering algorithms since networks unknown communities contain actually detectable structure last years lfr benchmark become standard benchmark experimental studies disjoint overlapping communities emergence massive networks handled main memory single computer new clustering schemes proposed advanced models computation since algorithms typically use hierarchical input representations quality results small benchmarks may generalizable larger instances produce large instances exceeding main memory propose generator external memory model computation follows lfr benchmark distributed ckb benchmark step similar direction however considers overlapping clusters uses different model communities contrast approach direct realization established lfr benchmark supports disjoint overlapping clusters random graphs prescribed degree sequence preliminary experiments identified generation random graphs prescribed degree sequence main issue transferring lfr benchmark setting terms algorithmic complexity runtime lfr benchmark uses fixed degree sequence model fdsm also known algorithm consists generating deterministic graph prescribed degree sequence randomizing graph using random edge switches edge switch two edges chosen uniformly random two endpoints swapped resulting graph still simple details see section edge switch seen transition markov chain markov chain irreducible symmetric aperiodic therefore converges uniform distribution also shown converge polynomial time maximum degree large compared number edges however analytical bound mixing time impractically high even comparably small graphs contains sum degrees power nine experimental results occurrence certain motifs networks suggest steps enough number edges results random connected graphs suggest average maximum path length link load converge swaps recently theoretical arguments experiments showed steps enough faster way realize given degree sequence configuration model problem loops may generated erased configuration model illegal edges deleted however alters graph properties since skewed degree distributions necessary lfr benchmark properly realized context question arises whether edge switches starting configuration model used uniformly sample simple graphs random contribution main contributions first external memory versions lfr benchmark fdsm defining notation introduce lfr benchmark detail focus fdsm describe realization two steps classic fdsm namely generating deterministic graph prescribed degree sequence section randomizing graph using random edge switches section steps form pipeline moving data one algorithm next section describe alternative approach generating uniform random simple graphs using configuration model edge rewiring sections describe algorithms remaining steps external memory lfr benchmark conclude experimental evaluation algorithms demonstrate version fdsm faster existing internal memory implementation scales well large instances compete distributed parallel algorithm compare original lfr implementation show significantly faster producing equivalent networks terms community detection algorithm performance graph properties also investigate mixing time give evidence alternative sampling scheme quickly yields uniform samples number swaps suggested original lfr implementation kept preliminaries notation define graph sequentially numbered nodes edges unless stated differently graphs assumed undirected unweighted called simple contains neither obtain unique representation undirected edge write contrast directed edge ordering shall used algorithmically carry meaning application degree sequence graph iff deg denote integer powerlaw distribution exponent values limits pld let integer random variable drawn pld proportional otherwise statement depending number said hold high probability satisfied probability least constant also refer table appendix contains summary commonly used definitions model use commonly accepted external memory model aggarwal vitter features memory hierarchy fast internal memory may hold data items slow disk unbounded size input output algorithm stored computation possible values measure algorithm performance number required transfers block consecutive items memory levels reading writing contiguous items disk requires scan sorting consecutive items triggers sort realistic values scan sort sorting complexity constitutes lower bound intuitively tasks degrees community sizes memberships sample edges edge edge remove rewire illegal edges community figure left sample node degrees community sizes two powerlaw distributions mixing parameter determines fraction edges assign node sufficiently large communities center sample graphs edges right lastly remove illegal edges respective global graph time forward processing let algorithm performing discrete events time iterations loop produce values reused following events data dependencies modeled using directed acyclic graph every node corresponds event edge indicates value produced required computing solution algorithm traverses topological order simplicity assume already ordered time forward processing tfp technique uses minimum priority queue provide means transport data implied iterate events increasing order receive messages sent claiming removing items priority inductively messages minimal priority amongst items stored event computes result sends every successor inserting priority using suited tfp incurs sort number messages sent lfr benchmark lfr benchmark describes generator random graphs featuring planted community structure powerlaw degree distribution powerlaw community size distribution revised version also introduces weighted directed graphs overlapping communities consider commonly used versions unweighted undirected graphs possibly overlapping communities parameters listed table fully supported revised generator changes original algorithm even initial scenario unweighted undirected graphs communities describe recent approach also used author implementation initially degrees number node community sizes memberships randomly sampled according supplied parameters observe number communities follows endogenously analysis assume nodes members communities implies depending mixing parameter every node features dext interi community edges din edges within communities algorithm assigns every node either communities random requested community sizes number communities per node realized desired internal degree din strictly smaller size community case overlapping communities internal degree evenly split among communities node part computation din splitting several communities use rounding avoid biases maximal community size grows number communities governed parameter pld dmin dmax pld smin smax definition number nodes produced degree distribution nodes typically random nodes belong communities remainder one membership size distribution communities typically mixing parameter fraction neighbors every node shall share community table parameters overlapping lfr typical values follow suggestions illustrated fig lfr benchmark generates graph using ext fdsm degree sequence dext order violate mixing parameter rewiring steps applied global graph replace edges two nodes sharing community analogously network sampled community overlapping case rewiring steps used remove edges exist multiple communities would result duplicate edges final graph realistic parameters graphs fit main memory assume global graph global graph large communities variant fdsm applicable implement using described sections deterministic edges degree sequence section introduce scheme takes positive degree possible outputs graph realizes degrees sequence called graphical matching simple graph exists havel hakimi gave inductive characterizations graphical sequences directly lead graph generator given connect first node degree minimal among nodes vertices emitting edges nodes remove decrement remaining degree every new neighbor yields updated sequence subsequently remove sort keeping track original positions able output correct node indices finally recurse positive entries remain every iteration size reduced least one resulting rounds implementation keep sequence ordered decrementing neighbors degrees internal memory solutions typically employ priority queues optimized integer keys approach incurs sort using since every edge triggers update pending degree least one endpoint propose variant emits stream edges lexicographical order fed streaming algorithm without disk additionally may used time test whether degree sequence graphical drop problematic edges yielding graphical sequence section thus consider internal emphasize storing output necessary application requires time scan number edges produced within pipeline generate monotonic degree sequence first sampling monotonic uniform sequence online based ideas applying inverse sampling technique carrying monotonicity yields required distribution thus additional sorting steps necessary graph variant due node maximal degree picked connected data structure instead maintaining degree every node individually compacts nodes equal degrees group yielding groups since monotonic nodes consecutive ids compaction performed streaming sequence stored doubly linked list group assigns degree nodes vbj vbj algorithm built around following invariants holding begin every iteration groups represent strictly monotonic degrees gaps node ids invariants allow bound memory dfootprint two steps first observe list size describes graph least edges due thus graphs arbitrary filling whole edges even pessimistic assumptions amounts edge list size realistic therefore even worst case whole data structure kept practical scenarios top probabilistic argument applies exist graphs fig lemma gives bound sampled powerlaw distribution refer also section experimental results lemma let degree sequence nodes sampled pld number unique degrees bounded high probability proof consider random variables sampled pld unordered degree sequence fix index due powerlaw distribution likely small degree even degrees realized occurrences would covered claim thus suffices bound number realized degrees larger first show total probability mass small argue asymptotically unaffected rare occurrences riemann zeta function satisfies exploit sum monotonicity bound two integrals order bound number occurrences define boolean indicator variables iff observe modelpbernoulli trials thus expected number high degrees chernoff inequality gives exponentially decreasing bound tail distribution sum thus holds high probability due lemma graph sampled powerlaw distribution computed high probability direct sampling group multinomial distribution beneficial lfr may used omit compaction phase applications single item represented three values two pointers total bytes per item assuming bit integers pointers suffice storing items result least edges storing two bytes per edge would require one petabyte standard tricks exploiting redundancy due used reduce memory footprint figure materialization degree sequence ddk maximizes memory consumption asymptotically node label corresponds vertex degree algorithm works rounds every iteration corresponds recursion step original formulation time extracts vertex smallest available minimal degree extraction achieved incrementing lowest node group decreasing size group becomes empty removed end iteration connect node nodes end let group smallest index connects two cases node connects nodes directly emit edges decrement degrees groups accordingly since degree remains unchanged may match decremented violation resolved merging groups due union contains consecutive ids suffices grow delete group connects number nodes group split two groups sizes respectively connect vertex nodes first fragment hence need decrease degree thus merge analogous may required see fig groups consumed wholly requested degree met input graphical since vast majority nodes low degrees sufficiently large random powerlaw degree sequence contains nodes materialized requested therefore explicitly ensure sampled degree sequence graphical rather correct negligible inconsistencies later ignoring unsatisfiable requests improving current formulation perform constant work per edge already optimal however introduce simple optimization improves constant factors gives accesses also allows test whether graphical time observe groups vicinity split merge call active frontier contrast stable groups gdd keep relative degree differences pending degrees nodes decremented one iteration become neighbors subsequently extracted nodes group eventually becomes active merge candidate thus update stable degrees every round rather maintain single global iteration counter count many iterations group remained stable group becomes stable iteration annotate adding activated iteration updated degree follows degree remains positive since enforces timely activation list uncompressed degree sequence merge split extract merge extract split extract merge extract extract initial situation group group splitting front group group splitting back group split degrees groups decreased group figure left values row correspond state beginning iteration groups visualized directly extraction head vertex number next symbol indicates new degree updates splitting merging takes place right consider two adjacent groups degrees split left right directly triggers merge number groups remains lemma optimized variant requires scan stored external memory list proof list requires scan execute sequence sequential read insertion deletion requests adjacent positions seeking necessary argue scans roughly twice starting simultaneously front back every iteration starts extracting node minimal degree corresponds accessing eventually deleting list first element list head block cached incur deleting head groups yielding scan whole execution true accesses back list minimal degree increases monotonically algorithm execution extracted node connected remaining vertices graphical sequence implies one group remains ignore simple base case asymptotically neglecting splitting merging distance list head active frontier decreases monotonically triggering scan merging described may necessary reactivate stable groups reload group behind active frontier towards end thus keep block containing frontier cached also block behind incur additional since scanning backwards already read reactivation stable groups hence incurs whole block consumed deleted since happen merges take place reactivations may trigger scan total splitting observe doubles size splitting group degree neighbor degree directly triggers another merge fig since split replaces one group two adjacent fragments differ degree exactly one second split one fragments increase size list edge switching central building block pipeline used randomize rewire existing graphs applies sequence edge swaps simple graph typically constant graph represented lexicographically ordered edge list stores pair omits every ordered edge illustrated fig swap encoded direction bit edge ids position edges supposed swapped switched edges denoted given defined fig input false true false true edges positions edge list illegal creates figure swap consists two edge ids direction flag edge ids describe induced subgraph left flag indicates incident nodes shuffled assume swap constituents drawn independently uniformly random thus sequence contain illegal swaps would introduce executed illegal swaps simply skipped order following tasks addressed gather nodes incident edges compute skip arises verify graph remains simple skip edge already exist update graph representation whole graph fits hash set per node storing neighbors used adjacency queries updates expected constant time executed swap expected time however model approach incurs per swap high probability graph constant improve situation split smaller runs swaps batchwise processed note two swaps within run depend edge contained one swap nodes incident edge may change first swap executed call source edge dependency since resulting graph remain simple dependency two swaps target edges executing creates removes edge created model types dependencies explicitly forward information dependent swaps using time forward processing illustrated fig executes several phases run roughly correspond four tasks outlined simplicity sake first assume swaps independent two swaps share source edge target edge explain dependencies handled independent swaps request nodes phase requests swap endpoints two edges positions requests executed load nodes phase combination implement task subsequently step simulate swaps computes corresponds fourth step load existence check target edges whether already exists step perform swaps executes swaps iff graph remains simple corresponds implement state involved edges materialized update graph phase communication different phases mostly realized via external memory independent swaps require communication shown top fig term sorter refers container two modes operation first phase items pushed sorter arbitrary order algorithm explicit switch filled data structure becomes elements provided lexicographically stream rewound time sorter functionally equivalent filling sorting reading back vector restricted access model reduces constant factors implementation runtime markers edges receive updates invalidedge edge state first swap edgemsg request nodes incident edge edgereq basic edge switching dependency handling sorter stream request nodes swap priority queue edge existence request existreq load nodes simulate swaps edge swap inform successor swap idsucc edge state updates successor edge exist info first swap existmsg edges processed swaps edgeupdates load existence perform swaps edge swap inform successor swap existsucc update graph edge edge state existence updates successor figure data flow run communication phases implemented via sorters use tfp brackets within phase represent type elements iterated multiple input streams used joined key request nodes load nodes goal two phases load every referenced edge iterate sequence swaps swap push two messages edge req edge req sorter edgereq message third entry encodes whether request issued first second edge swap information becomes relevant allow dependencies scans parallel edge list requests edgereq sorted edge ids request edge req edge edge node pair sent requesting swap pushing message edge msg sorter edgemsg additionally every edge push bit sequence invalidedge asserted iff edge received request edges considered invalid deleted updating graph since phases produce constant amount data per input element obtain complexity sort scan simulate swaps load existence two phases gather information required decide whether swap legal scans sequence swaps edgemsg parallel swap exactly two messages edge msg edge msg edgemsg information suffices compute switched edges test avoid remains check whether switched edges already exist thus push existence requests exist req exist req sorter existreq contrast request nodes use node pairs rather edge ids well defined afterwards parallel scan edge list existreq performed answer requests edge requested swap found message exist msg pushed sorter existmsg phases hence incur total sort scan perform swaps rewind edgemsg sorter jointly scan sequence swaps sorters edgemsg existmsg described simulation phase computes switched edges original state swap marked illegal switched edge existence info received via existmsg legal push switched edges sorter edgeupdates otherwise propagate unaltered source edges phase requires sort update edge list new edge list obtained merging original edge list updated edges edgeupdates triggering scan process skip edges flagged invalid bit stream invalidedge dependencies contrast earlier swaps may share source ids target edges case produces result sequential processing two swaps containing source edges detected load nodes phase case arrive multiple requests edge record dependencies explicit dependency chain see details simulation phase know yet whether swap executed therefore need consider cases swap executed existing edge prevented execution dynamically forward possible edge states using priority queue load existence phase detect whether several swaps might produce outcome case issue existence request edge simulation explicit dependency chain computed perform swaps phase forwards source edge states existence updates successor swaps using information dependency chains target edge dependencies consider case swap changes state edges respectively later second swap inquires existence either four edges obviously changed compared initial state extend simulation phase order track edge modifications push messages exist req exist req sorter edgereq also report original edges may change achieved using messages exist req may change exist req may change pushed sorter dependencies multiple messages received edge load existence phase case request first swap involved answered also every swap informed direct successor pushing message exist succ sorter existsucc yielding aforementioned dependency chain optimization may change requests end chain discarded since recipient exists perform swaps phase executes steps described earlier swap may receive successor every edge sent existence request informs successor state appropriate edge swap processed source edge dependencies consider two swaps share source edge dependency detected load nodes phase requests edge req edge req arrive edge case answer request build dependency chain described using messages succ pushed sorter idsucc simulation phase decide whether swap legal therefore sends every conflicting edge original state well updated state slot using swap receives multiple edge states per slot simulates swap every possible combination perform swaps phase operates described independent case computes swapped edges determines whether swap skipped successor exists new state pushed edgeupdates sorter rather forwarded successor tfp fashion way every invalidated edge receives exactly one update edgeupdates merging remains correct due second modification complexity increases number swaps target edge number quite low case let random variable expressing number swaps reference edge since every swap constitutes two independent bernoulli trials towards indicator binomially distributed yielding expected chain length input degree sequence shuffled sequence matched edges materialized resulting graph figure configuration model run degree sequence also swaps holds high probability based argument thus bound largest number edge states simulated high probability polylog assuming dependency chains observe converges towards independent poisson distribution large expected state space per edge experiments suggest bound also holds overlapping dependency chains section order keep dependency chains short splits sequence swaps runs equal size experimental results show run size suitable choice every run algorithm executes six phases described time graph updated mapping edge may change switching probabilities however remain unaltered due initial assumption uniformly distributed swaps thus triggers sort total high probability sampling random graphs prescribed degree sequence section propose alternative approach generate graph prescribed degree sequence contrast generates highly biased simple graph use configuration model sample random graph thus resulting graph may contain remove obtain simple graph configuration model let degree sequence nodes configuration model builds multiset node ids thought stubs produces total labelled node algorithm chooses two uniformly random creates edge according labels repeats last step remaining matched implementation procedure requires high probability constant therefore impractical fully external setting illustrated fig similar instead materialize multiset sequence node appears times subsequently sequence shuffled obtain random permutation sort sorting sequence uniform variate drawn finally scan shuffled sequence match pairs adjacent give upper bounds thep number introduced configuration model define hdi mean second moment sequence expected number already studied results stated following two lemmata random permutation obtained scan case affect complexity total pipeline lemma let degree sequence nodes expected number given hdi hdi hdi lemma let degree sequence nodes expected number bounded hdi hdi hdi let degree sequence drawn powerlaw distribution pld fixed bound number illegal edges function since entry independently drawn suffices give bounds expected value second moment underlying distribution general expected value second moment given respectively expressions bound two integrals respectively case ond moment using identity lower bound hdi obtain two following lemmata lemma let drawn pld expected number bounded lemma let drawn pld expected number bounded edge rewiring graphs consequence lemmata graphs generated using configuration model may contain order detect first sort edge list lexicographically types illegal edges detected single scan issue swap randomly selected partner edge similarly group parallel edges generate swaps random partner edges subsequently execute provisioned swaps using variant see process repeated illegal edges removed accelerate endgame double number swaps remaining illegal edge every iteration since employed remove parallel edges based targeted swaps needs process graphs analogous initial formulation forbid swaps introduce even would reduce multiplicity another edge nevertheless requires slight modifications graphs consider case existence inquired several times since sorted initial edge multiplicities counted scanning load existence phase order correctly process dependency chain forward possibly updated multiplicity information successor swaps annotate existence tokens exist msg counters multiplicity edge precisely perform swaps phase swap informed amongst others multiplicities edges incoming existence messages legal send requested edges multiplicities swapped state successor provided existsucc swapped state consists edges multiplicities incremented decremented otherwise forward edges multiplicities unchanged initial state optimization edges removed multiplicity zero omitted community assignment sake simplicity first restrict community assignment nonoverlapping case every node belongs exactly one community consider sequence community sizes sequence degrees let positive task find random surjective assignment every community assigned nodes requested every node becomes member sufficiently large community din ignoring constraint community size without constraint bipartite assignment sampled spirit configuration model section draw permutation nodes uniformly random assign nodes community ease later modifications prefer equivalent iterative formulation exists yet unassigned node draw community probability proportional number remaining free slots assign reduce community probability mass updating repeat construction first scheme unbiased equivalence approaches follows special case lemma implement random selection process efficiently based binary tree community corresponds leaf weight equal number free slots community inner nodes store total weight left subtree order pto draw community sample integer uniformly random tree total weight following tree according yields leaf corresponding community data structure based lazy evaluation dynamic probability distributions enables fully external algorithm sort however since store tree allowing algorithm needs scan triggering scan enforcing constraint community size enforce exploit monotonicity define max din index smallest community node may assigned since therefore monotonic computed online additional scan fully external setting scanning parallel order restrict random sampling communities reduce aforementioned random interval partial sum available computing generalize notation uniformity assignments subject follows lemma given let two nodes constraints let arbitrary community let assignment generated proof without loss generality assume one nodes tightest constraints case execute reach node constraints apply lemma inductively consider bipartite graph partition classes given communities nodes respectively edge corresponds assignment legal since streams single pass oblivious future values case neither become member therefore claim follows trivially consider case let number free slots community beginning round sum time definition assigns node community probability algorithm update number free slots thus iteration holds assigned otherwise number free slots reduced one step remains show claim follows transitivity true definition consider induction step ind hyp assignment overlapping communities overlapping case weight increases account nodes multiple memberships additional input sequence corresponding number memberships node shall din neighbors sample one community per node different ones since number memberships small duplication check repeated sampling easy case change complexity however possible near end execution less free communities memberships requested address issue switching offline strategy last assignments keep communities free slots last vertices legal assignment exists high probability offline strategy proceeds unable find different communities node case randomly picks earlier assignments swapping communities possible fully external setting complexity grows linearly number samples taken thus bounded sort however community memberships obtained lazily may assign node several times community corresponds bipartite assignment graph removed using rewiring technique detailed section merging repairing graphs global edge rewiring global graph materialized without taking community structure account therefore contain edges nodes share community edges removed increase mixing parameter accordance lfr use rewiring steps perform edge swap forbidden edge randomly selected partner since unlikely random swap introduces another illegal edge sufficiently many communities exist probabilistic approach effectively removes forbidden edges apply idea iteratively perform multiple rounds forbidden edges remain community assignment step outputs lexicographically ordered sequence pairs containing community node nodes join multiple communities several pairs exist based annotate every edge communities incident vertices scanning edge list twice sorted source nodes target nodes forbidden edge swap generated drawing random partner edge swap direction subsequently swaps executed using also emits set edges involved suffices restrict scan illegal edges set since edges contained legal complexity round needs sort selecting edges executing swaps number rounds usually small depends community size distribution smaller communities less likely edges inside community edge rewiring case overlapping communities edge generated part multiple clusters similarly section iteratively apply swaps remove parallel edges however selection random partners involved order violate community size distribution edges swap belong community easy achieve considering communities independently need consider whole merged graph detect forbidden edges order annotate edge community merge communities together one graph possibly contains parallel edges scan sorted edges select set parallel edges one candidates rewiring random partner community drawn edges sort edges selected candidates community counting edges per community sample random partners load second scan randomize order assign candidates community execution need considered need know edge exists also often update information accordingly together loaded edges also need store community ids uniquely identify update correct information steps possible external memory exploit fact significantly less communities nodes hence storing information per community internal memory possible assume candidates stored internal memory many candidates would simply consider round avoid expensive step sorting edges community every round using following observation scanning edges keep track many edges community seen far sort edge ids loaded every community keep pointer current position list every community allows load specific edges communities without need sort edges community complexity fully external rewiring requires sort initial step following round variant triggers scan per round number rounds usually small overall runtime spend step insignificant nevertheless described scheme las vegas algorithm exist unlikely instances mitigate issue allow small fraction edges removed detect slow convergence speed endgame also draw additional swaps uniformly random communities contain implementation implemented proposed algorithms based stxxl library providing implementations data structures parallel sorter priority queue among others applied following optimizations message types contain swap flag indicating swap edges targeted encoded single integer using least significant bit swap store flag significantly reduces memory volume yields simpler comparison operator since standard integer comparison already ensures correct lexicographic order instead storing reading sequence swaps several times exploit implementation pipeline structure directly issue edge requests every arriving swap since time edge ids read swap remaining direction flag stored efficient vector uses one bit per flag supports writing reading steps overlapped ongoing run instead storing edge sorted external edge list pair nodes store source node list targets vertex still supports sequential scan merge operations operations need almost halves volume scanning updating edge list execution several runs delay updating edge list combine load nodes phase next run reduces number scans per additional run three two use asynchronous stream adapters tasks streaming sorters generation random numbers adapters run parallel background preprocess buffer portions stream advance hand main thread besides parallel sorting asynchronous pipeline stage current implementation facilitates parallelism generation randomization graphs computed pleasingly parallel experimental results number repetitions per data point different random seeds denoted errorbars correspond unbiased estimation standard deviation lfr perform experiments based two different scenarios lin one setting maximal degrees community sizes scale linearly function parameters chosen dmin dmax smin smax const second setting keep community sizes degrees constant consider communities parameters chosen dmin dmax smin smax networks shown increasing average degrees become larger increasing maximum degree first setting lin increases average consider node member two communities connected nodes one neighbors also appears communities rewired relative frequency number unique elements run size run size run size number samples number edge configuration received swap figure left number distinct elements samples node degrees degree sequence taken pld right overhead induced tracing dependencies fraction swaps function number edge configurations receive simulation phase degree maximum community size means however significant proportion nodes belongs huge communities tightly knit due large number nodes low degree limited growth probably realistic exact parameters depend network model second parameter set const shows example much smaller maximum degrees community sizes chose parameters approximate degree distribution facebook network may consisted million active users reported note however strict powerlaw models unable accurately mimic facebook degree distribution show degree distribution users removing connections users similar one facebook users whole world supporting use one parameter set different graph sizes actual minimum degree facebook network smaller degrees significantly less prevalent power law degree sequence would suggest chose larger value maximum degree larger one reported facebook latter also arbitrarily enforced limit facebook expected average degree degree sequence slightly higher reported world parameters chosen median degree approximately matches worldwide facebook network similar first parameter set chose maximum community size slightly larger maximum degree nodes state size lemma bound internal memory consumption showing sequence numbers randomly sampled pld contains distinct values high probability order support lemma estimate hidden constants samples varying size taken distributions exponents time number unique elements computed averaged runs identical configurations different random seeds results illustrated fig support predictions small constants commonly used exponent find distinct elements sequence length dependencies whenever multiple swaps target edge simulates possible states able retrieve conflicting edges argued number dependencies thus state size remains manageable sequence swaps split sufficiently short runs found edges swaps runs minimize runtime large instances lin indicated fig setting swaps receive additional edge configurations simulation phase less consider four additional states similarly existence requests remain without dependencies runtime runtime original lfr number node number edges figure left runtime sysa original lfr implementation right runtime sysb graph edges average degree executing swaps performance benchmarks runtime measurements conducted following systems sysa inexpensive compute server intel xeon threads ram samsung pro sata ssd sysb commodity hardware intel core cpu threads ram samsung pro sata ssd since edge switching scales linearly number swaps case number runs measurements beyond runtime extrapolated progress verified errors stay within indicated margin using reference measurements without extrapolation performance implementation produces million edges per second sysa least edges include computation input degree sequence compaction step well writing output external memory performance figure presents runtime required sysb process swaps input graph edges average degree reference performance existing internal memory edge swap algorithm based authors implementation included report edge swapping process excluding precomputation achieve comparability removed connectivity tests fixed memory management issues adopted number swaps extended counters edge ids accumulated degrees bit integers order support experiments edges slows factor data structure exceeds available internal memory less observe analogous behavior machines larger ram faster instances edges graphs still fit main memory fdsm applications beyond synthetic graphs instance used real data assess statistical significance observations spirit execute undirected version crawled graph core obtain deleting nodes corresponding uncrawled performing swaps graph nodes edges feasible less sysb consider vertices unrealistically simple degree account nodes original graph nodes mixing mean swaps degree assortativity number triangles number swaps per edge nodes mixing mean swaps number swaps per edge figure left number triangles const right degree assortativity const order factor increased runtime compared plots shifted runtime phase relative execution algorithm incurs additional error along bhuiyan propose distributed edge switching algorithm evaluate compute cluster nodes equipped two intel xeon processors ram authors report perform swaps graph generated preferential attachment process less generate preferential attachment graph using generator matching aforementioned properties carried edge swaps using sysa observe slow machine number comparable cores internal memory performance qualitative comparison section describe alternative graph sampling method instead seeding emes highly biased graph using employ configuration model quickly generate random graph order obtain simple graph carry several runs fashion since scans edge list iteration runs swaps inefficient reason start subsequent markov chain early first identify generate swaps random partners second step introduce additional random swaps run contains least experimental comparison consider runtime yield sufficiently uniform random sample course uniformity hard quantify similarly related studies section estimate mixing times approaches follows starting common seed graph generate ensemble instances applying independent random sequences swaps process regularly export snapshots intermediate instances graph start seed graph apply algorithm carry swaps described snapshot compute number metrics average local clustering coefficient acc number triangles degree investigate distribution measures evolves within ensemble carry increasing number swaps omit results acc since less sensitive compared measures section illustrated fig appendix proxy measures converge within swaps small variance statistically significant change observed compared markov chose number yields execution times similar simple graphs preliminary experiments also included spectral properties extremal eigenvalues matrix closeness centrality fixed nodes measurement expensive compute yield qualitatively similar results decided include larger trials local cluster coeff degree assortativity triangle count covergence steps covergence steps number nodes local cluster coeff degree assortativity triangle count number nodes figure number swaps per edge ensembles graphs const left right converge due computation costs ensemble size reduced large graphs chain operations computed subset ensemble emhh generates biased instances special properties high number triangles correlated node degrees features output nearly match converged ensemble suggests number swaps obtain sufficiently uniform sample reduced due computational costs study carried multiple machines executing several tasks parallel hence absolute running times meaningful rather measure computational costs units time required carry swaps process accounts offset first data point number rounds required obtain simple graph depends degree distribution const fraction edges produced configuration model illegal requires rewiring runs case single swap used per round rewire illegal edge default mode operation rounds suffice number rewiring swaps per illegal edge doubled round larger graphs edges illegal need rewiring runs convergence similar spirit previous section indirectly investigate markov chain mixing time function number nodes generate ensembles compute graph metrics group measure search first snapshot measure mean within interval half standard deviation final values subsequently remains least three phases interpret proxy mixing time depicted fig measure shows systematic increase two orders magnitude considered hence seems plausible increase number swaps performed compared original implementation performance figure reports runtime original lfr implementation function number nodes faster graphs nodes feature approximately edges well domain implementation capable producing graphs edges using time budget original implementation generates graphs two orders magnitude smaller roughly spend global rewiring point less one edge invalid situation algorithm using random may yield alternatively could simply discard remaining invalid edges since constitute insignificant fraction mixing cluster louvain mixing cluster infomap orig networkit orig networkit number nodes number nodes figure adjusted rand measure ground truth disjoint clusters smin smax mixing cluster oslom overlap mixing cluster oslom overlap orig nmi nmi orig number nodes number nodes figure nmi oslom ground truth overlapping clusters per node qualitative comparison designing made sure closely follows lfr benchmark expect produce graphs following distribution original lfr generator order show experimentally achieved goal generated graphs identical parameters using original lfr implementation disjoint clusters also compare implementation part networkit using networkit evaluate results infomap louvain oslom three clustering algorithms compare using adjusted rand measure nmi examine average local clustering coefficient measure percentage closed triangles shows presence locally denser areas expected communities report measures graphs ranging nodes fig present selection results found appendix small differences within range random noise graphs generated emlfr two implementations note due computational costs edges one sample original implementation explains outliers fig similar results also observe performance clustering algorithms drops significantly graph size grows might due less clearly defined community structures since parameters scaled also due limits current clustering algorithms behavior clearly demonstrates necessity able study phenomenon even larger graphs develop algorithms able handle instances outlook conclusion propose first graph generator lfr benchmark fdsm challenging step involved materializes graph based prescribed degree distribution without virtually realistic parameters including generation powerlaw degree sequence writing output disk implementation generates avg local clustering coeff mixing orig networkit number nodes figure average local clustering coefficient disjoint clusters edges per second graphs exceeding main memory perturbs existing graphs edges based edge switches using sort demonstrate faster internal memory implementation even large instances still fitting main memory scales well beyond limited main memory compared distributed approach cluster cpus exhibits one cpu hence poses viable alternative implementation orders magnitude faster original lfr implementation large instances scales well graphs exceeding main memory generated graphs equivalent gave evidence indicating commonly accepted parameters derive length edge switching markov chain remain valid graph sizes approaching external memory domain used accelerate process currently yet fully exploit parallelism offered modern machines dependencies swaps make parallelization challenging preliminary experiments indicate extension possible run split smaller batches parallelized spirit another possibility speedup could use recently proposed curveball sampling algorithm graphs fixed degree sequence studies necessary establish whether really leads faster sampling practice underlying markov chain seems require less steps converge practice step expensive also combination curveball seems possible starting point clustering large graphs exceed main memory using external memory new clustering algorithms needed also evaluation results needs investigated since existing evaluation measures might easily computable external memory acknowledgment thank hannes seiwert mark ortmann valuable discussions references alok aggarwal jeffrey vitter complexity sorting related problems communications acm pages omer angel remco van der hofstad cecilia holmgren limit laws multiple edges configuration model corr url https lars arge buffer tree new technique optimal extended abstract algorithms data structures international workshop wads kingston ontario canada august proceedings pages url http david bader henning meyerhenke peter sanders christian schulz andrea kappes dorothea wagner encyclopedia social network analysis mining chapter benchmarking graph clustering partitioning pages springer new york url http andreas beckmann roman dementiev johannes singler building parallel pipelined external memory algorithm library ieee international symposium parallel distributed processing ipdps rome italy may pages url http jon louis bentley james saxe generating sorted lists random numbers acm trans math url http hasanuzzaman bhuiyan jiangzhuo chen maleq khan madhav marathe fast parallel algorithms achieve target visit rate heterogeneous graphs international conference parallel processing icpp minneapolis usa september pages url http vincent blondel guillaume renaud lambiotte etienne lefebvre fast unfolding communities large networks journal statistical mechanics theory experiment url http nazar buzun anton korshunov valeriy avanesov ilya filonenko ilya kozlov denis turdakov hangkyu kim egolp fast distributed community detection billionnode social networks ieee international conference data mining workshop pages dec corrie jacobien carstens annabell berger giovanni strona curveball new generation sampling algorithms graphs fixed degree sequence arxiv september kyrylo chykhradze anton korshunov nazar buzun roman pastukhov nikolay kuzyurin denis turdakov hangkyu kim distributed generation social graphs overlapping community structure complex networks proceedings workshop complex networks complenet pages springer international url http doi roman dementiev lutz kettner peter sanders stxxl standard template library xxl data sets pract url http roger eggleton derek allan holton simple multigraphic realizations degree sequences proceedings eighth australian conference combinatorial mathematics lecture notes mathematics pages springer url http scott emmons stephen kobourov mike gallant katy analysis network clustering algorithms cluster quality metrics scale plos one july url http alcides viamontes esquivel martin rosvall comparing network covers using mutual information url http santo fortunato community detection graphs physics reports url http santo fortunato darko hric community detection networks user guide physics reports url https christos gkantsidis milena mihail ellen zegura markov chain simulation method generating connected power law random graphs proceedings workshop algorithm engineering experiments alenex pages siam catherine greenhill matteo sfragara switch markov chain sampling irregular graphs digraphs corr url http seifollah hakimi realizability set integers degrees vertices linear graph journal society industrial applied mathematics michael hamann ulrich meyer manuel penschuck dorothea wagner generation massive graphs following lfr benchmark proceedings meeting algorithm engineering experiments alenex pages siam url http steve harenberg gonzalo bello gjeltema stephen ranshous jitendra harlalka ramona seay kanchana padmanabhan nagiza samatova community detection largescale networks survey empirical evaluation wiley interdisciplinary reviews computational statistics url http havel existenci pro matematiky url http lawrence hubert phipps arabie comparing partitions journal classification december url http marcus kaiser mean clustering coefficients role isolated nodes leafs clustering measures networks new journal physics url http panqanamala ramana kumar martin wainwright riccardo zecchina mathematical foundations complex networked information systems politecnico torino italy volume springer andrea lancichinetti santo fortunato benchmarks testing community detection algorithms directed weighted graphs overlapping communities phys rev jul source code available https url http andrea lancichinetti santo fortunato filippo radicchi benchmark graphs testing community detection algorithms phys rev oct url http andrea lancichinetti filippo radicchi ramasco santo fortunato finding statistically significant communities networks plos one april url http jure leskovec jon kleinberg christos faloutsos graphs time densification laws shrinking diameters possible explanations proceedings acm sigkdd international conference knowledge discovery data mining pages acm press url http anil maheshwari norbert zeh survey techniques designing algorithms algorithms memory hierarchies pages url http ulrich meyer manuel penschuck generating massive networks resource constraints proceedings eighteenth workshop algorithm engineering experiments alenex pages url http ulrich meyer peter sanders parallel single source shortest path algorithm algorithms esa annual european symposium venice italy august proceedings pages url http ulrich meyer peter sanders jop sibeyn editors algorithms memory hierarchies advanced lectures dagstuhl research seminar march volume lecture notes computer science springer ron milo nadav kashtan shalev itzkovitz mark newman uri alon uniform generation random graphs prescribed degree sequences eprint arxiv december arxiv rajeev motwani prabhakar raghavan randomized algorithms chapman mark newman networks introduction oxford university press new york usa rasmus pagh basic external memory data structures algorithms memory hierarchies pages springer jaideep ray ali pinar seshadhri yet stop markov chain generating random graphs algorithms models proceedings international workshop waw lecture notes computer science pages springer url https martin rosvall daniel axelsson carl bergstrom map equation european physical journal special topics url http peter sanders random permutations distributed external hierarchical memory inf process url http peter sanders fast priority queues cached memory journal experimental algorithmics jea wolfgang schlauch katharina zweig different flavors randomness comparing random graph models fixed degree sequences social network analysis mining url http christian staudt aleksejs sazonovs henning meyerhenke networkit tool suite complex network analysis network science lemur project web graph nov http johan ugander brian karrer lars backstrom cameron marlow anatomy facebook social graph corr url http fabien viger matthieu latapy fast generation random connected graphs prescribed degrees corr source code available https url http jeffrey scott vitter efficient algorithm sequential random sampling acm trans math url http doi jianping zeng hongfeng study graph partitioning schemes parallel graph community detection parallel computing url http james zhao expand contract sampling graphs given degrees combinatorial families corr url http summary definitions symbol dmin dmax din pld smin smax scan sort description section undirected simple edge implication section number items block transferred section degree nodes lfr benchmark section din degree node section degree sequence graph section degree support section number vertices graph section number edges graph section mixing parameter lfr benchmark ratio neighbors shall communities section number items fitting internal memory section powerlaw distribution exponent interval section size communities lfr benchmark section scan scan complexity section sort sort complexity section table definitions used paper comparing lfr implementations mixing cluster infomap orig networkit orig networkit nmi orig networkit mixing cluster louvain orig networkit mixing cluster louvain orig networkit nmi nmi orig networkit number nodes mixing orig networkit orig networkit mixing orig networkit number nodes orig networkit orig networkit number nodes number nodes mixing number nodes avg local clustering coeff number nodes avg local clustering coeff edges edges mixing number nodes orig networkit number nodes mixing mixing cluster louvain number nodes orig networkit number nodes mixing cluster louvain orig networkit number nodes mixing cluster louvain number nodes orig networkit number nodes mixing cluster louvain number nodes mixing cluster infomap number nodes orig networkit mixing cluster infomap nmi nmi nmi number nodes mixing cluster infomap edges orig networkit number nodes avg local clustering coeff mixing cluster infomap mixing cluster infomap mixing orig networkit number nodes comparison original lfr implementation networkit implementation solution values dmin dmax smin smax clustering performed using infomap louvain compared emitted generator using adjustedrandmeasure normalized mutual information nmi due computational costs graphs reduced multiplicity case original implementation may based single run accounts outliers mixing cluster oslom overlap mixing cluster oslom overlap orig orig orig orig mixing orig orig number nodes orig avg local clustering coeff number nodes mixing orig number nodes number nodes mixing mixing cluster oslom overlap number nodes nmi nmi orig mixing degree assortativity overlap orig number nodes orig mixing cluster oslom overlap orig orig number nodes nmi mixing cluster oslom overlap avg local clustering coeff number nodes mixing number nodes number nodes degree assortativity mixing degree assortativity overlap degree assortativity degree assortativity orig number nodes mixing degree assortativity overlap number nodes avg local clustering coeff avg local clustering coeff avg local clustering coeff orig number nodes orig number nodes mixing mixing cluster oslom overlap orig number nodes nmi nmi nmi mixing degree assortativity overlap mixing cluster oslom overlap orig number nodes mixing cluster oslom overlap orig number nodes orig number nodes mixing mixing degree assortativity overlap degree assortativity degree assortativity number nodes orig number nodes mixing degree assortativity overlap mixing number nodes avg local clustering coeff mixing number nodes avg local clustering coeff avg local clustering coeff number nodes degree assortativity avg local clustering coeff nmi orig nmi nmi mixing cluster oslom overlap orig mixing orig number nodes number nodes mixing degree assortativity overlap orig number nodes mixing degree assortativity overlap degree assortativity orig degree assortativity degree assortativity mixing degree assortativity overlap orig number nodes comparison original lfr implementation solution values values dmin dmax smin smax clustering performed using oslom compared groundtruth emitted generator using generalized normalized mutual information nmi comparing mean swaps number swaps per edge nodes mixing mean swaps nodes mixing number swaps per edge number swaps per edge nodes mixing mean swaps mean swaps number swaps per edge mean swaps nodes mixing mean swaps nodes mixing number swaps per edge number swaps per edge nodes mixing mean swaps number triangles number triangles degree assortativity degree assortativity number triangles nodes mixing degree assortativity degree assortativity number triangles number swaps per edge nodes mixing mean swaps number swaps per edge triangle count degree assortativity graph ensemble obtained applying random configuration model common seed graph refer section experimental details
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apr diversity optimization problem instance classification wanru gao optimisation logistics school computer science university adelaide adelaide australia samadhi nallaperuma department computer science university sheffield sheffield frank neumann optimisation logistics school computer science university adelaide adelaide australia april abstract understanding behaviour heuristic search methods challenge even holds simple local search methods traveling salesperson problem paper present general framework able construct diverse set instances hard easy given search heuristic diverse set obtained using evolutionary algorithm constructing hard easy instances diverse respect different features underlying problem examining constructed instance sets show many combinations two three features give good classification tsp instances terms whether hard solved introduction heuristic search methods local search simulated annealing evolutionary algorithms ant colony optimization shown successful various combinatorial optimization problems although usually come performance guarantees runtime approximation behaviour often perform well several situations understanding conditions optimization algorithms perform well essential automatic algorithm selection configuration effective algorithm design artificial intelligence operational research communities topic become major point interest analysis heuristic search algorithms become important part understanding type algorithms approach characterizes algorithms performance given problem based features problem instances thereby provides important tool bridging gap pure experimental investigations mathematical methods analysing performance search algorithms current methods analysis based constructing hard easy instances investigated search heuristic given optimization problem evolving instances using evolutionary algorithm evolutionary algorithm constructs problem instances examined algorithm either shows bad good approximation behaviour requires large small computational effort come good optimal solutions although evolutionary algorithm constructing instances usually run several times obtain large set hard easy instances question arises whether results terms features instances obtained give good characterization problem difficulty paper propose new approach constructing hard easy instances following recent work using evolutionary algorithms generating diverse sets instances high quality introduce evolutionary algorithm maximizes diversity obtained instances terms given feature approach allows generate set instances much diverse respect problem feature hand carrying process several features considered problem algorithm gives much better classification instances according difficulty solved considered algorithm show benefit approach compared previous methods consider classical algorithm tsp previous analyses already considered hard easy instances terms approximation ratio analyzed features hard easy instances obtained evolutionary algorithm experimental results new approach show diversity optimization features results improved coverage feature space classical instance generation methods particular results show combinations two features possible classify hard easy instances two clusters wider coverage feature space compared classical methods moreover combinations improve classification hard easy instances feature combinations furthermore classification model built using diverse instances classify tsp instances based hardness remainder paper organized follows firstly introduce euclidean tsp background feature based analysis afterwards state diversity optimization approach evolving instances according feature values report impact diversity optimization terms range feature values features value diverse easy hard instances consider combinations several features instance classification afterwards build classification model classify instances based hardness finally finish conclusions background consider classical euclidean traveling salesperson problem tsp example problem evolving hard easy instances diverse set features methodology applied optimization problem using tsp study advantage already investigated extensively different perspectives including area analysis input problem given set cities euclidean plane euclidean distances cities goal find hamiltonian cycle whose sum distances minimal candidate solution tsp often represented permutation cities goal find permutation minimizes tour length given investigations cities always normalized plane city interval following tsp instance always consists set points euclidean distances local search heuristics shown successful dealing tsp prominent local search operator operator resulting local search algorithm starts random permutation cities repeatedly checks whether removing two edges reconnecting two resulting paths two edges leads shorter tour improvement found carrying operation tour called locally optimal algorithm terminates key factor area analysis identify problem features contribution problem hardness particular algorithm problem combination achieved investigating hard easy instances problem using evolutionary algorithm possible evolve sets hard easy instances maximizing minimizing fitness tour length case tsp instance however none approaches considered diversity instances explicitly within study expect improve evolutionary algorithm based instance generation approach introducing diversity optimization structural features dependent underlying problem features groups used provide understanding algorithm performance tsp different feature classes established distance features mode features cluster features centroid features mst features angle features convex hull features feature values regarded indicators allow predict performance given algorithm given instance diversity optimization section introduce approach evolving diverse set easy hard instances diverse respect important problem features previous algorithm initialize population tsp instances approximation ratio least let produce offspring mutation add remove individual arg uniformly random repeat step termination criterion reached studies measure hardness given instance ratio solution quality obtained considered algorithm value optimal solution approximation ratio algorithm given instance defined value solution produced algorithm given instance opt value optimal solution instance within study tour length obtained given tsp instance opt optimal tour length obtain experiments using exact tsp solver concorde propose use evolutionary algorithm construct sets instances tsp quantified either easy hard terms approximation diverse respect underlying features produced problem instances evolutionary algorithm shown algorithm evolves instances diverse respect given features meet given approximation ratio thresholds algorithm initialized population consisting tsp instances approximation ratio least case generating diverse set hard instances case easy instances start population instances approximation ratio instances approximation ratio accepted next iteration iteration offspring produced selecting parents applying mutation selected individuals offsprings meet approximation threshold rejected immediately new parent population formed reducing set consisting parents offsprings satisfying approximation threshold set solutions achieved done removing instances one one based contribution diversity according considered feature core algorithm selection among individuals meeting threshold values approximation quality according feature values let elements features values furthermore assume feature values upper bounded assume holds diversity contribution instance population instances defined contribution based individuals population let individual set assigns diversity contribution individual based next smaller next larger feature values set individual otherwise implies individual feature value equal instances population gains furthermore individual unique smallest largest feature value always stays population working features tsp instances characterizing easy hard tsp instances studied consider features coming different feature classes shown well suited classification prediction features angle mean mean value angles made point two nearest neighbor points centroid mean distance centroid mean value distances points centroid chull area area covered convex hull cluster mean distance centroid mean value distances cluster centroids levels reachability mst depth mean mean depth minimum spanning tree nnds mean mean distance nearest neighbours mst dists mean mean distance minimum spanning tree refer reader detailed explanation feature carry diversity optimization approach features use evolutionary algorithm evolve feature diverse population instances meets approximation criteria instances given approximation ratio thresholds programs experiments written run environment use functions tspmeta package compute feature values setting evolutionary algorithm diversity optimization used experiments follows use parent offspring population size respectively algorithm executed instance five times different initial solutions set average tour length obtained examined instance sizes denoted number cities one instance based previous investigations initial experimental investigations set instances size instances size evolving hard instances use instances size respectively mutation operator picks step one city figure left boxplots centroid mean distance centroid feature values population consisting different hard easy tsp instances different number cities without diversity mechnism right boxplots cluster distance distance centroid feature values population consisting different hard easy tsp instances different number cities without diversity mechnism easy hard instances conventional approach diversity optimization indicated respectively given parent uniformly random changes choosing offset according standard deviation coordinates interval reset value parent based initial experiments use two mutation operators different values use probability probability mutation step evolutionary algorithm terminates generations allows obtain good diversity considered features features set easy hard instances generated results independent runs range feature values first evaluate diversity optimization approach terms diversity obtained respect single feature focusing single feature run provides insight possible range certain feature value hard easy instances previous study suggests differences possible range feature values easy hard instances study effect diversity optimization range features comparing instances generated diversity optimization instances generated conventional approach evolving hard instances based conventional evolutionary algorithm obtained instances mean approximation ratios easy instances mean approximation ratios figure left presents variation mean distance distances points centroid feature centroid mean distance centroid hard easy instances three considered sizes set consists instances generated independent runs shown figure left hard stances higher feature values easy instances instance sizes example instance size hard instances median value indicated red line easy instances respective range feature value hard instances easy instances instances generated diversity optimization easy hard instances indicated respectively difference median feature values hard easy instances similar instances generated conventional approach additionally range feature values hard easy instances significantly increased example instance size median value easy instances range hard instances size median range see figure left similarly figure right presents variation cluster distance centroid cluster distance centroid feature hard easy instances generated conventional approach indicated hard easy instances generated diversity optimization indicated general observations box plots quite similar observations mst dist mean shown figure left easy instances size range feature value conventional instances instances generated diversity optimization similarly hard instances range feature values increased diversity optimization approach shown figure right significant increase range instance sizes well improved ranges feature values observed considered features however due space limitations included paper results suggest diversity optimization approach resulted significant increase coverage feature space threshold approximation ratios method guarantees hardness instances approximation thresholds extreme mean approximation values obtained conventional method furthermore starting initial population duplicated instances hard coded threshold modified able achieve hard instances approximation ratio respectively instance size majority instances clustered small region feature space points dispersed across whole space evident median values similar values instances respect conventional approach significantly larger range feature value conventional approach failed explore certain regions feature space missed instances existing regions able discover instances spread whole feature space approach provides strong basis effective feature based prediction result increased ranges similar gap median feature values hard easy instances compared conventional instances strong overlap ranges features easy hard instances generated diversity optimization observed results mst dist mean cluster distance centroid shown figure similar pattern holds features well prevents good classification problem instances based single feature value figure plots feature combinations provide separation easy hard instances blue dots orange dots represent hard easy instances respectively classification based multiple features single feature capable clearly classifying instances combinations two three different features examined following analysis mainly focuses combinations previously introduced features diversity maximization single feature value firstly represent instances according combination two different features feature value space see figure example according observation discussion two features distance max angle mean considered together provide accurate classification hard easy instances whereas increasing diversity seven different feature values wider coverage space achieved separation easy hard instances obvious clusters dots representing hard easy instances overlapping shown left graphs figure large overlapping areas lying two groups instances another example separation given combination mst dists mean chull area measure mean distance minimum spanning tree area convex hull however number cities instance increases overlapping area becomes larger hard classification based figure plots feature combinations provide clear separation easy hard instances blue dots orange dots represent hard easy instances respectively examining different combinations two features seven features found combinations two features provide fair separation hard easy instances increasing diversity different feature values shown figure taking mst dists mean chull area features consideration separations spotted hard easy instances however combinations able give clear classification hard easy instances example figure neither combination features nnds mean centroid mean distance centroid features mst depth mean chull area shows clear classification instances different hardness moreover along instance size increment overlapping area dots standing hard easy instances grows since majority combinations capable classifying easy hard instances idea combining three different feature put forward analysis combination values three selected features plotted space considering third feature combination different combinations clear separations two groups instances good selection features results accurate classification instances combinations features measuring statistics minimum spanning tree always provide good separation hard easy instances shown figure figure although overlapping area two clusters hard easy instances spot areas dots instances certain hardness taken another feature value consideration combination able provide good separation give clear classification hard easy instances example illustrating included figure together additional feature mst dists mean combination features mst depth mean chull area shows clear separation easy hard instances comparing results shown left graph figure investigation combination combination found range feature values larger tsp instances smaller good combinations classifying hardness smaller instances may work larger instances centroid features performs well combining another feature classifying hardness instances cities show clear separation instance size study however exist combinations give good classification easy hard instances without regarding instance size example mst dists mean chull area nnds mean mst dists mean chull area mst depth mean diversity maximization multiple feature values order examine relationship feature combination hardness instances weighted population diversity based multiple features introduced weighted population diversity certain set features defined weighted sum normalised population diversity features contribution instance weighted population diversity defined denotes normalised contribution population diversity certain feature represents weight feature contribution individual population diversity certain feature normalised based maximum population diversity feature order reduce bias among different features weighted population diversity used algorithm gain insight relationship features combination instance quality parent offspring population sizes used experiments instance sizes examined still algorithm executed five times obtain approximation quality experiments execute generation previous since shown section combination three features able provide good separation hard easy instances good combinations chosen exploration weight distributions considered experiments hardness thresholds used experiments previous seven independent runs figure plots combining experiment results maximizing diversity features mst dists mean nnds mean chull area provides good separation easy hard instances hard easy instances represented blue dots orange dots respectively figure plots combining experiment results maximizing diversity features mst dists mean chull area centroid mean distance centroid provides good separation easy hard instances legend figure figure plots combining experiment results maximizing diversity features mst dists mean chull area mst depth mean provides good separation easy hard instances legend figure figure plots combining experiment results maximizing diversity features mst dists mean nnds mean chull area considering weighting provides good separation easy hard instances legend figure easy hard instances final solution sets put together therefore results set instances instance size hardness type previous experiments results plotted space compared previous experiments single feature discussed section weighted population diversity offers way examine overlapping area hard easy instances weighting technique takes consideration relationship different features examined since features independent others weighted population diversity considers multiple features time predictable weighted population diversity extreme value single feature may reach example shown figure focusing maximizing weighted population diversity combination features mst dists mean nnds mean chull area shown good combination separating hard easy instances comparison figure figure see although results maximizing weighted population diversity cover wider search space provides detailed insight intersection hard easy instances plots different instance sizes show combination three certain features provide clear separation hard easy instances overlapping areas search space clear combination features provide hints predicting hard easy instances instances classification using support vector machine support vector machines svms supervised learning models machine learning used classification regression outliers detection order quantify separation instances different hardness based feature values svm models constructed combination features linear svm linear classifier first model tried classifying dataset svm linear classifiers separate data maximum margin termed optimal separating plots figure clear none datasets linearly separable taken maximizing margin minimizing number misclassified data points consideration svm used classification let accn training accuracy feature combination separating hard easy instances size define accn ratio number instances correctly classified model total number instances dataset classification experiments done library training data svm models population instances generated section training accuracy regarded quantified measurement separation hard easy instances feature combinations used classification combinations combinations discussed section experiment results combinations lie range average accuracy combination lie average value case instances city number combination results lying range average combinations average accuracy equal larger instance size range average combination whereas combination lie scope average although combinations show better accuracy separation hard easy instances combinations significant difference acc combinations combinations moreover general low accuracy implies high possibility linear models suitable separating hard easy instances based feature combinations move applying kernel function mapping feature combination nonlinear classification rbf kernel linearly features become linearly separable mapped higher dimension feature space radial basis function rbf kernel one kernel function used svm classification two parameters need selected applying rbf cost parameter setting rbf crucial since increasing leads accurate separation training data time causes svms generated quantifying separation rate hard easy instances rather classifying instances initial trials set tests avoid parameter setting may best parameters certain feature combination svm classifying helps gain understanding separation hard easy instances generated previous experiments based condition table show accuracy different two features three features combination hard easy instances separation rbf kernel svm certain parameter setting generate model separating dataset average accuracy feature space instance size respectively whereas three features svm parameter setting provides separation average accuracy instance size respectively results concluded better separations hard easy instances feature space feature angle mean angle mean angle mean angle mean angle mean angle mean centroid mean distance centroid centroid mean distance centroid centroid mean distance centroid centroid mean distance centroid centroid mean distance centroid chull area chull area chull area chull area cluster mean distance centroid cluster mean distance centroid cluster mean distance centroid mst depth mean mst depth mean nnds mean feature centroid mean distance centroid chull area cluster mean distance centroid mst depth mean nnds mean mst dists mean chull area cluster mean distance centroid mst depth mean nnds mean mst dists mean cluster mean distance centroid mst depth mean nnds mean mst dists mean mst depth mean nnds mean mst dists mean nnds mean mst dists mean mst dists mean table table lists accuracy svm rbf kernel separating hard easy instances different space feature angle mean angle mean angle mean angle mean angle mean angle mean angle mean angle mean angle mean angle mean angle mean angle mean angle mean angle mean angle mean centroid mean distance centroid centroid mean distance centroid centroid mean distance centroid centroid mean distance centroid centroid mean distance centroid centroid mean distance centroid centroid mean distance centroid centroid mean distance centroid centroid mean distance centroid centroid mean distance centroid chull area chull area chull area chull area chull area chull area cluster mean distance centroid cluster mean distance centroid cluster mean distance centroid mst depth mean feature centroid mean distance centroid centroid mean distance centroid centroid mean distance centroid centroid mean distance centroid centroid mean distance centroid chull area chull area chull area chull area cluster mean distance centroid cluster mean distance centroid cluster mean distance centroid mst depth mean mst depth mean nnds mean chull area chull area chull area chull area cluster mean distance centroid cluster mean distance centroid cluster mean distance centroid mst depth mean mst depth mean nnds mean cluster mean distance centroid cluster mean distance centroid cluster mean distance centroid mst depth mean mst depth mean nnds mean mst depth mean mst depth mean nnds mean nnds mean feature chull area cluster mean distance centroid mst depth mean nnds mean mst dists mean cluster mean distance centroid mst depth mean nnds mean mst dists mean mst depth mean nnds mean mst dists mean nnds mean mst dists mean mst dists mean cluster mean distance centroid mst depth mean nnds mean mst dists mean mst depth mean nnds mean mst dists mean nnds mean mst dists mean mst dists mean mst depth mean nnds mean mst dists mean nnds mean mst dists mean mst dists mean nnds mean mst dists mean mst dists mean mst dists mean table table lists accuracy svm rbf kernel separating hard easy instances different space conclusions paper introduced new methodology evolving instances diverse respect feature sets optimization problem hand using diversity optimization approach shown easy hard instances obtained approach covers much wider range feature space previous methods diversity optimization approach provides instances diverse respect investigated features proposed population diversity measurements provide good evaluation diverse single multiple feature values experimental investigations tsp shown large set diverse instances classified quite well easy hard instances considering suitable combination multiple features provide guidance predication next step particular svm classification model built diverse instances classify tsp instances based problem hardness provides strong basis future performance prediction models lead automatic algorithm selection configuration building models would require experimentation determine minimal set strong features predict performance accurately references applegate cook dash rohe solution vehicle routing problem informs journal computing apr bonet koenig editors proceedings aaai conference artificial intelligence january austin texas usa aaai press cortes vapnik networks machine learning croes method solving problems operations research eggensperger hutter hoos efficient benchmarking hyperparameter optimizers via surrogates bonet koenig pages englert worst case probabilistic analysis algorithm tsp algorithmica feurer springenberg hutter initializing bayesian hyperparameter optimization via bonet koenig pages gunn support vector machines classification regression hutter hoos algorithm runtime prediction methods evaluation artif neumann witt theoretical analysis two aco approaches traveling salesman problem swarm intelligence mersmann bischl trautmann wagner bossek neumann novel approach characterize algorithm performance traveling salesperson problem annals mathematics artificial intelligence meyer dimitriadou hornik weingessel leisch misc functions department statistics probability theory group formerly wien package version nallaperuma wagner neumann parameter prediction based features evolved instances ant colony optimization traveling salesperson problem ppsn xiii international conference ljubljana slovenia september proceedings pages nallaperuma wagner neumann bischl mersmann trautmann comparison local search christofides algorithm travelling salesperson problem foga pages neumann witt bioinspired computation combinatorial optimization algorithms computational complexity new york new york usa edition core team language environment statistical computing foundation statistical computing vienna austria lopes measuring instance difficulty combinatorial optimization problems computers van hemert lim understanding tsp difficulty learning evolved instances international conference learning intelligent optimization lion lion pages springer ulrich bader thiele defining optimizing diversity measures multiobjective search schaefer cotta kolodziej rudolph editors ppsn volume lecture notes computer science pages springer ulrich bader zitzler integrating decision space diversity multiobjective search pelikan branke editors gecco pages acm van hemert evolving combinatorial problem instances difficult solve evolutionary computation vilalta drissi perspective view survey artificial intelligence review hutter hoos satzilla algorithm selection sat journal artificial intelligence research june
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variants plane diameter petr clment dimitrios sep abstract plane diameter completion problem asks given plane graph positive integer spanning subgraph plane graph diameter examine two variants problem input comes another parameter first variant called bpdc upper bounds total number edges added second called bfpdc upper bounds number additional edges per face prove problems first even graphs second even graphs paper give parameterized algorithms problems run log steps introduction chung problem introduced following problem find optimum way add edges given graph resulting graph minimum diameter notice problems defined paper directly assume simple graph loops contribute diameter graph holds take simple edges instead multiple ones problem proved aim obtain graph diameter later shown even completion problem also know completion parameterized planar graphs dejter fellows introduced planar diameter completion problem asks whether possible obtain planar graph first author supported european research council european union seventh framework programme erc grant agreement second author supported project countgraph ref collateral project rancongraph ref berlin mathematical school research third author european union european social fund esf greek national funds operational program education lifelong learning national strategic reference framework nsrf research funding program aristeia emails requile sedthilk department informatics university bergen bergen norway freie berlin institut mathematik und informatik berlin germany algco project team cnrs lirmm france department mathematics university athens athens greece computer technology institute press diophantus patras greece diameter given planar graph edge additions known whether planar diameter completion admits polynomial time algorithm dejter fellows showed parameterized planar diameter completion fixed parameter tractable proof based fact problem closed taking minors robertson seymour theorem algorithm implies set graphs characterized finite set forbidden minors fact along algorithm implies exists algorithm deciding whether plane graph plane completion diameter using parameterized complexity means planar diameter completion fpt parameterized make result constructive one requires set forbidden minors unknown find constructive parameterized problem remains major open problem parameterized algorithm design results denote sphere plane graph mean simple planar graph vertex set edge set drawn two edges embedding intersect plane graph plane completion simply completion another plane graph spanning subgraph completion plane graph completion completion plane graph completion edges added face paper consider variants plane diameter completion problem plane diameter completion pdc input plane graph output completion diameter notice important difference pdc aforementioned problems consider plane graphs aim reduce diameter given embedding planar graph preserving embedding particular interested following variants bounded budget pdc bpdc input plane graph question completion diameter also completion bounded pdc bfpdc input plane graph question completion diameter also completion examine complexity two problems hardness results following theorem bpdc bfpdc moreover bpdc even graphs bfpdc even graphs hardness results proved section using series reductions departing planar problem shown lichtenstein results theorem prompt examine parameterized problems consider following general problem bounded budget bdc bbfpdc input plane graph question completion diameter also completion completion notice bbfpdc yields bfpdc bbfpdc yields bpdc main result bbfpdc fixed parameter tractable belongs parameterized class fpt parameterized log theorem possible construct bbfpdc algorithm statement rest paper use function otherwise main ideas algorithm theorem following first observe pdc variants bounded branchwidth definition branchwidth see section typical approach case derive either expressing problem monadic second order logic msol using courcelle theorem design dynamic programming algorithm problem however completion problems really plausible logic quantify existing edges vertices graph completion edges also indicates design dynamic programming algorithm problems general easy task paper show tackle problem bbfpdc special cases bpdc bfpdc approach deal input part complicated graph additional edges namely cylindrical enhancement see section definition informally sufficiently large cylindrical grids placed inside faces internally vertex disjoint paths grids used emulate edges solution original problem placed inside corresponding faces thus enhancement reduce bbfpdc new problem certified suitable additional edges roughly partition consists edges added completion edges link edges boundary face parameterized complexity refer reader inserted edges useless rest additional edges new problem asks partition simulates bounded diameter completion good news long number edges per face added bounded case bbfpdc new graph still bounded branchwidth possible new instance quantify graph however even circumstances express new problem monadic second order logic easy reasons decided follow technical approach designing dynamic programming algorithm leads better complexity bounds theorem algorithm quite involved due technicalities translation bbfpdc new problem runs decomposition plane embedding tables encode partial solution behaving inside closed disk whose boundary meets edges stress encoding takes account topological embedding combinatorial structure decompositions well necessary combinatorial structures encoding presented section dynamic programming algorithms presented section technical part paper definitions preliminaries given graph denote respectively set vertices respectively edges graph subgraph graph denote also case say spanning subgraph set vertices set edges graph graph graph obtained removal elements set edges define graph whose vertex set consists endpoints edges whose edge set distance diameter let graph let weighting edges given two vertices call every path endpoints also define min max connected infinite graph unweighted use distg diam instead plane graphs simplify notations plane graphs consider plane graph union points embedding corresponding vertices edges way subgraph seen graph faces plane graph connected components set vertex edge resp plane graph incident face incident resp lies boundary two faces adjacent common incident edges denote set faces degree face number edges incident bridges count double number maximum degree face given face define graph whose set points boundary whose vertices vertices incident set open disc homeomorphic also closed disk closure open disk branch decomposition given graph vertices branch decomposition pair tree internal vertices degree three bijection set leaves edges every edge define middle set mid follows two connected components let hie set mid width maximum order middle sets edges max branchwidth minimum width branch decomposition denoted grid annulus graph obtained cartesian product cycle vertices path vertices need following result proposition let planar graph integers either minor isomorphic branch decomposition width central feature pdc problem variants bounded branchwidth lemma exists constant pdc holds graphs bpdc bfpdc bbfpdc proof examine case pdc bpdc bfpdc bbfpdc also pdc notice first completion diameter also completion diameter notice also every completion grid annulus diameter therefore pdc contain minor proposition branchwidth bounded linear function lemma follows graph minor graph obtained applying edge contractions subgraph reduction cylindrical enhancements let positive integers define graph annulus cartesian product path vertices cycle vertices notice uniquely embeddable homeomorphism plane exactly two faces faces incident edges incident vertices degree call one faces interior exterior call vertices incident interior exterior base roof given edge base define ceilings set edges contains whose dual edges form minimum length path duals interior exterior face cylindrical enhancement plane graph let plane graph next give definition graph let let connected components cji denote number edges agreeing number bridge edges count twice cji consists one vertex add copy embedding cji contained interior contained exterior figure edges colored red add cji edges around disks figure base number connected components cji trivial case cji consists one vertex add edges way resulting embedding remains plane set consecutive vertices base cji connected vertex cji observe one way add edges fulfill restrictions notice set always induces path resulting graph except case cji consists single vertex induces cycle later case pick maximal path cycle denote example figure bold paths innermost cycle apply enhancement connected component boundary face denote resulting graph call face one connected components notice cji roof previously added let consecutive vertices roof add inside copy base subset let partition roof parts consisting consecutive base vertices example figure annulus one edges middle figure base innermost cycle add edges depicted interconnecting edges figure connecting vertex vertex way resulting figure example cylindrical enhancement inside face graph connected components boundaries disks embedding remains plane unique way done apply enhancement face denote resulting graph notice uniquely defined definition depends choice sets always consider arbitrary choice call cylindrical enhancement construction directly follows say edge expansion edge edge also denote graph created contract expansion edges ceilings added construction drawings let connected plane graph denote graph obtained draw together dual dual edges intersecting single point introduce vertex intersection points recursively define every next proposition direct consequence lemma proposition exists constant every connected plane graph holds corollary every connected plane graph holds lemma connected plane graph minor dlog proof notice first minor enough observe every minor minor following lemma indicates cylindrical enhancements considerably increase branchwidth graph lemma constant plane graph proof let graph created add vertex non trivial face connected components arbitrarily pick vertex make adjacent path internal vertices branchwidth graph maximum branchwidth connected components follows also easy see minor lemma minor dlog corollary follows edge colorings new edges let two plane graphs subgraph let given define function say following conditions hold path endpoints consists edges every face contains edges given refer elements respectively also call edges first step towards algorithm reduce bbfpdc problem given plane graph open set define graph whose edge set consists edges subsets whose vertex set consists endpoints disjoint paths let graph say two paths disjoint none internal vertices path vertex given collection pairwise disjoint paths define endpoints path proofs following proposition found proposition let graph let completion every face collection disjoint paths graph max ghf lemma let plane graph bbfpdc max proof assume first bbfpdc let completion diameter also completion completion means every graph hhf contains edges graph new contains edges proposition collection internally disjoint paths max let set edges obtained every pick one edge paths let let new observe new max construction satisfies conditions definition max required let new max max construct graph removing max edges resulting graph contract edges easy observe completion also completion completion structures dynamic programming dynamic programming algorithm need variant branchwidth plane graphs whose middle sets additional topological properties decomposition let plane graph arc subset plane homeomorphic circle called noose meets vertices also set arc noose connected component trivial case arcs decomposition triple branch decomposition function mapping cyclic orderings vertices every noose following properties satisfied meets every face contained one closed disks bounded contained definition branch decomposition cyclic ordering voe defined clockwise traversal embedding denote voe always assume vertices clockwise enumerated according denote set containing arcs also use notation boundary arc consists vertices also define embedding occurring add arcs edges face called internal incident arc also face face marginal properly included face dynamic programming require hand optimal decomposition done combining main result see also summarized following proposition exists algorithm input plane graph outputs decomposition width reports next step define series combinatorial structures necessary dynamic programming given two sets denote set functions given set integer say pair partition collection pairwise disjoint subsets function mapping integers integers case partition corresponds pair empty function function whose domain empty let two finite sets given define quintuple partition partition graph possibly loops otherwise fusions restrictions let two partitions sets respectively define follows say set contains let transitive closure contains equivalence classes define follows let define min fusion partitions pair partition denoted given partition set given subset define restriction partition denote also define intersection partition denote notice always dynamic programming following result main algorithmic contribution paper lemma exists algorithm given input plane graphs subgraph decomposition width decides whether log log steps proof use notation old new old new choose arbitrary edge subdivide adding new vertex vnew update adding new vertex adjacent vnew root vertex extend setting call edges incident leaves except edge vnew edge called internal denote set denote internal edges also call let tree forest contain leaf let edges images via leaves also leaves denote observe edge define children two edges belong connected component contain root share common endpoint also edge define closed disk bounded finally edge set mid venew new veold old eenew new eeold old distance signatures dependency graphs let eenew vertex define vector function min define graph may contain loops two necessarily distinct vertices connected edge exist notice set eenew elements new assign unique edge set intuitively corresponds partition elements vertices part signature moreover existence edge graph two parts implies contain vertices one part whose bigger tables aim give dynamic programming algorithm running describe table containing information partial solutions problem graph way table edge computed using tables two children size table depend final answer derived table define function mapping collection configurations particular iff exists eenew following hold connected components contains vertex veold otherwise pair encodes connected components contain vertices registers number vertices veold information important control condition partition two arcs belong set say incident marginal face moreover equal number edges inside encodes partial faces embedding inside correspond number contain useful order guarantee algorithm faces stop marginal contain required condition graph graph recall collection different distance vectors vertices notice also might pairs vertices whose bigger order completion diameter two vertices become connected step algorithm paths passing outside check possibility enough know distance vectors encoded set moreover fact still far away inside certified existence edge loop distance vectors pair min information complementary one stored registers distances vertices inside see used order compute distance vectors well dependencies steps algorithm path endpoints veold consists edges ensures condition satisfied current graph every internal face contains edges ensures condition holds internal faces either two vertices demand two vertices far away inside chance come close obtain final graph condition satisfied fact already stored edge two distance vectors possibility may come close step algorithm concerns graph depends distance vectors vertices edges inside internal faces clearly last condition becomes void information helps control condition algorithm notice case graph correspond step graph denoted bounding set characteristics next step bound notice first means log instantiations log instantiations previously log noticed different instantiations moreover log instantiations instantiations conclude exists function moreover log log characteristic function root edge observe enew edge colorable indeed happens conditions become void conditions imply satisfies conditions respectively definition colorability new computation tables show compute give definition case leaf following given define otherwise suppose ael eeold eenew veold set contains single edge loop eenew veold venew assume edge children collection given join join procedure depicted notice symmetric difference ael aer consists endpoints arcs also set xef xel xer procedure join input two collections cel cer xel ael xer aer output collection set every pair qel qer cel cer merge qel qer void let merge qel qer return remains describe routine merge assume receives inputs xel ael xer aer respectively procedure merge qel qer returns constructed follows return void otherwise controls number contained let return void compute fusion connected components hel vertices vel ver makes sure none created components contains let computes fusion restricted boundary let return void let compute function fel fer fel fer whose description given latter take disjoint union graphs remove every edge let obtained graph edge holds every return void consider function every following every set identify vertices least one pair adjacent add loop vertex created identification let resulting graph notice definition function present definition function used description tables dynamic programming procedure given set define ordq mod ordering given define fel fer fel fer distinguishing following cases xer fel xel fer min min xef xel mod otherwise xel fel xer fer min xef xel mod otherwise one two vertices xer xer xef fel fer min min ordq xef fel mod otherwise min ordq xef equality belong different sets otherwise function previous case xer xel min min xef equality xer mod otherwise belong different sets xer xel min xef function previous case exactly one say belongs xer xer xef min min min ord xef function two previous cases case belongs xer xer xef swap positions equation belong xer xer xef min min min ordq xef previous equality mod otehrwise running time analysis remains prove procedure join runs log steps recall exists function therefore merge called step times first computationally step merge step function computed notice entries values require running permutations subsets xef facts imply computation takes steps steps deal graphs vertices running time join claimed one position prove main algorithmic result paper proof theorem given input bbfpdc consider graph max whose construction takes steps lemma run algorithm proposition input answer proposition therefore lemma safely report algorithm proposition outputs decomposition width call dynamic programming algorithm lemma input lemma provides answer bbfpdc log log instance log steps completes proof theorem proofs section show bounded budget plane diameter completion bounded plane diameter completion problems npcomplete consider graphs graphs embedded plane graph exactly one unbounded face called outer face faces called inner faces take mind every graph many embeddings number faces correspond face embedding chosen outer face problems equivalently restated graphs choose embeddings facilitate presentation result section also need additional terminology walk graph sequence vertices edges edges pairwise distinct walk walk closed length walk number edges walk say walk path pairwise distinct possible exception cycle closed path write denote walk omitting edges recall problem given boolean formula clauses literals variables asks whether assignment satisfies write literal resp interval instance define graphs follows vertex set contains either respectively let variables instance planar let plane embedding let also define bipartite graph graph vertex set edge set incident consider following special variant satisfiability plane satisfiability connectivity variables input boolean formula clauses literals variables planar plane embedding connected output possible satisfy show problem hard lemma plane satisfiability connectivity variables npcomplete proof straightforward see plane satisfiability connectivity variables show reduce planar problem restricted instances planar problem shown lichtenstein let variables instance planar plane graph construct plane embedding well known done polynomial time classical algorithm hopcroft tarjan algorithm boyer myrvold consequently consider variables modify suppose variable occurs clauses cjp without loss generality assume edges cjp ordered clockwise shown fig perform following modifications replace new variables replace cjk construct clauses assume cik modify current plane graph shown fig demonstrate constructions plane embeddings figures instead long technical formal descriptions denote obtained boolean formula plane graph respectively construction plane embedding cjp cjp figure modification modification show satisfied satisfying assignment suppose variables assigned values true assign value variables replace straightforward verify true assignment assume true values variables observe variables replace value satisfy remains observe value true construction cjp figure construction assumed contains contains cjp contains observe variable occurs clauses occurs least positive least negation implies plane embedding constructed splitting variable vertices shown fig clearly constructed polynomial time claim connected see observe constructed replacing cycle see fig respectively graph new faces inner faces cycles faces correspond faces denote inner face notice contains edges cycles follows vertex adjacent vertex hence prove connectivity sufficient show two vertices consider dual recall two vertices adjacent corresponding faces adjacent straightforward observe dual plane graph always connected hence show two vertices sufficient prove holds two adjacent vertices adjacent faces suppose face corresponding face vertex lies boundary construction edge lies boundaries vertex adjacent assume faces distinct adjacent faces correspond faces common vertex boundaries implies adjacent already proved therefore completes proof connectedness proof lemma proof main result need special gadgets introduce prove properties useful let positive integer construct graph follows see fig construct vertices vertex construct path xir length xir construct cycle xrj construct edge assume xrj say vertices inner vertices gadget let plane graph face let facial walk say obtained attaching web constructed adding copy vertices gadget identified vertices names facial walk embedding shown fig notice vertices facial walk occur several times lemma let plane graph face facial walk length let plane graph obtained attaching web figure construction figure attachment web two vertices distg moreover shortest inner vertices attached vertex vertex proof let facial walk prove sufficient observe length greater length shortest lies boundary definition immediately implies let positive integer graph defined follows see fig construct vertices construct path xir length xih construct cycle xrj construct edges say vertices inner vertices gadget also say root pole let plane graph let vertex incident face triangle facial walk let also positive integer say figure construction obtained attaching mast height rooted constructed adding copy vertices gadget identified vertices names facial walk embedding need properties summarized following straightforward lemma lemma let positive integer let plane graph let vertex incident face triangle facial walk let also plane graph obtained attaching mast height rooted two vertices distg moreover shortest inner vertices attached vertex iii pole inner vertices ready prove main result section proof theorem straightforward see bpdc bfpdc show reduce plane satisfiability connectivity variables shown lemma first consider bpdc let instance plane satisfiability connectivity variables boolean formula clauses literals variables planar plane embedding connected recall bipartite graph bipartition vertex set set faces edge incident face notice degh select arbitrary vertex using connectedness find polynomial time tree shortest search assume rooted defines relation let set leaves let max distt figure construction gadgets construct plane graph follows construct copy vertex crate vertex embedded face iii denote parent child construct vertices edges embed shown fig denote inner face cycle inner face cycle denote parent construct vertices edges embed shown fig denote inner face cycle inner face cycle resp replace edge resp length distt denote constructed stage graph observe connected hence face facial walk denote length longest facial walk proceed construction face distinct faces attach web denote constructed stage graph notice due attached webs vii select face obtained graph incident attach mast height rooted notice boundary triangle attached webs viii attach mast height distt rooted face facial walk vertex select face triangle boundary incident face always exists due attached webs attach mast height rooted notice obtained graph attachments masts destroy also faces degree faces degree complete construction instance bpdc set show plane satisfiability connectivity variables bpdc suppose plane satisfiability connectivity variables assume variables values true true add edge parent embed edge respectively add edge embed edge alse denote obtained graph show diam construction vertex lemma vertex distance vertex hence vertex observe also distt show consider five cases case case vertices mast attached face lemma height mast case vertex mast attached face lemma mast rooted suppose mast rooted vertex case vertices distinct masts attached faces rooted respectively distt distt distt clearly bounds hold remains consider last case case vertices masts attached faces mast rooted mast rooted suppose distt distt assume clause contains literal value true let literal case contains true symmetric notice true vertex distt construction selection added edges distt distt suppose bpdc let set edges graph obtained addition diameter faces degree faces degree edge boundary embedded face using observation denote graphs obtained respectively addition let pole mast rooted diam particular holds poles masts consider masts rooted mast rooted denote pole lemma distt lemma conclude distt distt shortest contain vertices obtain every edge lies unique parent holds leaf either parent let variable true alse otherwise show assignment satisfies consider clause simplify notations assume contains literals cases contains two literals literals negations variables considered way let pole mast rooted vertex lemma lemma therefore let xih xih construction min xih distt let xih distt follows xih distt immediately implies xih parent definition xih true therefore true holds conclude true complete proof bpdc remains observe constructed polynomial time show bfpdc use similar arguments let instance plane satisfiability connectivity variables boolean formula clauses literals variables planar plane embedding connected pick arbitrary vertex find tree rooted shortest set leaves let max distt figure construction gadgets construct plane graph similarly construction difference steps iii replaced following steps denote parent child construct vertices edges embed shown fig denote inner face cycle denote parent construct vertices edges embed shown fig denote inner face cycle observe obtained deletion vertices notice obtained graph faces degree faces degree complete construction instance bfpdc set show plane satisfiability connectivity variables bfpdc suppose plane satisfiability connectivity variables assume variables values true true add edge parent embed edge respectively add edge embed edge alse denote obtained graph exactly arguments proof inequality diam diam suppose bfpdc let set edges graph obtained addition diameter faces degree faces degree edge boundary embedded face one edge embedded let pole mast rooted diam particular holds poles masts consider masts rooted mast rooted denote pole arguments used proof bpdc obtain implies either parent let variable true alse otherwise prove assignment satisfies use arguments follows fact clause pole mast rooted vertex complete proof bpdc remains observe constructed polynomial time proved bpdc planar graphs whitney theorem see two plane embeddings plane graphs equivalent gives following corollary bounded budget planar diameter completion input planar graph integers output possible obtain planar graph diameter adding edges corollary bounded budget planar diameter completion planar graphs discussion remark algorithm still works classic pdc problem input graph bounded define following problem bounded face bdc fpdc input plane graph question possible add edges resulting embedding remains plane diameter directly following corollary theorem log theorem possible construct fpdc algorithm construct pdc parameterized remains insisting open problem reason approach apply least directly pdc long completion may add arbitrary number edges face guarantee dynamic programming algorithm applied graph bounded branchwidth believe approach particular machinery dynamic programming algorithm might useful investigations problem problems paper defined plane graphs however one may also consider counterparts problems pdc bpdc asking input planar combinatorial graphs without particular embedding similarly counterpart also defined case bfpdc ask whether completion embedding new edges per face parameterized problems known fpt results however approach fails design corresponding algorithms strongly requires embedding input graph reason believe even versions bpdc bfpdc challenging general planar diameter completion problem acknowledgement would like thank anonymous referees earlier version paper remarks suggestions improved presentation paper references john boyer wendy myrvold cutting edge simplified planarity edge addition graph algorithms dimitris chatzidimitriou archontia giannopoulou spyros maniatis clment requil dimitrios thilikos dimitris zoros fixed parameter algorithms completion problems planar graphs manuscript chung diameters graphs old problems new results congressus numerantium bruno courcelle expression graph properties graph transformations monadic logic handbook graph grammars pages italo dejter michael fellows improving diameter planar graph manuscript may reinhard diestel graph theory edition volume graduate texts mathematics springer frederic dorn eelko penninkx hans bodlaender fedor fomin efficient exact algorithms planar graphs exploiting sphere cut decompositions algorithmica rodney downey michael fellows fundamentals parameterized complexity texts computer science springer yong gao donovan hare james nastos parametric complexity graph diameter augmentation disc appl hisao tamaki optimal planar graphs time acm transactions algorithms hisao tamaki improved bounds planar branchwidth respect largest grid minor size algorithms computation international symposium isaac pages john hopcroft robert endre tarjan efficient planarity testing acm athanassios koutsonas dimitrios thilikos planar feedback vertex set face cover combinatorial bounds subexponential algorithms algorithmica thomas mccormick david edge addition problems oper res david lichtenstein planar formulae uses siam journal computing neil robertson paul seymour graph minors xiii disjoint paths problem comb theory ser neil robertson paul seymour graph minors xiii disjoint paths problem combin theory ser neil robertson paul seymour graph minors wagner conjecture combin theory ser anneke schoone hans bodlaender jan van leeuwen diameter increase caused edge deletion journal graph theory paul seymour robin thomas call routing ratcatcher combinatorica
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centralized local algorithm sparse spanning graph problem christoph lenzen reut levi jul july abstract constructing sparse spanning subgraph fundamental primitive graph theory paper study problem centralized local model goal decide whether edge part spanning subgraph examining small part input yet answers must globally consistent independent prior queries unfortunately maximally sparse spanning subgraphs spanning trees constructed efficiently model therefore settle spanning subgraph containing edges number vertices given parameter achieve query complexity poly maximum degree input graph algorithm first arbitrary bounded degree graphs moreover achieve additional property algorithm outputs spanner distances approximately preserved high probability deleted edge path log log hops output connects endpoints introduction operating large graphs often impractical infeasible hold entire graph local memory processing unit run even slower algorithms even iii single processing unit perform computations sequentially constraints inspired centralized local model essentially views input stored likely distributed database provides query access external processing units minimize coordination overhead system furthermore required shared memory communication querying processes except shared randomness provided alongside access input idea run algorithms extract useful global properties graph examine input graph locally upon demand applications studying graphs model leads need query access variety structures like independent dominating sets case crucial locally evaluating whether node participates set consistent evaluation nodes task local decisions coordinated implicitly via structure input examined little possible shared randomness mpi informatics germany clenzen mpi informatics germany rlevi hides polylogarithmic factors nonetheless budding field brought forth number elegant algorithms solving maximal independent set hypergraph coloring approximate maximum matching approximate minimum vertex cover bipartite graphs work consider another basic graph structure sparse spanning subgraphs task select subset edges connected input graph output still connected edges mean input parameter number selected edges denotes number nodes one may see relaxed version problem outputting spanning tree graph rigid requirement looking fast algorithms cycle single edge deleted necessitates first verify input graph fact line definition algorithm local sparse spanning graph lssg algorithm given parameter query access incidence list representation connected graph vertices degree provides oracle access subgraph connected high probability determined internal randomness providing oracle access mean input returns whether sequence edges answers consistently respect interested llsg algorithms given edge perform queries possible observe item implies answers llsg algorithm queries depend previously asked queries note relaxing requiring tree output makes possible ask additional guarantees general met spanning tree instead merely preserving connectivity becomes possible maintain distances small factors subgraphs known sparse multiplicative spanners fact choosing yields spanners edges away trees contribution give first lssg algorithm centralized local model runs arbitrary graphs achieve query complexity poly per edge moreover guarantee edge selected spanner path log log hops consisting edges selected spanner referred stretch log log simplicity assume moment constants algorithm combines following key ideas classify edges expanding sufficiently many roughly nodes within log hops endpoints edges efficiently simulate standard distributed spanner algorithm small query complexity solutions running time log known probability least arbitrary constant chosen upfront node set induced expanding edges construct partition voronoi cells respect roughly randomly sampled centers voronoi cells spanned trees depth log expanding nodes closest center within log hops finding closest center query complexity refine partition voronoi cells clusters nodes simply let node singleton cluster subtree spanning tree cell contains nodes construction query complexity constructing complete cluster yet ensures clusters total due low depth trees moreover cluster completely contained voronoi cell remains select edges interconnect voronoi cells main challenge properties partition crucial keep number selected edges small expectation mark subset expected size clusters marking voronoi cell thereby constituent clusters probability try ensure clusters select edge adjacent marked voronoi cell marked voronoi cell adjacent adjacent cluster select one edge connecting cluster adjacent cell main issue previous step afford construct adjacent cluster preventing guaranteeing circumvent obstacle identifying adjacent clusters cell keeping edge purpose satisfies certain minimality requirement respect rank cell used symmetry breaking purposes way avoid construction adjacent clusters instead needing determine rank voronoi cells way maintain query complexity however entails inductive argument ensuring connectivity also affects stretch choosing voronoi cell ranks uniformly random ensure length inductive chain bounded log together depth voronoi cell trees log stretch spanner algorithm edges also log yields total bound stretch scheme finally note place routine wrapper verifying number globally selected edges significantly exceed expectation case wrapper starts process since attempt success probability constant verifier succeeds get within log attempts bound number selected edges satisfied routine terminates relation property testing observed testing one sidederror model reduced lssg problem reduction follows obtain tester works time czumaj studied problem freeness one freeness complexity therefore complexity better however would like point lssg algorithm gives stronger guarantee error tester lssg algorithm used find edges cycle belongs witness violation provided constrast error tester merely guarantees find single cycle instances perhaps importantly approach proves useful testing minors recently fichtenberger built work construct error testers minor related work problem finding sparse spanning subgraph centralized local model first studied authors show lower bound queries constant see also survey rubinfeld also present upper bound nearly tight query complexity graphs good expansion properties however general bounded degree graphs algorithm might query entire graph completing single call oracle also provide efficient algorithm graphs later improved algorithm presented achieves query complexity polynomial independent stretch factor algorithm also independent depends size excluded minor characterization query complexity problem presented specifically work provides upper bound builds algorithm query complexity independent however families graphs roughly speaking sufficiently everywhere hand show family graphs expansion properties slightly better local algorithm must query complexity depends distributed local model ram vicari study problem provide algorithm runs min log rounds denotes diameter input graph algorithm achieves sparsity property breaking short cycles preliminaries graphs consider undirected known degree bound assume query access incidence list representation namely vertex index possible obtain ith neighbor performing query graph less neighbors special symbol returned without loss generality assume graphs simple contain neither loops parallel number vertices graph assume vertex unique simplicity also denote total order ids given two distinct ids decide whether let graph denote distance two vertices vertex integer let denote set vertices distance graph clear context shall use shorthands respectively total order vertices induces total order edges graph following straightforward manner min min min min max max total order vertices also induces order vertices visited breadth first search bfs starting given vertex whenever refer bfs mean performed according order answer always negative default rejecting first edge two nodes figure partition graph cells clusters black lines borders voronoi cells whose centers black fillings red edges belong bfs trees spanning clusters dashed gray lines edges red circles indicate singleton clusters node red child roots subtrees form cluster children black whenever referring one orders may refer rank element respective order simply index respective element listing elements ascendingly respect order graph pair disjoint subsets vertices let def clear context omit subscript say pair subsets vertices adjacent algorithm works promise begin describing lssg algorithm works following promise input graph sample uniformly random log log log log def let constant determined later every def let minr promised words assume neighborhood every vertex contains least vertices first fix simple partition underlying partition def centers pick set vertices random shall refer vertices centers vertex center denoted center closest amongst centers break ties centers according center voronoi cells voronoi cell vertex denoted vor set vertices additionally assign cell random rank uniformly random total order cells note carefully rank cell thus differs rank center given identifier assigned randomly remark determine rank cell shared randomness cell identifier simply use identifier center clusters voronoi cell consider bfs tree spanning rooted respective center every let denote parent bfs tree center every let denote subtree bfs tree remove edge simply entire tree consider voronoi cell cell contains vertices cluster vertices voronoi cell cell otherwise two cases contains least vertices cluster singleton otherwise unique ancestor including contains less vertices contains least vertices cluster set vertices cluster let denote center vertices vertices cluster center let vor denote voronoi cell vertices describes partition voronoi cells refinement partition clusters see figure illustration edge set spanner initially contains voronoi cell vor edges bfs tree spans vor bfs tree rooted center vor spanning subgraph induced vor see section details clearly edges also span clusters next describe edges add order connect clusters marked clusters def center marked independently probability center marked say voronoi cell marked clusters cell marked well every marked cluster define denoted set clusters consists clusters adjacent cluster participating edge minimum rank vor also belongs thus adjacent vor vor marked unique cluster vor participates see figure visualization figure illustration marked clusters clusters clusters thick red black ovals marked unmarked cells respectively thin circles clusters cluster comprises entire cell thick edges ones minimum rank incident clusters dotted edges meet criterion arrows red edges indicate participation respective adjacent marked cluster note participate adjacent marked cell vor exclusively participates connected edge minimum rank vor marked blue constituent clusters also participate vor edges clusters saying connect two adjacent subsets vertices mean add def minimum ranked edge cluster define adjacent centers vor vor vor set voronoi cells adjacent definition explicitly excludes vor need connect voronoi cell next describe connect clusters idea make sure every marked cluster clusters participate respective remain connected clusters adjacent marked cluster make sure keep connected adjacent voronoi cells formally connect every cluster every adjacent marked cluster cluster participating cell adjacent marked connect adjacent cell suppose cluster adjacent cluster adjacent marked cell vor denote unique cluster cell vor participates connect following conditions hold vor minimum rank amongst vor vor figure illustration third edge selection rule example thick black edge minimum rank vor minimum rank however rank vor smaller vor hence dashed edge selected select direct edge connecting due first rule minimum ranked edge vor also figure showcases third rule sparsity lemma number clusters denoted proof first observe due promise follows every recall terminology subsection consider therefore cluster singleton say vertex special every child holds inductive argument follows ancestor special vertex since every pair special vertices vertex disjoint obtain special vertices since every special vertex ancestors total number vertices bounded observe cluster either singleton contains node iii entire voronoi cell bounded number clusters type immediately get bound number type clusters number type iii clusters bounded number voronoi cells showing desired bound lemma exp proof number edges add due bfs trees voronoi cells number edges taken due condition times number marked clusters expectation marked clusters yielding edges expectation since obtain let cluster number edges adjacent taken due condition bounded total number number exactly number marked clusters thus total number edges taken due condition bounded observe probability cluster adjacent marked cell hence adjacent marked cell using union bound clusters follows cluster without adjacent marked cell satisfies probabilty least probability event bound violated contribute expectation therefore total number edges taken due condition bounded conclude total number edges expectation desired connectivity stretch lemma connected proof recall contains spanning tree every voronoi cell hence suffices show connect pair voronoi cells path vertices moreover facts connected voronoi cells partition imply sufficient prove pair adjacent voronoi cells accordingly let vor two cells vor consider clusters vor edge minimum rank vor adjacent marked cell condition implies selected thus may assume adjacent marked cell accordingly exists marked cluster participating rank minimum vor vor selected condition done otherwise observe connected edge minimum rank selected condition therefore suffices show vor gets connected let cell minimum rank among vor vor let cluster satisfying edge minimum rank note connected saw connected cluster vor adjacent selects edge minimum rank condition overall see sufficient show vor gets connected smaller rank repeat reasoning inductively step either succeed establishing connectivity vor vori determine cell smaller rank vori connected vori sequence voronoi cells descending ranks must finite induction halts finitely many steps induction invariant connected vori establishes connectivity vor completing proof lemma denote gvor graph obtained contracting voronoi cells hvor subgraph obtained cells ranks uniformly random hvor spanner gvor stretch log proof recall proof lemma established connectivity inductive argument step increased number traversed voronoi cells two hence suffices show induction halts log steps see observe first gvor independent ranks assigned voronoi cells pick pair adjacent cells vor neighbors gvor perform induction assigning ranks high low needed step according following process step query rank cells given answer rank ranks cells rank least revealed well step begin querying rank vori consider cluster vori adjacent satisfying edge minimum rank vori also assume without loss generality adjacent marked cluster participating otherwise connects directly terminate process ranks cells adjacent already revealed process terminates otherwise query rank cells whose rank still unrevealed set cell queried cluster minimum rank continue next step claim step either process terminates rank half rank vori probability least verify observe beginning step cell center whose rank revealed far rank uniformly distributed rank vori probability least rank vori adjacent cells whose ranks revealed yet process terminates hence regardless whether process terminates claim holds chernoff bound conclude process terminates within log steps bounded number voronoi cells trivially bounded union bound pairs cells vor get desired guarantee corollary spanner stretch log log proof due promise spanning trees voronoi cells depth log hence edge within voronoi cell claim holds moreover edge connecting different voronoi cells lemma path length log hvor connecting respective cells navigating log hops traversed cell obtain suitable path length log log algorithm general graphs use combination algorithm section algorithm elkin neiman spanners call vertex remote respect set centers neighborhood include center fix let denote set remote vertices def respect abbreviate step first query observe statement holds first step run algorithm section subgraph induced added algorithm outputs edge second step run algorithm elkin neiman subgraph induced added algorithm outputs algorithm proceeds follows given integer vertex draws according exponential distribution parameter parameter controls success probability algorithm vertex receives every vertex within distance stores neighbor shortest path denoted edges added spanner every probability least holds claim choose algorithm succeeds require following lemma implies total number edges add second step expectation lemma proof lemma every exp third step add edges following lemma implies expected number edges added third step lemma exp proof observe edge one integer vice versa edge random choice probability event edge included desired bound follows linearity expectation stretch factor consider edge removed removed algorithm section applied subgraph induced applying lemma connected component get path length log log lemma choice parameters path length log third step arrive following corollary corollary algorithm guarantees stretch log log satisfies exp local algorithm section prove main theorem algorithm described connected graphs simply apply connected component algorithm lssg general graphs input output whether compute output algorithm elkin neiman running connected component subgraph induced return true false otherwise return true otherwise proceed according section nodes ignored vor vor return true bfs tree vor false otherwise otherwise let denote clusters respectively return true least one following conditions hold symmetrically false otherwise marked cluster minimum rank amongst edges participating namely clusters adjacent marked case take minimum rank amongst edges vor iii exists marked cluster participating following holds vor minimum rank amongst vor vor minimum rank amongst edges vor theorem algorithm lssg algorithm graph vertices maximum degree query complexity space complexity length random seed running time poly proof correctness algorithm follows previous sections shall prove complexity claimed analyze complexity terms additional factors depend polynomially following claims hold simultaneously vertices recall vertex remote contain center due sampling probability centers center found exploring vertices therefore decide vertex whether query time complexity moreover without additional cost respective subroutine return center explore bfs fashion step need determine since every vertex obtain query time complexity step total accordingly step query time complexity algorithm proceeds section claim reconstructing clusters determining centers adjacent nodes deciding whether bfs edge vor vor parent bfs vor rooted vice versa takes queries time show first reconstruct clusters efficiently consider determine neighbor determine whether node discarded center distance assuming must least one neighbor distance satisfies otherwise contradiction tiebreaking rule centers among candidates know one minimum rank parent bfs tree vor rooted due rule bfs construction otherwise use subroutine partially explore bfs vor given node determine parent children vor query time complexity conclude determine whether query complexity partially completely exploring determine completely collect information ancestors determine cluster finally repeat procedure reconstruct cluster determine nodes adjacent either cluster whether centers query time complexity operation total cases respectively information determine whether vor vor vor vor whether vor vor whether marked marked whether minimum rank whether adjacent marked cluster none adjacent nodes centers marked whether minimum rank vor whether marked cluster adjacent participates vor minimum rank vor vor minimum rank vor note since degree bounded number vertices probability cell marked number participates words perform necessary checks decide whether algorithm section requires random bits selection centers marked clusters emulation algorithm elkin neiman suffices random variables independent outcome algorithm vertex depends random variables vertices vertices thus overall random bits poly factors sufficient references alon rubinfeld vardi xie local computation algorithms proceedings annual symposium discrete algorithms soda pages artur czumaj oded goldreich dana ron seshadhri asaf shapira christian sohler finding cycles trees sublinear time random structures algorithms michael elkin ofer neiman efficient algorithms constructing sparse spanners emulators proceedings annual symposium discrete algorithms soda barcelona spain hotel porta fira january pages even medina ron deterministic stateless centralized local algorithms bounded degree graphs algorithms esa annual european symposium wroclaw poland september proceedings pages feige mansour schapire learning inference presence corrupted inputs proceedings conference learning theory colt paris france july pages hendrik fichtenberger reut levi yadu vasudev maximilian testing minorfreeness bounded degree graphs error unpublished manuscript levi moshkovitz ron rubinfeld shapira constructing near spanning trees local inspections random structures algorithms pages levi ron time partition oracle graphs excluded minor acm trans algorithms levi ron rubinfeld local algorithms sparse spanning graphs proceedings eighteenth international workshop randomization computation random pages levi ron rubinfeld local algorithms sparse spanning graphs corr levi rubinfeld yodpinyanee local computation algorithms graphs degrees algorithmica pages mansour rubinstein vardi xie converting online algorithms local computation algorithms automata languages programming international colloquium icalp pages mansour vardi local computation approximation scheme maximum matching proceedings sixteenth international workshop approximation algorithms combinatorial optimization problems approx pages peleg graph spanners journal graph theory peleg ullman optimal synchronizer hypercube siam journal computing ram vicari distributed small connected spanning subgraph breaking diameter bound technical report rubinfeld tamir vardi xie fast local computation algorithms proceedings second symposium innovations computer science ics pages ronitt rubinfeld locally compute sparse connected subgraphs computer science theory applications international computer science symposium russia csr kazan russia june proceedings pages
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adaptive sampling scheme efficiently train fully convolutional networks semantic segmentation lorenz eoin jorge dec tig cmic university college london london innersight labs london abstract deep convolutional neural networks cnns shown excellent performance object recognition tasks dense classification problems semantic segmentation however training deep neural networks large sparse datasets still challenging require large amounts computation memory work address task performing semantic segmentation large data sets medical images propose adaptive sampling scheme uses error maps generated throughout training focus sampling difficult regions resulting improved learning contribution threefold give detailed description proposed sampling algorithm speed improve learning performance large images propose deep dual path cnn captures information fine coarse scales resulting network large field view high resolution outputs show method able attain new results visceral anatomy benchmark introduction paper addresses problem efficiently training convolutional neural networks cnns large imbalanced datasets propose training strategy adaptively samples training data effectively speed training avoid data contains little extra information work investigate problem automatic segmentation high resolution scans several deep learning techniques recently proposed segmentation medical datasets overcome problem dealing large datasets computed tomography volumes commonly dimension previous approaches train cnn cropped region interest reduces size individual training images around fold reducing size training images fit memory network trained effectively selected data however identifying regions interest requires additional step may easy many applications also training cnns cropped images limits field view cnn subsequently lorenz berger introduce unwanted image boundary induced effects testing applications training cnns large images problem includes segmentation histology datasets segmentation aerial images example aerial image segmentation training cnn segment ships difficult large portions image contain water provide little information training resulting slow learning ideas address already proposed example fixed handcrafted weight map used help learn small separation borders touching cells biomedical image segmentation work proposed sampling scheme ends dynamically learning weight mapping making generally applicable many applications curriculum learning derivative methods like learning build intuition rather considering samples simultaneously algorithm presented training data meaningful order facilitates learning ideas already successfully applied image classification ordering images easy hard training also problem weakly supervised semantic segmentation similar ideas applied predictions previous training iterations used iteratively learn segmentation maps single class label per image focus paper fully supervised semantic segmentation representative training set available dense manual label annotations challenge lies efficiently learning large datasets give detailed account implementation straightforward extension existing cnn segmentation system present segmentation results visceral anatomy benchmark methods neural network architecture dual path network architecture build several previous ideas compared network outlined develop architecture replacing standard convolution layers popular resnet blocks increase maximum network depth layers layers deeper network sampled pathway input resolution original resolution obtain large receptive field size whilst maintaining deep high resolution pathway compromise resolution pooling layers architecture results total parameters sketch architecture given figure figure numbers inside round brackets give input dimensions block training stage dimensions chosen carefully balance memory usage processing speed testing dimensions may chosen large fit memory take advantage fully convolutional inference numbers square brackets refer number feature maps used layer proposed configuration allows large number samples isample adaptive sampling fully convolutional networks patches per batch ensure balanced class sampling effective optimization whilst maintaining deep wide enough network capture high variability spatial semantics data blocks labeled conv standard convolutional layers kernel size blocks labeled res block standard bottleneck resnet blocks respectively detailed fully connected layer preceded dropout layer probability softmax used final classification layer rationale deep low resolution path increase receptive field allow complex higher level features learned organ positioned relation structures minimize memory footprint high definition path chosen slightly shorter low resolution path seems reasonable path learn texture information likely require fewer layers details training given section fig proposed dual path cnn architecture coronal slice scan overlaid segmentation output described section organs visible slice lungs green liver red spleen light blue psoas major muscle dark blue kidneys brown bladder yellow lorenz berger adaptive sampling strategy problem class imbalance described dealt choosing small patch sizes evenly sampling class weighted loss functions methods either load whole image gpu memory feasible large images select small subset patches lead inefficient training sparse datasets overcome issues propose simple sampling algorithm algorithm isample adaptive sampling algorithm initialize error maps every image training data cnn training training epoch filling batch patches pick image training set pick class corresponding label map pick patch image centered location accept patch batch end loss batch update current cnn weights end select subset images label maps training set update error maps cnn end end algorithm random number drawn uniform distribution refers error map ith training image error maps easily calculated either epoch concurrently training process cnn cnn map cnn predictions full training image evaluated using current weights outputting probability true class label position examples error maps produced throughout training show figure additional parameter controls strength isample scheme setting corresponds choosing patches based entirely amount error currently produce network condition always satisfied left standard sampling scheme accepts every chosen patch results shown paper chosen since interested using isample scheme full effect detailed investigations best set parameter different datasets varying amounts sparsity left future work subset images label maps training set may chosen line quickly introduce isample scheme training amount computational resources available isample adaptive sampling fully convolutional networks experiments access four gpus three used train cnn continuously one gpu used parallel continuously perform full predictions validation dataset training dataset full dice scores validation dataset full error maps training dataset could calculated future extension work could log loss individual batches training use information get estimate network difficulty modification would avoid need allocate additional resources segmenting training data instead could used speed training however practical reasons increasing availability large cards found issue also access dice scores calculated full images throughout training helpful development since true dice scores provide meaningful information dice scores calculated individual batches biased sampling scheme size patches error maps produced also useful debugging purposes development results trained validated tested automatic segmentation method contrast enhanced scans visceral anatomy dataset made training scans unseen testing scans currently available download scans form heterogeneous dataset various topological changes patients manual segmentations available number different anatomical structures randomly split training set scans training scans validation also present results online submission unseen test dataset illustrative purposes first experiment section focuses segmenting kidneys full body scans section present results simultaneously segmenting multiple organs data cnn training setup training perform data augmentation patches resolution also rotate patch degrees set voxels values greater values less divide values constant factor standard deviation dataset use glorot initializations convolution layers batchnorm layers use initializations technique described impose weight decay size convolutional layers except last fully convolutional layer final softmax using techniques described make use large batch sizes large learning rates use sgd nestrov momentum set initial learning rate batch contains patches sampled one randomly selected scans training set run epoch batches also employ learning rate lorenz berger warm schedule described first epochs use standard loss function segmenting kidneys full body scans experiment use labels kidneys train cnn resulting simple two class foreground kidneys background everything else segmentation problem figure shows curves training loss mean validation dice score segmented kidneys throughout training averaged three separate runs blue curve represents training runs patches sampled randomly evenly background kidney foreground red curve represents training runs patches sampled using proposed sampling algorithm isample adaptively selects difficult patches training progresses loss higher seen figure figure sampler achieves faster generalization current results indicate final generalization cnn trained proposed sampling scheme slightly improved sparse segmentation setup kidneys make voxels within whole scan table shows average dice scores achieved dual cnn without isample throughout training using isample scheme cnn able achieve dice score training iterations close end training performance dice score dual cnn without isample use fast training cnns using isample useful debugging evaluating changes neural network architectures experiments run using computational resources quickly estimate end performance network dual cnn dual cnn isample mean validation dice training loss epoch dual cnn dual cnn isample epoch fig training loss mean validation dice scores averaged runs figure show coronal slices training error volume calculated algorithm seen figure initially significant produced isample adaptive sampling fully convolutional networks cnn prediction epoch example misclassifying aorta part background class similar intensity values kidneys training epoch figure shows error much lower cnn learned aorta part background class however subtle regions collecting system large vessels within kidney see small hole true segmentation left kidney figure still produce high errors focused training required optimize weights correctly classified also remains high error around border kidneys result sampling process selecting patches border region thus ends learning train network similar loss border weighted loss function designed fig coronal slices raw scan training set kidney segmentation overlaid onto scan error map foreground background classification training scan epochs epochs error maps white corresponds voxels incorrectly classified black correctly classified voxels dual cnn isample method kidney dice scores dual cnn validation data left right dual cnn isample validation data left right dual cnn isample crf test data left right wang test data left right vincent test data left right gass test data left right fig mean dice scores standard deviations different number iterations throughout training dice scores standard deviations available different methods automatically segmenting kidneys visceral enhanced dataset table shows dice scores segmenting kidneys using different methods proposed method isample performs significantly better without also submitted method addition crf postprocessing step segment test dataset achieved top score lorenz berger segmenting left right kidneys inference full size scans takes seconds using four tesla gpu cards ram segmentation extended previously described algorithm include classification output trained model main organs available visceral dataset output segmentation maps maximum class probability voxel applying filter retains largest connected binary object within segmentation thus removing small objects segmentation output one validation scans shown figure results proposed method methods also summarized given table method dual cnn isample val dual cnn val jimenez vincent agreement aorta lung kidney pmajor liver abdom spleen sternum trachea bladder table dice scores different automatic segmentation methods agreement results visceral dataset note evaluation service containing test data closed time running experiments able evaluate method test data thus making direct comparisons previous methods difficult previously mentioned trained method data scans validated remaining scans evaluated kidney cnn test data found testing dataset gave better dice scores validation set therefore confident results table representative soon test data set made available update results also organs lung kidney pmajor psoas major abdom rectus abdominis give mean dice scores left right organs experiment modified training schedule initial learning rate batch contains patches sampled randomly selected scans training set run epoch batches halve learning rate every epochs dual cnn without using isample scheme average organ dice slightly underperformed compared using isample average organ dice however difference far less notable previous experiments shown table hypothesize background class segmentation problem split background organs lung liver etc thus making dataset isample adaptive sampling fully convolutional networks especially background class easier sample potential benefit using isample method therefore problem dependent conclusion proposed evaluated sampling scheme deal large images scans shown section sampler enables fast training results indicate final generalization performance improved inline previous research shows positive effect curriculum learning optimization end performance machine learning systems experimental results suggests algorithm gives new state art performance aorta lung kidney rectus abdominis spleen sternum trachea bladder visceral anatomy benchmark improves upon human agreement scores following organs aorta lung kidney psoas major rectus abdominis spleen sternum trachea bladder encouraging results pave way using cnns robust automatic segmentation within clinical practice surgical planning references kamnitsas ledig newcombe simpson kane menon rueckert glocker efficient cnn fully connected crf accurate brain lesion segmentation medical image analysis vol christ elshaer ettlinger tatavarty bickel bilic rempfler armbruster hofmann danastasi automatic liver lesion segmentation using cascaded fully convolutional neural networks conditional random fields international conference medical image computing intervention springer ronneberger fischer brox convolutional networks biomedical image segmentation international conference medical image computing intervention springer dou chen jin qin heng deeply supervised network automatic liver segmentation volumes international conference medical image computing intervention springer jia wang zhang lai eric chang large scale tissue histopathology image classification segmentation visualization via deep convolutional activation features bmc bioinformatics vol zhang yao zhang feng zhang ship detection remote sensing international archives photogrammetry remote sensing spatial information sciences vol bengio louradour collobert weston curriculum learning proceedings annual international conference machine learning acm lorenz berger kumar packer koller learning latent variable models advances neural information processing systems avramova curriculum learning deep convolutional neural networks chen gupta webly supervised learning convolutional networks proceedings ieee international conference computer vision liu shi zhao jia augmented feedback semantic segmentation image level supervision european conference computer vision springer zhao shen shi jia icnet semantic segmentation images arxiv preprint lin milan shen reid refinenet refinement networks identity mappings semantic segmentation arxiv preprint zhang ren sun deep residual learning image recognition proceedings ieee conference computer vision pattern recognition fidon ekanayake kitchen ourselin vercauteren generalised wasserstein dice score imbalanced segmentation using holistic convolutional networks arxiv preprint sudre vercauteren ourselin cardoso generalised dice overlap deep learning loss function highly unbalanced segmentations arxiv preprint toro krenn gruenberg taha winterstein eggel goksel jakab cloudbased evaluation anatomical structure segmentation landmark detection algorithms visceral anatomy benchmarks ieee transactions medical imaging vol glorot bengio understanding difficulty training deep feedforward neural networks proceedings thirteenth international conference artificial intelligence statistics goyal girshick noordhuis wesolowski kyrola tulloch jia accurate large minibatch sgd training imagenet hour arxiv preprint nesterov introductory lectures convex optimization basic course vol springer science business media wang smedby segmentation using shape model guided local phase analysis international conference medical image computing intervention springer vincent guillard bowes fully automatic segmentation prostate using active appearance models miccai grand challenge prostate image segmentation vol gass szekely goksel segmentation landmark localization images large field view international miccai workshop medical computer vision springer koltun efficient inference fully connected crfs gaussian edge potentials advances neural information processing systems isample adaptive sampling fully convolutional networks del toro hierarchic based segmentation anatomical structures evaluation visceral anatomy benchmarks international miccai workshop medical computer vision springer valette sdika desvignes automatic multiorgan segmentation via clustering graph cut using spatial relations hierarchicallyregistered atlases international miccai workshop medical computer vision springer
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nov group regularized estimation structural hierarchy yiyuan zhifeng wang jiang department statistics florida state university abstract variable selection models including interactions explanatory variables often needs obey certain hierarchical constraints weak strong structural hierarchy requires existence interaction term implies least one associated main effects present model lately problem attracted lot attention existing computational algorithms converge slow even moderate number predictors moreover contrast rich literature ordinary variable selection lack statistical theory show reasonably low error rates hierarchical variable selection work investigates new class estimators make use multiple group penalties capture structural parsimony give minimax lower bounds strong weak hierarchical variable selection show proposed estimators enjoy sharp rate oracle inequalities algorithm developed guaranteed convergence global optimality simulations real data experiments demonstrate efficiency efficacy proposed approach keywords hierarchical variable selection scalable computation oracle inequality minimax optimality introduction statistical applications often noticed additive model including main effects inadequate including terms interactions particular often great help prediction modeling sometimes interactions may independent interest one example moderation analysis behavioral sciences cohen paper focus full quadratic model interactions taken account let raw predictor matrix response vector assume following nonlinear additive regression model denotes hadamard product ber predictors posing challenge variable selection even moderate moreover scenario statisticians often interested obtaining model satisfying certain logical relations structural hierarchy discussed nelder mccullagh nelder hamada hierarchy natural requirement gene regulatory network studies davidson erwin banded covariance matrix estimation bien lagged variable selection time series hierarchical variable selection leads reduced number variables measurement referred practical sparsity bien instance model consisting may parsimonious practitioners model involving setting two types hierarchy chipman bien strong hierarchy weak hierarchy let coefficient means interaction term exists model associated main effects must present requires inclusion interaction implies least one associated main effects added model show section relatively easy realize compared invariant linear transformations predictors peixoto primary concern work nontrivial task maintain hierarchy model selection using conventional approaches lasso tibshirani may violate well refer nelder peixoto bickel hao zhang procedures however might hoc greedy paper focuses approaches past works direction include shim choi vanish radchenko james bien shim reparametrizes enforces sparsity formulation motivating could also use however corresponding optimization problem nonconvex computational algorithm shim quite slow problems vanish one main motivations work discussed detail section recent breakthrough hierarchical variable selection one key ideas enforce magnitude constraint coefficients make hierarchy naturally hold vector coefficients predictors handle nonconvex constraint bien rephrased parts dropped product constraints quality convex relaxation seems theoretical justification literature experience excellent performance main effects strong large miss interaction effects become computationally prohibitive large datasets example take days obtain solution path work propose study group regularized estimation structural hierarchy gresh theory able establish nonasymptotic oracle inequalities show error rates proposed estimators minimax optimal logarithm factors come new recipe conquer theoretical difficulties analyzing overlapping regularization terms pursuing structural parsimony moreover develop computational algorithm guarantees convergences iterates function values efficient also simple implement rest paper organized follows notation symbols introduced section section presents general framework gresh fast computational algorithm theoretical support given section section builds oracle inequalities gresh section shows minimax optimal rates section simulation studies real data analysis conducted show prediction accuracy computational efficiency proposed approach technical proofs given appendices notation introduce convenient notation symbols used paper first matrix define kai kai vec respectively vec standard vectorization operator spectral norm frobenius norm denoted kakf respectively vector divided groups representing subvector defined kaj following two operators diag introduced notational simplicity square matrix aij diag ann vector diag defined diagonal matrix diagonal entries given define diag diag diag ann use denote submatrix rows columns indexed respectively arbitrary define vec coefficient vector denotes column standing set cardinality define clearly addition equals number nonzero elements given true signal following abbreviated symbols used paper frequently use concatenated coefficient matrix convenience column denoted given stand respectively let raw predictor matrix define consists interactions includes predictors quadratic model given subset abbreviate also easy see diag vec vec two real numbers means holds multiplicative numerical constant two equally sized matrices aij bij means aij bij group regularized estimation structural hierarchy simplicity assume exists intercept term model written diag response vector design matrix consisting main effects describe general framework hierarchical variable selection referred group regularized estimation structural hierarchy gresh gresh two different types depending objects regularize denoting squared error loss diag first type given min regularization parameters function satisfying property implies vector instance take function kxkr simply identity function first penalty imposes elementwise sparsity second group penalty enforces column sparsity argue two penalties constraint used strong hierarchical variable selection indeed sparsity comes second alone implies probability property function thus symmetry condition indicates hence coefficient zero consequently whenever removed model automatically obeyed describe reasoning follows without symmetry constraint complete argument first line dose hold interestingly guaranteed therefore gives relatively simpler problem pointed reviewer model contains intercept centering response raw predictors make vanish due presence nonlinear terms diag used approximate scenario least one relevant substituting suffices focus convex forms gresh work surely penalty penalty replaced nonconvex alternatives see gresh related methods literature makes special case one formulations corresponds single regularization parameter used another instance given min bien incorrectly described vanish radchenko james form without symmetry condition focus theoretical computational studies gresh matter fact radchenko james defined vanish different way motivates another type gresh min defined similarly argue keeps hierarchy takes form penalty part become kbj considered radchenko james vanish constructs main effects interactions two small sets orthonormal basis functions functional regression setting pose restriction design matrix arbitrarily large key difference two types gresh penalties imposed coefficients terms common practice calling shrinkage method predictors reasonable use common regularization parameter penalizing different coefficients way builds model normalized predictors interactions amounts forming overall design first performing standardization equivalent general type gresh preferable answer given section gresh offers general schemes hierarchical variable selection ordinary lasso group lasso since appears penalties well symmetry constraint main goal paper tackle computational theoretical challenges arising overlapping regularization terms high dimensions computation would like develop fast scalable algorithms section theory treat penalties constraint jointly derive sharp error bound gresh intriguing challenging sections computation perhaps natural think using alternating direction method multipliers admm boyd deal computational challenge admm recently gains popularity among statisticians fact bien designed algorithm based admm one main ingredients augmented lagrangian min lagrange multiplier matrix given constant sometimes referred penalty parameter although admm enjoys nice convergence properties theory practically large enough obtain solution good statistical accuracy often larger value slower primal convergence example package hiernet version computing recommends algorithm may take several days compute single solution path empirical schemes vary iteration hoc always behave well section consider slightly general optimization problem includes types gresh particular instances min diag regularization vector matrices let assume developing algorithm since algorithm applies general necessarily symmetric without loss generality penalties imposed overlapping groups variables worth noting symmetry constraint considerably complicates grouping structure without variable groups shown follow tree structure efficient algorithms developed jenatton simon algorithm follows different track admm details presented algorithm step updates results linearizationbased surrogate function step carries bauschke combettes use two concretely real number proximity operators given sgn sgn representing sign function vector defined componentwise multivariate version given otherwise gresh algorithm easy implement involves complicated matrix operations matrix inversion moreover contain sensitive algorithmic parameters like admm needs line search theorem provides universal theoretical choice guarantee global optimality particular strict iterate convergence addition convergence established considerably stronger every accumulation point type conclusion many numerical studies clarity assume inner iteration runs till convergence unnecessary see remark theorem suppose starting point sequence iterates converges globally optimal solution remark conclusion theorem holds type penalties well zou hastie owen associated proximity operators see hierarchical variable selection adding shrinkage particularly helpful compensate model collinearity remark neither convergence iterates optimality guarantee requires full convergence inner loop see appendix algorithm gresh algorithm solving general problem inputs data regularization parameters initialization large enough say repeat diag vec vec repeat iii end end end end end end convergence convergence output detail various stopping criteria employed schmidt experience running steps say usually suffices remark algorithm theorem extended beyond quadratic loss functions takes binomial deviance classification problems first step algorithm becomes vec diag vec exp extends componentwise vectors show theoretically choosing guarantees convergence algorithm remark recommend applying nesterov first acceleration implementations nesterov detail uses momentum update step vec diag vec vec diag vec empirically number iterations reduced comparison form analysis section given use denote submatrix given vec vec abbreviated respectively ambiguity setting standard treatments stochastic term give sharp error rates particular applying vec vec commonly used literature bickel lounici negahban van geer would yield prediction error bound order log ironically much worse error rate lasso group lasso analysis relies two interrelated inequalities derived statistical computational properties gresh estimators see appendix technical detail first let consider problem redefined vec let global minimizer interested prediction accuracy measured min kxb vec predictive learning perspective always legitimate evaluating quality estimator regardless ratio guarantee small predictor errors using convex method design matrix must satisfy certain incoherence conditions one popular restricted eigenvalue assumption bickel lounici following give extension hierarchy setting restricted cone defined penalties less intuitive technically much less demanding condition used proof assumption given constant satisfying jec following inequality holds vec rate choices regularization parameters play major role prediction choose according log log large constants quite different typical choice penalization see remark detail following theorem states oracle inequality well model cardinality bound gresh estimators convenience use abbreviated symbols estimate reference signal theorem assume let global minimizer sufficiently large constants following oracle inequality holds log provided satisfies constant furthermore regularity condition overall sparsity obtained model controlled log remark letting obtain error bound larger log omitting constant factors indicates gresh guarantees give error rate low lasso existence bias term makes results applicable approximately sparse signals practical significance theorem require spectral norms design matrices bounded assumed example zhang huang bickel addition true signal reference signal theorem need obey remark widely acknowledged penalty parameter grouped penalty adjusted group size yuan lin fact would order log lounici wei huang light fact groups size perhaps surprisingly parameter choice becomes suboptimal hierarchical variable selection fact due presence multiple penalties show proof suffices suppress noise turn leads reduced error rate novel finding owing careful treatment stochastic term generally applicable overlap group lasso jenatton conclusion essentially rate also facilitates parameter tuning since one needs search along grid remark theorem extended vec mean bounded covers noise distributions high probability form results prediction error obtained well without expectation additive term hold probability least min universal constants high probability moreover appendix show adapt proof deliver coordinatewise error bound used recovering sparsity pattern true signal remark version without symmetry condition following lines proof theorem show error rate order jew log jgw jew otherwise associated regularity condition uses jew jgw place respectively require symmetric details reported paper similarly derive oracle inequality gresh estimators let columns equals similarly defined corresponding coefficients denoted satisfy vec vec let global minimizer scaled problem vec aforementioned problem reduced equal general assumption given positive constants satisfying following inequality holds vec theorem conditions theorem place hold error bounds two types gresh order regularity conditions place different requirements design performed extensive simulation studies compare found usually holds suggests penalization basis terms seems appropriate coefficients therefore recommend regularization hierarchical variable selection minimax lower bound error rate comparison section show minimax sense error rate obtained theorem minimax optimal logarithmic factors consider two signal classes hierarchy joint sparsity obeys obeys jgw jew pjg recall definitions jgw jew remark following theorem let nondecreasing loss function regularity assumptions study minimax lower bounds strong weak hierarchical variable selection assumption satisfying symmetric vec holds positive constant assumption satisfying jew jgw vec holds positive constant theorem strong hierarchy assume vec satisfied exist positive constants depending sup inf cpo denotes estimator log log weak hierarchy let model assumption holds replaced replaced log ejg log give examples illustrate conclusion using indicator function know estimator log log occurs positive probability mild conditions theorem shows risk bounded multiplicative constant easy see minimax rates larger error rate obtained theorem comparison popular methods follows see benefits hierarchical variable selection context lasso solves minb bickel estimator prediction error order log optimization problem defined minb automatically maintains error rate lounici general clear winner two let turn particularly interesting case existence main effect model indicate associated interactions must relevant scenario lasso always outperforms although possess property gresh achieves low error rate guarantees hierarchy error rate proved always beat large values considered theorem yet even worst case gresh logarithmic factor worse practical data analysis performance loss gresh degenerates table error rate comparison lasso gresh constant factors omitted lasso gresh minimax log log log log ejg experiments simulations part perform simulation studies compare performance gresh type terms prediction accuracy selection consistency computational efficiency use toeplitz design generate main predictors correlation given true coefficients symmetric generated according following three setups example interactions relevant response variable satisfied example model involves main interaction effects obeys example true model strong main effects satisfies regularization parameters tuned separate large validation dataset containing observations need perform full twodimensional grid search find optimal parameters gresh rather motivated theorem set chose according experience convex nature problem pathwise computation warm starts used variable selection ridge regression model always refitted used prediction official package hiernet implemented set restricted ridge refitting substantially enhances accuracy make fair comparison gresh use error tolerance number grid values algorithmic parameters hiernet set default values given setup repeat experiment times evaluate performance algorithm according measures defined test error err mean squared error true mean estimate robustness stability report median test error runs joint detection rate fraction among experiments missing rate false alarm rate mean mean respectively path computational cost average running time algorithm seconds experiments run cpu memory windows table table summarize statistical computational results table gresh behaved equally well example model contains main effects gresh faster example example two methods show differences see test errors table statistical performance gresh measured test error joint detection rate missing rate false alarm rate simulation data numbers multiplied err err err gresh err err err gresh table path computation costs gresh computational times seconds unless otherwise specified gresh hours hours hours joint identification rates example also noticed gresh often gave parsimonious model main effects weak example may miss genuine interaction effects overall gresh showed comparable better test errors fact observed even satisfied results shown table suspect performance differences gresh largely result fact compares overall realize groups term basis select main effects computational times table show scalability algorithm varies variables total became computationally prohibitive also evidenced lim hastie gresh offered impressive computation gains experiment comparison admm part shows directly applying admm give scalable algorithm solving optimization problem large number groups large group size detailed algorithm design given appendix set admm compared algorithm results reported table table statistical performances two algorithms close reasonable solve optimization problem however admm much slower experiments admm became infeasible larger table statistical performance gresh admm measured test error joint detection rate missing rate false alarm rate err err err gresh admm err err err gresh admm table path computation costs gresh admm gresh admm real data example performed hierarchical variable selection california housing data pace barry dataset consists summary characteristics neighborhoods california response variable median house value neighborhood following hastie obtained eight predictor variables median income housing median age average number rooms bedrooms per household population average occupancy latitude longitude denoted medinc age avgrms avgbdrms popu avgoccu lat long respectively similar ravikumar radchenko james nuisance features generated standard gaussian random variables added make problem challenging full quadratic model enlarged dataset contains unknowns prevent getting error estimates used hierarchical procedure outer performance evaluation inner selective cvs parameter tuning managed run gresh hierarchical variable selection estimates local ridge fitting described section took approximately one half days complete experiment hours gresh median mean test errors models obtained respectively average number selected variables gresh gave median mean test errors respectively selected variables average half model size help reader get intuition selection frequencies predictors display heat maps figure two heat maps top panel include variables bottom panel shows heat maps restricted original covariates interactions according figure methods successfully removed artificially added noisy features average nuisance covariates exist models obtained gresh models heat maps gresh however neater selection results less parsimonious perhaps difficult interpret nonlinear terms gresh include interaction medinc age addition quadratic effects medinc age avgbdrms popu associated interaction terms never got selected gresh insignificance popu confirmed elaborate analysis based gradient boosting hastie main main main main figure top panel heat maps left gresh right california housing data bottom panel heat maps left gresh right restricted original variables interactions appendices proofs main theorems proof theorem denote object function define vec diag vec simplify notation sometimes write ambiguity given simple algebra minimization subject reduces min vec diag vec otherwise lower semicontinuous convex function lemma let consider following iterative procedure end end end end end end till convergence sequence sequence converge globally optimal solution prove convergence algorithm first notice following fact lemma given therefore mapping iteration algorithm use tool provided opial nonexpansive operators prove strict convergence first fix point set mapping due convexity kkt conditions mapping also asymptotically regular sense starting point property implied following lemma lemma sequence generated algorithm moreover uniformly bounded actually require optimality minimizing proof lemma satisfying makes hold need run inner loop till convergence moreover get asymptotic regularity uniform boundedness suffices terminate inner loop constant satisfying exactly solves satisfied opial conditions satisfied sequence unique limit point fixed point algorithm next prove optimality fixed point regarding problem lemma satisfies kkt conditions fixed point property established substituting vec diag vec vec diag vec exactly kkt conditions convex problem hence global minimizer problem similarly global optimality require solving exactly following use denote let arbitrary optimal solution proof lemma optimal satisfies relax construction obtain follows letting get must also global minimizer proof lemma first consider min let simple algebra last term depend based lemma difficult show global minimizer given similarly globally optimal solution given applying theorem bauschke combettes guarantees strict convergence iterates global optimality limit point proof lemma conclusion follows fact proximity operator associated convex function proof lemma optimality convexity penalties easy see follows construction hand noticing gradient respect diag vec taylor expansion yields furthermore conclusion follows proof theorem proof use denote universal constants necessarily occurrence throughout proof short short short prove theorem less restrictive condition first global minimizer convex symmetric necessarily satisfying based model assumption easy see diag vec introduce vec vec denote orthogonal projection matrix onto column space diag diag representing respectively applying triangle inequalities jec treat key term nontrivial derive two inequalities statistical analysis computational analysis first bound cardinality measures goal step show large constant supj negative high probability log log epjg reference signal satisfies strengthened log log following lines proof lemma notational convenience extend defined index set given binary matrix satisfying vec vec define given lemma given sup exp universal constants omit dependence brevity define rjg sup log log rjg log log log log log set lemma rjg exp exp log log exp follows also easy see sufficiently large occurs probability least next derive inequality based computational optimality due convexity problem stationary point diag lagrangian multipliers kkt conditions nonzero probability satisfies vec similarly letting satisfies sgn jkc ljr sgn vec sgn respectively adding equations cancel ljr sgn vec sgn vec notice sgn always squaring sides summing obtain vec vec follows inequality combining optimization inequality statistical inp equality yield bound fact log get log plugging gives jec log assume following condition holds jec vec large enough vec log log hence get log choose constants satisfying say complete proof show implies consider two cases case jec trivial suppose reverse inequality holds holds finally also plug resulting follows proof theorem follows lines details omitted proof lemma first notice fixed standard tail bound gives exp laurent massart lemma easy see sup sup sup sup denotes index set contains many elements multinomial coefficient bounded jpg ling approximation log jpg log similarly supj involve terms respectively log log ejg log log ejg applying union bound gives desired result proof theorem proof based general reduction scheme chapter tsybakov key design proper least favorable signals different situations consider two cases log log define signal subclass otherwise log small constant chosen later satisfies thus stirling approximation log log log cje log universal constant let hamming distance lemma rigollet tsybakov exists subset log log universal constants follows restricted conditional number assumption vec vec positive constant gaussian models divergence vec denoted vec denoted vec vec let assumption used therefore log log ejg combining choosing sufficiently small value apply theorem tsybakov get desired lower bound log log define signal subclass log small constant afterward treatment similar details omitted redefine otherwise otherwise jew log ejg rest follows lines proof coordinatewise error bound support recovery part show gresh estimators recover sparsity pattern true signal vec high probability result type lounici ravikumar assume stringent irrepresentable conditions mutual coherence conditions discussions see zhang assumption given satisfying following inequality holds vec jec vec theorem assume take log log suppose satisfies let global minimizer define vec vec sufficiently large constants probability least estimate satisfies vec addition minimum signal strength satisfies vec probability least recover true support signal strength conditions like must imposed interestingly compared theorem lounici hierarchical variable selection accommodate smaller signals group variable selection proof proof theorem get following inequality probability least jec log universal constant log positive constants determined also recall regularity condition vec vec log log taking choosing give vec log conclusion follows admm algorithm describe admm algorithm solving problem recall apply admm rewrite min diag mented lagrangian formed use two lagrangian multiplier matrices penalty parameter based proof theorem solve problems proximity operators details omitted full admm algorithm given follows algorithm admm algorithm inputs data regularization parameters repeat vec vec end end end convergence output references bauschke combettes algorithm two monotone operators pacific journal optimization bickel ritov tsybakov simultaneous analysis lasso dantzig selector annals statistics bickel ritov tsybakov hierarchical selection variables sparse regression borrowing strength theory powering festschrift lawrence brown institute mathematical statistics bien bunea xiao convex banding covariance matrix journal american statistical association bien taylor tibshirani lasso hierarchical interactions annals statistics boyd parikh chu peleato eckstein distributed optimization statistical learning via alternating direction method multipliers foundations trends machine learning chipman bayesian variable selection related predictors canadian journal statistics choi zhu variable selection strong heredity constraint oracle property journal american statistical association cohen cohen west aiken applied multiple analysis behavioral sciences routledge davidson erwin gene regulatory networks evolution animal body plans science hamada analysis designed experiments complex aliasing journal quality technology hao zhang interaction screening dimensional data journal american statistical association hastie tibshirani friedman elements statistical learning new york causality network learning journal machine learning research jenatton mairal obozinski bach proximal methods hierarchical sparse coding journal machine learning research laurent massart adaptive estimation quadratic functional model selection annals statistics lim hastie learning interactions via hierarchical grouplasso regularization journal computational graphical statistics lounici convergence rate sign concentration property lasso dantzig estimators electronic journal statistics lounici pontil van geer tsybakov oracle inequalities optimal inference group sparsity annals statistics mccullagh nelder generalized linear models london england chapman hall negahban ravikumar wainwright unified framework analysis decomposable regularizers statistical science nelder reformulation linear models journal royal statistical society series general nesterov gradient methods minimizing composite objective function technical report catholique louvain center operations research econometrics core opial weak convergence sequence successive approximations nonexpansive mappings bulletin american mathematical society owen robust hybrid lasso ridge regression contemporary mathematics pace barry sparse spatial autoregressions statistics probability letters peixoto hierarchical variable selection polynomial regression models american statistician peixoto property polynomial regression models american statistician radchenko james variable selection using adaptive nonlinear interaction structures high dimensions journal american statistical association ravikumar wainwright lafferty highdimensional ising model selection using logistic regression annals statistics ravikumar liu lafferty wasserman spam sparse additive models advances neural information processing systems mit press rigollet tsybakov exponential screening optimal rates sparse estimation annals statistics schmidt roux bach convergence rates inexact methods convex optimization shawetaylor zemel bartlett pereira weinberger eds advances neural information processing systems iterative algorithm fitting nonconvex penalized generalized linear models grouped predictors computational statistics data analysis learning topology dynamics large recurrent neural networks ieee transactions signal processing simon friedman hastie tibshirani sparsegroup lasso journal computational graphical statistics tibshirani regression shrinkage selection via lasso journal royal statistical society series methodological tsybakov introduction nonparametric estimation springer series statistics new york springer van geer weakly decomposable regularization penalties structured sparsity scandinavian journal statistics wei huang consistent group selection highdimensional linear regression bernoulli devlin ringquist trucco roeder screen clean tool identifying interactions association studies genetic epidemiology yuan lin model selection estimation regression grouped variables journal royal statistical society series statistical methodology zhang huang sparsity bias lasso selection linear regression annals statistics zhang sharp performance bounds least squares regression regularization annals statistics zou hastie regularization variable selection via elastic net journal royal statistical society series statistical methodology
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making deficit marathon run iztok fister suash janez iztok university maribor faculty electrical engineering computer science smetanova maribor slovenia email university maribor faculty mechanical engineering smetanova maribor slovenia may educational consultant ranchi jharkhand india distinguished professorial associate decision sciences modelling program victoria university melbourne australia university maribor faculty electrical engineering computer science smetanova maribor slovenia dated february predict final result athlete marathon run thoroughly eternal desire trainer usually achieved result weaker predicted one due objective environmental conditions well subjective factors athlete malaise therefore making deficit predicted achieved results main ingredient analysis performed trainers competition analysis search parts marathon course athlete lost time paper proposes automatic making deficit using differential evolution algorithm case study results obtained wearable athlete real marathon analyzed first experiments differential evolution show possibility using method future introduction running marathon challenging task every athlete good efficient preparation marathon complex process must performed many time competition marathon ordinary running competition epic distance roots ancient greece marathon name coined legend soldier philippides ran battle marathon athens order announce victory united greeks persian force nowadays big cities organize marathon competitions annually attract lot people athletes want finish marathon others higher goals line amateur athletes satisfied achievement around four hours competitive amateur athletes try beat magic line three hours although magic result set many years ago running marathon race sub three hours still considered good result amateur athletes however would like reach result sub three hours need proper sports training also lot knowledge race course additionally awareness characteristics athlete body crucial step evolution runners faster eat slow pace hills significant questions runners ask planning running pace making deficit time marathon run demanding comes play final results predicted sports trainers slower achieved one seconds thus experienced athletes also calculate pace kilometer race many weeks race start order predict final achievement however achieved result distinguish predicted objective well subjective factors objective factors represent environmental conditions weather temperature wind position even humidity subjective factors refer athlete conditions feelings race day nowadays predicting final result little bit easier using modern mobile technology group count professional consist also global positioning system gps heart rate measuring sensors cadence meters etc additionally comprehensive software support intelligent systems also arisen recently paper present novel solution making deficit marathon run based data collected worn athlete run primary aim study help athletes achieve predicted final results seconds show parts predefined course could improve intermediate pace predicted results could reached solution implemented using differential evolution calculates improvement intermediate pace based configuration running course improvements considered constraints base traditional rule marathon run seconds caught running course flat secondary aim approach also help athletes analysis already performed marathon race case study focus situation athlete tries achieve magic marathon time three hours paper structured remainder follows section making deficit marathon run formulated constraint satisfaction problem csp section iii devoted differential evolution section illustrates case study case athlete three hearts marathon taken consideration experiments presented second part section paper concluded section work done reviewed directions outlined research problem formulation making deficit time formulated solution specified vector problem variables representing intermediate pace deficit observed kilometer additionally variable bound constraint attached problem variable limits proper values variable interval lower upper bounds respectively case marathon race length whole course equals length vector consequently elements problem defined generalized form follows max subject denotes deficit time must made relation valid thus following inequality constraints must satisfied ensures problem solvable although upper bounds set negation corresponding lower bounds general upper bounds fixed zero mean negative values intermediate pace deficits allowed study lower bound values calculated configuration course according following assumptions flat sec alt downhill sec uphill sec alt average altitude obtained appropriate kilometer speculation behind assumptions speed course flat downward retain speed running upward quantitative values estimation deficit time also taken real marathon practice holds deficit time maximum seconds one kilometer course flat iii differential evolution differential evolution evolutionary algorithm appropriate continuous combinatorial optimization introduced storn price consists vectors representing candidate solutions follows element solution interval denote lower upper bounds variable respectively variation operator supports differential mutation differential crossover particular differential mutation selects two solutions randomly adds scaled difference third solution mutation expressed follows denotes scaling factor positive real number scales rate modification randomly selected values interval note typically interval used community differential crossover uniform crossover employed trial vector built parameter values copied two different solutions mathematically crossover expressed follows randj jrand otherwise controls fraction parameters copied trial solution note relation jrand ensures trial vector different original solution differential selection fact generalized selection expressed mathematically follows otherwise technical sense crossover mutation performed several ways differential evolution therefore specific notation used describe varieties methods also strategies generally example denotes base vector selected randomly vector difference added number modified parameters mutant vector follows binomial distribution making deficit time using modifying original solve problem making deficit time marathon run let call new variant marathon simply mde relatively simple implements variable bounds implicitly therefore main problem remains set appropriate variable bounds elements solution vectors answer question given intermediate pace athlete taken consideration intermediate paces nowadays obtained using mobile worn marathon runner run four phases modifying original making deficit time marathon run follows defining objective preparing bounds problem variables mapping solution decision space problem space evaluating solution problem space indeed defining objective refers expression define total deficit time must made preparing bounds problem variables bases tracked data referring information altitude overcome marathon runner kilometer according data appropriate boundary values calculated according mapping solution decision space counterpart problem space straightforward obeys following equation multiplying ensures given result seconds extended significant digit milliseconds decimal point finally fitness function presented used study order simplify solving problem last element vector represent whole kilometer excluded optimization taken uphill part although downhill practice case case study three hearts marathon three hearts marathon organized annually city radenci slovenia since years organizers changed marathon course slightly marathon course consisted two kilometer long laps seen figure goal fig three hearts marathon course athlete study run marathon sub hours however run marathon time intermediate pace average minutes per kilometer figure better less athlete plan run first kilometers little bit faster others although course flat average focus enough fig average pace kilometer ascents slight uphill large enough lose power especially bad weather conditions time still good enough achieving result hours last kilometers pretty hard overcome formerly broken nail abrasions heat many blisters therefore athlete one minute slower finish planned fig ascents descents course three hearts marathon proposed approach making deficit time works follows objective expressed min sec altitude graph marathon course problem variable bounds prepared presented figure graph figure presents differences altitudes meters according run distances kilometers divided two parts denoting vertical line although parts must theory assumption hold practice gps technology used far ideal however tracked data accurate enough purposes study results analysis altitude data follows gps receiver detected flat parts uphill downhills last meters downward essentially difference number uphill downhills consequence errors gps receiver hand serious uphill downhills marathon course seen maximum minimum differences altitude meters fact difference average ascent average descent values end kilometer higher meter declared uphill downhill study bounds problem variables prepared based analysis altitude data according results phase illustrated table columns lower bounds upper bounds sum lower bounds problem variables whereby condition problem solvability satisfied last two phases mapping evaluating solution performed using mde experiments mde coded python programming language strategy employed input algorithm data obtained three hearts marathon dimension problem population size set scaling factor crossover rate lower upper bounds set interval algorithm terminated either fitness function fulfilled objective thus constraints satisfied times number fitness function evaluations exceeded one run mde performed study obtained results mde algorithm presented table consists six columns first column counts kilometers marathon run second illustrates actual intermediate pace already presented figure third column depicts predicted intermediate pace obtained decrementing actual intermediate pace intermediate pace deficit found first run presented column four last values obtained conducting mde algorithm already mentioned last two columns present lower upper bounds table results making deficit three hearts marathon distance actual pace sec predicted pace sec difference lower bounds upper bounds sec sec sec total discussion first glance conducted experiments contrast principles stochastic optimization interested mean values obtained number runs strictly speaking true results obtained aforementioned point view however aim study twofold one hand show mde algorithm employed making deficit marathon race hand prove many solutions problem assumptions justified finding numerous feasible solutions one run question arises could say solutions found really optimal however answer question must answered professionals sports trainers coaches together athletes conclusion paper reports first successful application mde making deficit marathon run based history data obtained corresponding marathon course making deficit marathon run complex task usually performed professional trainers even athletes defined problem constraint problem solved mde preliminary experiments shown proposed approach may used however still many open directions research problem determine appropriate solution specific athlete huge set solutions line context dependent information need accumulated basis solution proposed mde algorithm automatically however extending input data heart rate monitor might also increase prediction possibilities proposed algorithm das mullick suganthan recent advances differential updated survey swarm evolutionary computation eiben smith introduction evolutionary computing volume springer fister fister fong fister widespread mobile devices applications drafting detection triathlons journal emerging technologies web intelligence fister suganthan perc fister computational intelligence sports challenges opportunities within new research domain applied mathematics computation fister yang fister brest fister brief review algorithms optimization vestnik storn price differential simple efficient heuristic global optimization continuous spaces journal global optimization
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hybrid precoder combiner design low resolution phase shifters mmwave mimo systems oct zihuan wang student member ieee ming senior member ieee qian liu member ieee lee swindlehurst fellow ieee wave mmwave communications considered key technology next generation cellular systems networks advances providing wider bandwidth current wireless networks economical hybrid precoding combining transceivers often proposed mmwave massive mimo systems overcome severe propagation loss mmwave channels one major shortcoming existing solutions lies assumption infinite phase shifters pss realize analog beamformers however pss typically adopted practice reduce hardware cost power consumption motivated fact paper investigate practical design hybrid precoders combiners pss mmwave mimo systems particular propose iterative algorithm successively designs analog precoder combiner pair data stream aiming conditionally maximizing spectral efficiency digital precoder combiner computed based obtained effective baseband channel enhance spectral efficiency effort achieve even large antenna array also investigate design hybrid beamformers resolution binary pss present novel binary analog precoder combiner optimization algorithm quadratic complexity number antennas proposed hybrid beamforming design extended multiuser mimo communication systems simulation results demonstrate performance advantages proposed algorithms compared existing lowresolution hybrid beamforming designs particularly onebit resolution scenario index wave mmwave communications hybrid precoder mimo phase shifters quantization ntroduction past decade witnessed exponential growth data traffic along rapid proliferation wireless dezihuan wang ming school information communication engineering dalian university technology dalian liaoning china wangzihuan mli qian liu school computer science technology dalian university technology dalian liaoning china qianliu swindlehurst center pervasive communications computing university california irvine usa also institute advanced study technical university munich munchen germany swindle paper supported national natural science foundation china grant fundamental research funds central universities grant vices flood mobile traffic significantly exacerbated spectrum congestion current frequency bands therefore stimulated intensive interest exploiting new spectrum bands wireless communications millimeter wave mmwave wireless communications operating frequency bands ghz demonstrated promising candidate fundamentally solve spectrum congestion problem however challenges always come along opportunities mmwave communications still need overcome several technical difficulties deployment negative result increase carrier frequency propagation loss mmwave bands much higher conventional frequency bands ghz due atmospheric absorption rain attenuation low penetration positive perspective smaller wavelength mmwave signals allows large antenna array packed small physical dimension aid techniques massive mimo systems large antenna array provide sufficient beamforming gain overcome severe propagation loss mmwave channels also enables simultaneous transmission multiple data streams resulting significant improvements spectral efficiency mimo systems operating conventional cellular frequency bands precoder combiner completely realized digital domain adjusting magnitude phase baseband signals however conventional schemes require large number expensive radio frequency chains converters adcs converters dacs since mmwave communication systems operate much higher carrier frequencies wider bandwidths enormous cost power consumption required chains make adoption precoding combining schemes impractical mmwave systems recently economical hybrid precoders combiners advocated promising approach tackle issue hybrid precoding approaches adopt large number phase shifters pss implement analog precoders compensate severe mmwave bands small number chains dacs realize lowdimensional digital precoders provide necessary flexibility perform advanced techniques investigation hybrid precoder combiner design attracted extensive attention recent years potential energy efficiency mmwave mimo communications major challenges designing hybrid precoders practical constraints associated analog components requirement analog precoding implemented constant modulus pss thus hybrid precoder design typically requires solution various matrix factorization problems constant modulus constraints particular popular solution maximize spectral efficiency transmission minimize euclidean distance hybrid precoder fulldigital precoder hybrid precoder design partiallyconnected architectures also studied due special characteristics mmwave channels codebookbased hybrid precoder designs commonly proposed columns analog precoder selected certain candidate vectors array response vectors channel discrete fourier transform dft beamformers extensions hybrid beamformer design multiuser mmwave mimo systems also investigated aforementioned existing hybrid precoder combiner designs generally assume infinite pss used implementing analog beamformers order achieve satisfactory performance close scheme however implementing pss mmwave frequencies would significantly increase energy consumption complexity required hardware circuits obviously impractical employ pss mmwave systems realworld analog beamformers implemented lowresolution pss consequently important research direction exploration signal processing techniques hybrid architectures mitigate loss beamforming accuracy due pss straightforward approach obtain beamformer design analog beamformer first directly quantize phase term finite set however solution becomes inefficient pss low resolution alternative solution hybrid beamforming pss design however pss size codebook small resulting performance satisfactory sohrabi proposed iteratively design hybrid precoder maximize spectral efficiency however performance algorithm often suffers quantized pss applied paper first consider problem designing hybrid precoders combiners pss mmwave mimo system objective proposed algorithm minimize performance loss caused pss maintaining low computational complexity achieve goal propose successively design analog precoder combiner pair data stream aiming conditionally maximizing spectral efficiency iterative phase ing algorithm introduced implement analog precoder combiner pair digital precoder combiner computed based obtained effective baseband channel enhance spectral efficiency note power consumption cost proportional resolution example resolution mmwave frequencies requires resolution needs effort achieve maximum hardware efficiency also investigate design hybrid beamformers resolution binary pss inspired findings present binary analog precoder combiner optimization algorithm approximation equivalent channel algorithm quadratic complexity number antennas achieve almost performance optimal exhaustive search method finally investigation hybrid precoders combiners extended multiuser mmwave mimo systems numerical results simulation section demonstrate proposed algorithms offer performance improvement compared existing lowresolution hybrid beamforming schemes especially resolution scenario notation following notation used throughout paper boldface letters indicate column vectors matrices respectively denote transpose operations respectively represents statistical expectation extracts real part complex number sign denotes sign operator angle represents phase complex number indicates identity matrix denotes set complex numbers denotes determinant matrix denotes cardinality set kak magnitude norm scalar vector respectively kakf denotes frobenius norm matrix finally adopt matrix indexing notation denotes column matrix denotes element row column matrix denotes element vector ystem odel roblem ormulation mmwave mimo system model first consider mmwave mimo system using hybrid precoder combiner pss illustrated fig transmitter employs antennas ntrf chains simultaneously transmit data streams receiver equipped antennas nrrf chains ensure efficiency mmwave communication limited number chains number data streams number chains constrained ntrf nrrf transmitted symbols first processed baseband digital precoder fbb cnt domain via chains precoded analog precoder frf dimension ntrf baseband digital precoder fbb enables amplitude phase modifications elements analog precoder fig mmwave mimo system using hybrid precoder combiner frf implemented pss constant amplitude quantized phases frf phase quantized number bits control phase denote constraint set analog precoder follows frf obviously larger number bits leads finer resolution pss potentially better performance also results higher hardware complexity power consumption transmitted signal written following form frf fbb symbol vector ssh ins represents transmit power power constraint enforced normalizing fbb kfrf fbb consider propagation channel yields following received signal hfrf fbb ins wbb wrf hfrf fbb frf wrf wbb wrf nrrf analog combiner whose elements constraint frf wrf thus wrf wrf wrf wbb noise covariwhere wbb ance matrix combining aim jointly design digital beamformers fbb wbb well analog beamformers frf wrf maximize spectral efficiency arg max wbb wrf received signal vector channel matrix inr complex gaussian noise vector corrupting received signal receiver employs analog combiner implemented pss digital combiner using nrrf chains process received signal signal spatial processing form wrf hfrf fbb wbb wrf wbb hardware constraint aim jointly design hybrid precoder combiner mmwave mimo system gaussian symbols transmitted mmwave mimo channel achievable spectral efficiency given frf wrf kfrf fbb kwrf wbb obviously optimization problem nphard problem next section attempt decompose original problem series seek solution satisfactory performance iii esolution ybrid recoder ombiner esign wbb nrrf digital baseband combiner combiner matrices normalized kwrf wbb problem formulation consider practical scenario pss reduce power consumption complexity simplify joint hybrid precoder combiner design objective problem decomposed two separate optimizations first focus joint design analog precoder frf combiner wrf effective baseband channel associated obtained optimal analog precoder combiner digital precoder fbb combiner wbb computed maximize spectral efficiency analog precoder combiner design observe assumption high snr achievable spectral efficiency approximated wbb wrf hfrf fbb frf wrf wbb snr mmwave systems typically low snr high enough justify approximation addition verified mimo systems optimal analog beamformers approximately orthogonal frf intrf enables assume fbb normalization factor similarly wrf wrf wbb ins wbb wbb ins wbb let simplified hfrf wrf wrf hfrf follows since square matrices therefore analog precoder combiner design pss approximately reformulated frf wrf arg max wrf hfrf frf wrf unfortunately optimization problem still exponential complexity therefore propose decompose difficult optimization problem series chain pair considered one one analog precoder combiner pair successively designed particular define singular value decomposition svd unitary matrix unitary matrix rectangular diagonal matrix singular values due sparse nature mmwave channel matrix typically low rank particular effective rank channel serves upper bound number data streams channel support thus assume channel well approximated retaining strongest components objective converted wrf frf wrf hfrf next write analog precoding combining matrices frf frf wrf wrf respectively frf wrf analog precoder combiner pair data stream furthermore denote frf precoding matrix excluding precoder vector frf wrf combining matrix excluding lth combiner vector wrf formulation transformed presented top following page small scalar assure invertibility thus objective reformulated wrf hfrf hfrf wrf wrf frf define channel matrix frf wrf according frf wrf known problem reformulated finding corresponding precoder frf combiner wrf pair conditionally maximize achievable spectral efficiency frf wrf arg max wrf frf frf wrf motivates propose iterative algorithm starts appropriate initial precoding combining matrices successively designs frf wrf according updated algorithm converges complexity obtaining optimal solution iteration reduced still high practically solve problem follows present iterative phase matching algorithm searches conditionally optimal phase element analog precoder frf combiner wrf specifically first design analog precoder frf assuming analog combiner wrf fixed let phase element analog precoder frf let phase element analog combiner wrf temporarily remove discrete phase constraint optimal continuous phase element analog precoder frf given following proposition whose proof provided appendix proposition given phases analog combiner wrf phases analog precoder frf optimal continuous phase element analog wrf frf log log log log precoder frf frf frf wrf frf wrf wrf frf frf wrf frf wrf frf wrf frf wrf frf wrf frf wrf ins log log frf frf wrf ins frf frf wrf wrf angle finding optimal continuous phase reconsider discrete phase constraint find optimal phase quantization arg min similarly analog precoder frf determined optimal continuous phase element wrf angle algorithm iterative phase matching algorithm lowresolution analog precoder combiner design input output wrf initialize frf wrf obtain frf wrf wrf update frf wrf convergence obtain quantized phase end obtain quantized phase end end construct frf wrf end wrf wrf construct frf goto step convergence frf wrf optimal phase obtained arg min motivated iterative procedure design precoder frf combiner wrf straightforward appropriate initial design precoder frf finding conditionally optimal phases obtained design combiner wrf finding conditionally optimal phases alternate designs frf wrf iteratively obtained phase element frf wrf change convergence achieved note since precoder combiner design step objective function monotonically thus proposed algorithm guaranteed converge least locally optimal solution summarize proposed joint analog precoder combiner design algorithm digital precoder combiner design analog pairs detere mined obtain effective baseband channel frf wrf wrf baseband precoder combiner design define svd effective baseband channel unitary matrices diagonal matrix singular values enhance spectral efficiency baseband digital precoder combiner employed finally baseband precoder combiner normalized wbb fbb wbb kwrf joint optimization problem decoupled individually designing analog precoder frf combiner wrf frf arg max frf esolution nalog recoder ombiner esign wrf arg max wrf frf wrf previous section proposed novel hybrid beamformer design maximizing spectral efficiency mmwave mimo system analog precoder combiner implemented pss order achieve maximum hardware efficiency section focus design analog precoders combiners using resolution binary pss maximally reduce power consumption simplify hardware complexity although iterative phase matching algorithm proposed previous section also applied simpler approach possible case therefore section present efficient resolution analog beamformer design achieve good performance much lower complexity follow procedure hybrid beamforming design proposed previous section modify optimization problem attempts determine analog precoder combiner pair particularly reformulate analog beamformer design problem constraint resolution pss wrf frf max frf wrf arg frf wrf optimization problem solved exhaustive search exponential complexity would possible large antenna arrays therefore following attempt develop efficient resolution beamformer design polynomial complexity number antennas first define svd left right singular vectors respectively largest singular value objective rewritten frf wrf frf utilize approximation keeping strongest term optimization function approximated frf wrf max wrf frf arg frf wrf two optimization problems require singular vectors associated largest singular value quickly obtained power iteration algorithm instead complete svd calculation however solving exhaustive search still exponential complexity number antennas order reduce complexity without significant loss performance propose construct smaller dimension candidate beamformer set optimal beamformer found linear complexity following present algorithm precoder design example combiner design follows procedure introduce auxiliary variable reformulate optimization problem frf arg max frf frf arg max frf frf cos denotes phase obviously given corresponding binary precoder maximizes frf sign cos conditionally optimal frf given shown show always construct set candidate binary precoders guarantee frf maximization carried set candidates without loss performance first define angles map angles rearranged ascending order periodicity cosine function maximization problem respect carried interval length construct optimal must located one since full interval length therefore optimization problem solved examining separately assuming optimal corresponding optimal binary obtained sign cos form given inverse sorting maps rearrange corresponding elements obtain based relationship defined achieve conditionally optimal precoder case since optimal must located one subintervals obtain conditionally optimal precoders examining construct candidate precoder set must contain optimal precoder frf therefore without loss performance problem transformed equivalent maximization task set frf frf arg max frf linear complexity similarly also construct candidate analog combiner set obtain wrf procedure solution returned based approximation equivalent channel approximation may cause performance degradation revisit original problem therefore order enhance performance propose jointly select precoder combiner candidate sets frf wrf frf arg max frf wrf may return better solution quadratic complexity analog beamformer design resolution pss summarized algorithm ybrid recoder ombiner esign ultiuser wave mimo ystems section consider mmwave multiuser mimo uplink system extend hybrid precoder combiner designs proposed previous sections multiuser system algorithm resolution analog beamformer design input output frf wrf calculate svd define angles map ascending order obtain obtain based inverse mapping obtain end construct construct similar procedure steps find optimal frf wrf system model problem formulation consider multiuser mmwave mimo uplink system presented fig equipped antennas nrf chains simultaneously serves mobile users due power consumption hardware limitations mobile user antennas single chain transmit one data stream assume number chains equal number users nrf let frf analog precoder user element frf constant magnitude lowt resolution discrete phases frf transmitted signal user precoding formulated frf symbol user user transmit power let cnr denote uplink channel user received signal written frf inr complex gaussian noise first applies analog combining matrix wrf wrf wrf process received signal analog combiner wrf corresponding user also implemented pss wrf baseband digital combiner wbb employed retrieve information user let wrf wbb denote hybrid combiner corresponding user combining process estimated symbol user expressed wkh frf wkh frf wkh fig multiuser mmwave mimo system using hybrid precoder combiner given received signal ratio sinr user expressed wkh frf wkh frf achievable multiuser uplink system log aim jointly design analog precoders combiners implemented pss well digital combiners maximize uplink multiuser system wrf wbb frf arg max frf log wrf hybrid precoder combiner design obviously optimization problem directly solved thus adopt approach similar propose successively design analog beamformer pair user aiming enhancing channel gain well suppressing interference baseband combiner calculated mitigate interference maximize particular first user analog precoder combiner pair designed maximize corresponding channel gain formulated follows wrf frf arg max wrf frf frf wrf analog precoder combiner design problem efficiently solved algorithm presented sec pss utilized algorithm proposed sec resolution pss available analog precoders frf combiners wrf remaining users successively designed iterative procedure iteration attempt find analog beamformer pair suppresses interference users whose analog beamformers already determined achieve goal channel user whose combiner calculated projected onto space orthogonal collection previously designed analog combiners approach leads orthogonal analog combiners suppress interference specifically design user analog beamformer pair first extract orthonormal components viously determined analog combiners wrf procedure wrf wrf note wrf wrf analog combiner calculated first user combiner components removed user channel obtain modified channel inr analog beamfinally based modified channel former pair user found solving following optimization using algorithms proposed previous frf wrf finding analog beamformers users effective baseband channel user obtained hek wrf frf minimum mean square error mmse baseband digital combiner user employed suppress interference wbb wrf wrf hek hek proposed hybrid precoder combiner design multiuser mmwave systems summarized algorithm wrf wrf inr obtain wrf frf solving max wrf frf arg frf wrf frf wrf wrf end obtain digital combiners wbb wrf wbb wrf hek imulation esults section provide simulation results proposed joint hybrid precoder combiner designs lowresolution pss mmwave systems well multiuser mmwave systems mmwave channels expected sparse limited number propagation paths simulations adopt geometric channel model paths particular mmwave channel formulated snr fig spectral efficiency versus snr ntrf nrrf spectral efficiency algorithm hybrid precoder combiner design multiuser mmwave systems input output frf wrf wbb obtain wrf frf solving arg max wrf cdm cdm hbf hbf proposed proposed spectral efficiency sections wrf frf arg max wrf frf cdm cdm hbf hbf proposed proposed number antenna fig spectral efficiency versus number antenna ntrf nrrf snr independent identically distributed complex gains propagation path ray angles departure aod angles arrival aoa respectively finally array response vectors depend antenna array geometry assume commonly used uniform linear arrays ulas employed transmit antenna array response vector receive antenna array response vector written sin sin sin sin respectively signal wavelength distance antenna elements following simulations consider environment scatterers exhaustive search proposed spectral efficiency spectral efficiency cdm cdm hbf hbf proposed proposed spectral efficiency cdm hbf proposed snr fig spectral efficiency versus number iteration ntrf nrrf snr number iterations resolution bits fig spectral efficiency versus resolution pss ntrf nrrf snr transmitter receiver antenna spacing simulation results mmwave system first consider mmwave communication system transmitter receiver equipped ulas number chains transmitter receiver ntrf nrrf number data streams also assumed fig shows average spectral efficiency versus snr channel realizations evaluate spectral efficiency algorithm proposed sec iii case resolution pss algorithm proposed sec case resolution pss comparison purposes also plot spectral efficiency two lowresolution hybrid beamformer designs coordinate descent method cdm algorithm hybrid beamforming hbf algorithm best knowledge fig spectral efficiency versus snr algorithm achieves best performance lowresolution pss existing literature performance fully digital approach using beamforming hybrid beamforming scheme pss using phase extraction algorithm also included performance benchmarks fig illustrates proposed algorithm outperforms competitors particularly case resolution pss moreover observed proposed algorithm achieves performance close optimal beamforming hybrid beamforming pss additional simulation validation fig illustrates spectral efficiency versus number antennas similar conclusions drawn order illustrate convergence proposed algorithm show spectral efficiency versus number iterations fig also includes algorithms comparison observed proposed algorithms converge faster two iterative schemes highly favorable property fig show spectral efficiency function illustrate impact resolution pss spectral efficiency expected increasing resolution improve system performance using bits sufficient closely approach performance ideal unquantized case beyond additional cost complexity associated using pss warranted given marginal increase spectral efficiency moreover proposed algorithms outperform two beamforming methods resolutions examine impact approximations used deriving proposed resolution hybrid beamformer scheme fig compare optimal exhaustive search approach number antennas transmitter receiver chosen number data streams relatively simple case examined due exponential complexity exhaustive search method see fig spectral efficiency achieved three algorithms approaches size beamsteering codebook set quantization bits fairness comparison observed fig proposed lowresolution hybrid beamforming design outperforms three algorithms using resolution pss moreover performance resolution pss also comparable fig shows versus number users fig see proposed algorithm resolution pss always outperforms algorithms furthermore even resolution pss proposed algorithm still achieve competitive performance compared approaches proposed proposed snr vii onclusions fig spectral efficiency versus snr ntrf proposed proposed paper considered problem hybrid precoder combiner design mmwave mimo systems lowresolution quantized pss proposed efficient iterative algorithm successively designs analog precoder combiner pair data stream digital precoder combiner computed based obtained effective baseband channel enhance spectral efficiency design hybrid beamformers multiuser mimo communication systems also investigated simulation results verified effectiveness proposed algorithms particularly scenarios onebit resolution phase shifters used ppendix roof roposition number users optimization problem equivalently formulated fig spectral efficiency versus ntrf snr proposed algorithm optimal exhaustive search method suggesting proposed hybrid beamforming algorithm resolution pss provide optimal performance simulation results multiuser mmwave system next evaluate performance proposed lowresolution beamformer algorithm multiuser uplink system assume users equipped antennas one chain transmit single data stream antennas nrrf chains fig illustrates versus snr various hybrid beamformer designs particular include three multiuser hybrid beamforming approaches comparison hybrid beamforming hybrid beamforming iii iterative hybrid beamforming max discarding constant coefficient transformed max since term involve summation index put outside first summation resulting max obviously optimal value makes phases first second term equal obtain largest amplitude proved eferences khan introduction mobile broadband systems ieee commun vol june rappaport sun mayzus zhao azar wang wong schulz samimi gutierrez millimeter wave mobile communications cellular work ieee access vol lee swindlehurst ayanoglu heydari capolino massive mimo next wireless revolution ieee commun vol rappapport maccartney samimi sun wideband propagation measurements channel models future wireless communication system design ieee trans vol heath rangan roh sayeed overview signal processing techniques millimeter wave mimo systems ieee sel topics signal vol april shen zhang letaief alternating minimization algorithms hybrid precoding millimeter wave mimo systems ieee sel topics signal vol april rusu heath low complexity hybrid precoding strategies millimeter wave communication systems ieee trans wireless vol rusu heath hybrid precoders combiners mmwave mimo systems power proc ieee global communication conference globecom washington chen iterative hybrid transceiver design algorithm millimeter wave mimo systems ieee wireless commun vol june dong hybrid processing massive mimo systems via matrix decomposition ieee trans signal vol gao dai han heath hybrid analog digital precoding mmwave mimo systems large antenna arrays ieee sel areas vol april dai gao quan han hybrid analog digital precoding downlink mmwave massive mimo systems proc ieee int conf commun icc london june huang transceiver design hybrid architecture mimo systems ieee access vol ayach rajagopal heath spatially sparse precoding millimeter wave mimo systems ieee trans wireless vol mar alkhateeb ayach leus heath channel estimation hybrid precoding millimeter wave cellular systems ieee sel topics signal vol rusu heath low complexity hybrid sparse precoding combining millimeter wave mimo systems ieee int conf commun icc london june chen efficient beamforming algorithm massive mimo systems ieee trans veh appear gao dai yuen wang beamforming based tabu search algorithm massive mimo systems ieee trans veh vol july han rowell antenna systems hybrid analog digital beamforming millimeter wave ieee commun vol masouros hybrid mumimo transmission virtual path selection ieee commun vol liang dong hybrid precoding massive multiuser mimo systems ieee wireless commun vol masouros hybrid precoding combining design mimo based svd ieee int conf commun icc paris france may kim lee hybrid processing communication systems mimo interference channels ieee trans veh vol june alkhateeb leus heath limited feedback hybrid precoding millimeter wave systems ieee trans wireless vol nguyen hybrid mmse precoding mmwave multiuser mimo systems ieee int conf commun icc kuala lumpur malaysia may wang tian liu iterative hybrid precoder combiner design mmwave multiuser mimo systems ieee commun vol july bogale haghighat vandendorpe number chains phase shifters scheduling design hybrid beamforming ieee trans wireless vol may poon taghivand supporting enabling circuits antenna arrays wireless communications proc ieee vol july rusu alkhateeb hybrid mimo architectures millimeter wave communications phase shifters switches ieee access vol march chen hyrbrid beamforming discrete phase shifters massive mimo systems ieee trans veh appear sohrabi hybrid beamforming phase shifters mimo systems proc ieee workshop signal process adv wireless commun spawc stockholm sweden june sohrabi hybrid digital analog beamforming design antenna arrays ieee sel topics signal vol april karystinos pados adaptive design binary spreading codes ieee trans inf theory vol golub van loan matrix computations baltimore usa jhu press
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phaseless compressive sensing using partial support information dec zhiyong jun department mathematics mathematical statistics university sweden december abstract study recovery conditions weighted minimization signal reconstruction phaseless compressive sensing measurements partial support information available strong restricted isometry property condition provided ensure stable recovery moreover present weighted null space property sufficient necessary condition success phaseless recovery via weighted minimization numerical experiments conducted illustrate results keywords phaseless compressive sensing partial support information strong restricted isometry property weighted null space property introduction compressive sensing aims recover unknown signal underdetermined linear measurements see comprehensive view known phase retrieval phaseless compressive sensing phase information phaseless compressive sensing problem recently attracted considerable research interests many algorithms proposed solve problem existing literature include name specifically goal phaseless compressive sensing recover unimodular scaling constant noisy magnitude measurements measurement matrix noise term sparse compressible stable recovery guaranteed solving following minimization problem min subject provided measurement matrix satisfies strong restricted isometry property srip noiseless case first sufficient necessary condition presented proposing new version null space property phase retrieval problem paper generalize existing theoretical framework phaseless compressive sensing incorporate partial support information consider case estimate support signal available follow similar notations arguments arbitrary signal let best approximation minimizes vectors let support let support estimate subset cardinality parameter determines ratio size estimated support size actual support corresponding author support parameter determines ratio number indices support accurately estimated size prior support information adopt weighted minimization min subject incorporate present srip condition weighted null space property condition guarantee success recovery via weighted minimization problem paper organized follows section introduce definition srip present stable recovery condition tool section sufficient necessary weighted null space property condition real sparse noise free phase retrieval given section numerical experiments presented illustrate theoretical results finally section devoted conclusion throughout paper vector denote norm kxkp weighted norm matrix denotes norm set denote cardinality vector called entries nonzero supp denotes index set nonzero entries denote index set matrix index set denote rows indices kept srip recover sparse signals via minimization classical compressive sensing setting introduced notion restricted isometry property rip established sufficient condition say matrix satisfies rip order exists constant vectors cai zhang proved rip order guarantee exact recovery noiseless case stable recovery noisy case via minimization condition sharp see details recently chen generalized sharp rip condition weighted minimization problem partial support information incorporated first present following useful lemma extension result lemma let suppose satisfies rip order max kaz remark note solution weighted minimization problem min subject kaz kaz therefore lemma extension theorem letting proof follows almost procedure proof theorem section via replacing letting order repeat leave details addition result also generalizes lemma special case noise term lemma play crucial role establishing stable phaseless recovery result via weighted minimization later address phaseless compressive sensing problem stronger version rip needed definition provided follows definition srip say matrix strong restricted isometry property srip order bounds min kai max kai holds vectors say strong lower restricted isometry property order bound lower bound holds similarly say strong upper restricted isometry property order bound upper bound holds next present conditions stable recovery via weighted minimization using srip theorem let adopt notations lemma assume satisfies srip order bounds max solution satisfies min constants defined lemma remark proved gaussian matrices log satisfy srip order high probability thus stable recovery result achieved using gaussian measurement matrix appropriate number measurements remark note weight assuming exactly theorem reduces theorem satisfies srip order bounds max signal arg similarly let noise term theorem goes theorem remark sufficient condition theorem identical theorem theorem constants see theorem addition sufficient condition theorem weaker theorem theorem case constants illustrate constants change set max different values figure plots set plot set plots fix note shows decreases increases means sufficient condition becomes weaker increases sufficient condition becomes stronger increases increases instance support estimate accurate standard minimization opposite conclusion holds case addition increases constant decreases meanwhile constant smaller proof theorem solution divide index set two subsets sign haj sign haj sign haj haj implies kat kat either use fact kat obtain kat weights weights weights figure comparison constants various plots set plot set plots fix since satisfies srip order bounds max therefore definition srip implies satisfies rip order max thus using lemma similarly obtain corresponding result proof theorem completed weighted null space property section consider noiseless weighted minimization problem min subject denote kernel space denote vector space definition matrix satisfies null space property order nonzero holds kht kht complementary index set restriction remark obviously weight weighted null space property reduces classical null space property according specific setting expression equivalent kht kht khg kht see arguments known signal recovered via weighted minimization problem measurement matrix weighted null space property order state follows see lemma given every vector holds arg min satisfies null space property order next extend lemma following theorem phaseless compressive sensing signal reconstruction theorem following statements equivalent arg min every holds nonzero satisfying remark theorem reduces theorem since otherwise expression equivalent proof theorem proof follows proof theorem minor modifications first show assume false exist nonzero set obviously since either hai hai words note otherwise would contradiction follows obtain contradiction thus holds next prove let fixed set consider following weighted minimization problem min subject solution denoted claim exists may exist equality holds prove claim assume first note statement implies classical weighted null space property order see nonzero set therefore statement implies consequence lemma exist hai hai similarly next nothing prove assume exist set set obviously furthermore statement proves proof completed simulations section present simple numerical experiments illustrate benefits using weighted minimization recover sparse compressible signals partial prior support information available phaseless compressive sensing case order facilitate computation follow nonstandard noise model ati xxt noise term weighted minimization goes subject min adopt compressive phase retrieval via lifting cprl algorithm developed solve phaseless recovery problem using lifting technique problem rewritten semidefinite program sdp specifically given ground truth signal let xxt induced semidefinite matrix denote ati linear operator weight matrix diag phaseless vector recovery problem cast following matrix recovery problem min subject rank course still problem due rank constraint lifting approach addresses issue replacing rank leads sdp min subject design parameter estimate finally found computing decomposition recovered matrix via singular value decomposition recovery performance assessed average reconstruction signal noise ratio snr experiments snr measured given snr min true signal recovered signal experiments fix parameter experiments measurements noisy set noise sparse case first consider case exactly sparse ambient dimension fixed sparsity sparse signals generated choosing nonzero positions uniformly random choosing nonzero values standard normal distribution nonzero positions recovery done via using support estimate size figure shows recovery performances different increasing number measurements noise free noisy cases observed best recovery achieved small whereas results lowest snr cases hand performance recovery algorithms better large small case results lowest snr performance gaps different particularly large seems medium achieves best recovery noise free case perfect recovery achieved long number measurements large enough also expected settings comparing noise free case lower snr noisy case findings largely consistent theoretical results provided section snr snr snr number measurements number measurements number measurements snr snr snr number measurements number measurements number measurements figure performance weighted recovery terms snr averaged experiments sparse signals varying number measurements left right noise free compressible case generate signal whose coefficients decay like kind signal sparse well approximated exactly sparse signal experiment set use best approximation fix sparse case phaseless recovery results presented figure shows average mediate value results best recovery general smaller favours better reconstruction results opposite conclusion holds case therefore expected behaviors occur exactly sparse case also occur compressible case snr snr snr number measurements number measurements number measurements number measurements snr snr snr number measurements number measurements figure performance weighted recovery terms snr averaged experiments compressible signals varying number measurements left right noise free conclusion paper established sufficient srip condition sufficient necessary weighted null space property condition phaseless compressive sensing using partial support information via weighted minimization conducted numerical experiments illustrate theoretical results problems left future work consider signal reconstruction case challenging generalize present results signal case besides interesting construct measurement matrix satisfying weighted null space property given directly acknowledgements work supported swedish research council grant references cai zhang sparse representation polytope recovery sparse signals matrices ieee transactions information theory candes eldar strohmer voroninski phase retrieval via matrix completion siam review candes soltanolkotabi phase retrieval via wirtinger flow theory algorithms ieee transactions information theory candes strohmer voroninski phaselift exact stable signal recovery magnitude measurements via convex programming communications pure applied mathematics candes tao decoding linear programming ieee transactions information theory chen recovery signals high order rip condition via prior support information arxiv preprint chen candes solving random quadratic systems equations nearly easy solving linear systems advances neural information processing systems eldar kutyniok compressed sensing theory applications cambridge university press foucart rauhut mathematical introduction compressive sensing new york usa friedlander mansour saab yilmaz recovering compressively sampled signals using partial support information ieee transactions information theory gao wang stable signal recovery phaseless measurements journal fourier analysis applications gao phaseless recovery using gaussnewton method ieee transactions signal processing mansour saab recovery analysis weighted using null space property applied computational harmonic analysis netrapalli jain sanghavi phase retrieval using alternating minimization ieee transactions signal processing ohlsson yang dong sastry compressive phase retrieval squared output measurements via semidefinite programming arxiv preprint shechtman beck eldar gespar efficient phase retrieval sparse signals ieee transactions signal processing voroninski strong restricted isometry property application phaseless compressed sensing applied computational harmonic analysis wang phase retrieval sparse signals applied computational harmonic analysis zhou xiu wang kong exact recovery sparse signal via weighted minimization arxiv preprint zhou recovery analysis weighted mixed minimization arxiv preprint
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inferential results new measure inequality youri davydova francesca greselinb laboratoire paul des sciences technologies lille lille jun lille france saint petersburg state university saint petersburg russia dipartimento statistica metodi quantitativi milano bicocca milan italy abstract paper derive inferential results new index inequality specifically defined capturing significant changes observed left right tail income distributions latter shifts apparent fact many countries like germany france last decades concern many policy makers propose two empirical estimators index show asymptotically equivalent afterwards adopt one estimator prove consistency asymptotic normality finally introduce empirical estimator variance provide conditions show convergence finite theoretical value analysis real data net income bank italy survey income wealth also presented base obtained inferential results keywords phrases income inequality lorenz curve gini index consistency asymptotic normality economic inequality confidence interval nonparametric estimator introduction view measuring economic inequality society suppose interested instance incomes let income random variable non negatively supported cdf next define inf quantile suppose possesses finite mean lorenz curve introduced lorenz irreplaceable tool domain defined curve expresses share income possessed poorer part population expressed firstly pietra english translation available pietra mathematically formulated gastwirth following also employ provides share income owned richer population obviously curve obtained applying central symmetry respect center unit square shown figure allows also rephrase gini recall gini rephrased weighted average comparisons made figure curves among mean income poorest overall mean greselin greselin dealing skewed distributions case many economic size distributions median preferred mean way gastwirth proposed modify gini accordingly recently motivated observed shifts toward extreme values income distributions new focus introduced gastwirth almost contemporarily davydov greselin policy makers nowadays interested understanding happens critical portions population significant changes observed left right tail income distributions countries like germany france last decades notice classical lorenz curve provides useful pointwise information reference poorest people hand approaches approaches display variation within upper end top distribution clearly novel approach consider equally sized opposite groups population compare mean income aiming contrasting economic position group poorer one richest following inequality curve introduced case perfect equality fraction population mean income hence income distribution moves toward variability mean income income richest people moving far mean poorest part population raises toward hence represent pointwise measure inequality population plotting naturally summarize information given curve single measure inequality taking expected value notice area observed inequality curve curve perfect equality horizontal line passing origin axes structure paper follows section introduces two estimators new inequality measure provide reasons selecting view main purpose third section core paper states main inferential results detail show consistency estimators state asymptotical distribution asymptotic negligibility difference also introduce empirical estimator variance establish convergence finite variance estimator lemmas useful inferential theory presented section along proof section shows inferential results employed develop analysis real income data final considerations given section estimators economic data entire complete population rarely available studies based data obtained sample surveys hence usually estimate summary measures samples introduce two empirical estimators estimating first one derived natural way say replacing population cdf mean value empirical counterparts considering empirical lorenz curve say dual follows set second estimator defined terms order statistics sample drawn therefore define expresses ratio mean income poorest richest elements sample asympwe show later theorem two estimators suitable developing inferential results totically equivalent estimator much simpler comes implement code analysis real data inferential results section present main results starting consistency next state asymptotic normal distribution deal estimator variance estimation finally show asymptotic equivalence two estimators unless explicitly stated otherwise assume throughout cdf continuous function natural choice modeling income wealth distributions many economic size distributions consistency consistent estimator theorem proof normalized definition empirical lorenz curve dual say useful introduce absolute versions given may rephrase converges probability uniformly see goldie approach converges probability uniformly due lebesgue dominated convergence theorem get thesis asympthotic normality estimator theorem moment finite asymptotic representation denotes random variable converges probability weight function asymptotically corollary conditions theorem normally distributed proof follows immediately applying central limit theorem proof theorem get definition remainder term given later show lemma respectively order proof follows approach greselin pasquazzi zitikis state asymptotic normality zenga inequality index zenga hence proceed analysis first two terms using vervaat process dual know mathematical historical details vervaat process see zitikis davydov zitikis greselin denoting uniform empirical process using one property vervaat process namely nvn find bound first term follows later show lemma deal second term obtain using similar arguments lemma show therefore notice first term right hand side equation could rewritten second term right hand side equation using change variable obtain completes proof theorem convergence empirical variance deal theoretical variance empirical counterpart let minimum value support value analogously let maximum value support notice may therefore without loss generality take theorem assume proof proof composed three steps step show probability almost given begin study first part related know uniformly convergence probability almost probability probability exists constant hence lebesgue theorem gives replaced consider second part takes place observe probability almost probability therefore using lebesgue theorem get completes proof step given show probability due previous step every know converges probability shown probability lim sup sup using follows sup lim sup hence lim sup sup observing function right hand side integrable apply lebesgue dominated convergence theorem prove step complete proof theorem need obtain bound integrals use following delicate estimation exists positive constant indeed hence every exists constant depending jointly give estimation similarly new random variable let introduce new probability space taking values due strong law large numbers hence probability lim sup observing integral converges defining takes form lim sup evidently replacing obtain theoretical counterparts collecting bounds convergence stated finally bounds three steps lim sup lim sup lim sup lim sup taking arrive established consistency asymptotic normality estimator defined would like prove similar properties second estimator prove asymptotic focus difference negligibility theorem moment finite proving theorem worth state two useful corollaries corollary moment finite theoretical variance corollary assumptions also empirical counterpart given true replace proof theorem let modulus continuity interval given sup let used inequalities hold true bounds get get max max therefore due obtain sufficient state convergence probability chosen proofs lemma conditions theorem proof estimate splitting integral two parts choosing look getting bound initially deal first part given provided consider lefthand term set sup know see greselin therefore employing inequality related vervaat process choosing obtain lebesgue dominated convergence theorem integral hence righthand term may estimated follows set latter quantity finite due existence moment complete analysis deal second part given bound found following steps continue proof finding bound second term therefore setting sup observe following two equalities hold true recalling due get remark finally get begin inequality use following bound latter integrand recalling exploiting get integrating hence obtain desired bound previous estimates follows fixing let let get lim sup finally gives lemma conditions theorem proof start definition recalled convenience observing using rewrite proof established following proof lemma minor modifications lemma conditions theorem proof estimate splitting two integrals follows let consider first term observe assume otherwise may replace appropriately chosen hence choosing arrive recalling deal second term observing obtain lemma conditions theorem proof deal previous lemma splitting follows observing initially consider finally result comes observing exists constant due assumption second hence first moment finite new inequality measure real data purpose section show real data application theoretical results obtained previous sections employ bank italy survey household income wealth hereafter named acronym shiw dataset published survey contains information household income wealth year covering households individuals sample representative italian population composed million households million individuals shiw provides information individual personal income tax net income contain corresponding gross income employ updated version microsimulation model described morini pellegrino estimate latter taxpayer comparison results microsimulation model official statistics published italian ministry finance shows distribution gross income net tax according bands gross income type employment close empirical analysis develop based observed net income shiv tax data gross income arise microsimulation model appreciate asymptotic results section empirical estimator calculate four types confidence intervals normal basic percentile bca confidence intervals drawing bootstrap samples empirical estimator evaluated sample figure show histograms obtained values considering gross income panel net income panel taxes panel inequality estimators skewed distribution case low sample size accuracy normal approximation apparent due large sample size check quality first order approximation figure shows plots obtained three cases gross income panel net income panel taxes figure histograms theoretical quantiles sample quantiles sample quantiles sample quantiles panel theoretical quantiles theoretical quantiles gaussian quantiles gross income panel net income figure plots panel taxes panel normal basic percentile bca gross income net income taxes gross income net income taxes figure confidence intervals finally figure observe four methods constructing confidence intervals substantial agreement agree assuring statistically significant increase inequality passing net income gross income redistributive effect taxation gross income taxes recall adjusted percentile methods also named bca calculating confidence limits inherently accurate basic bootstrap percentile methods davison hinkley concluding remarks moved considerations nowadays many developed countries critical extremes portions population facing great reshaping economic situation new index measuring inequality proposed davydov greselin cited paper discussion properties index given motivate introduction literature show descriptive features inferential results index still missing paper first contribution fill gap proposing two empirical estimators shown asymptotic equivalence consistency asymptotic normality first estimator derived also proved convergence empirical estimator variance finite theoretical value finally used new statistical inferential results analyze data net income bank italy survey household income wealth compare gross income taxes references atkinson measurement inequality journal economic theory cobham sumner putting gini back bottle palma measure inequality technical report kings college london davydov zitikis convex rearrangements random elements asymptotic methods stochastics vol fields institute communications american mathematical society providence usa davydov greselin comparisons poorest richest measure inequality revision davison hinkley bootstrap methods application vol cambridge university press gastwirth measures economic inequality focusing status lower middle income groups statistics public policy gastwirth measures inequality reassessing increase income inequality sweden journal international association official statistics gini sulla misura della concentrazione della dei caratteri atti del reale istituto veneto scienze lettere arti english translation gini measurement concentration variability characters metron greselin equal poorer richer unequal economic quality control greselin pasquazzi zitikis zenga new index economic inequality estimation analysis incomes italy journal probability statistics greselin pasquazzi zitikis contrasting gini zenga indices economic inequality journal applied statistics greselin puri zitikis processes statistics measuring economic inequality actuarial risks statistics interface goldie convergence theorems empirical lorenz curves inverses advances applied probability lorenz methods measuring concentration wealth journal american statistical association morini pellegrino personal income tax reforms genetic algorithm proach european journal operational research first online https pietra delle relazioni tra gli indici note atti del reale istituto veneto scienze lettere arti pietra relationships variability indices note metron zenga inequality curve inequality index based ratios lower upper arithmetic means statistica applicazioni zitikis vervaat process szyszkowicz asymptotic methods probability statistics elsevier science amsterdam
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scalable morphology control embodied machines nick josh vytas hod department computational biology biological statistics cornell university ithaca department computer science university wyoming laramie department computer science university vermont burlington intelligent robotics group intelligent systems division nasa mountain view department mechanical engineering columbia university new york dec sculpts body plans nervous systems agents together time contrast robotics robot body plan usually designed hand control policies optimized fixed design task simultaneously morphology controller embodied robot remained challenge psychology theory embodied cognition posits behavior arises close coupling body plan sensorimotor control suggests two subsystems difficult evolutionary changes morphology tend adversely impact sensorimotor control leading overall decrease behavioral performance examine hypothesis demonstrate technique morphological innovation protection temporarily reduces selection pressure recently individuals thus enabling evolution time readapt new morphology subsequent control policy mutations show potential method avoid local optima converge similar highly fit morphologies across widely varying initial conditions sustaining fitness improvements optimization technique admittedly first many steps must taken achieve scalable optimization embodied machines hope theoretical insight cause evolutionary stagnation current methods help enable automation robot design behavioral training simultaneously providing testbed investigate theory embodied cognition ntroduction designing agile autonomous machines longstanding challenge field robotics animals including humans served examples inspiration many researchers meticulously painstakingly attempt reverse engineer biological organisms navigate even dynamic rugged unpredictable environments relative ease however another competing approach use evolutionary algorithms search robotic designs behaviors without presupposing designs behaviors may methods often take inspiration evolutionary method rather exact specifications given organism produced use evolutionary algorithm automated design comes many benefits removes costly endeavour determining traits given organism specific biological niche useful design features provide beneficial functions instantiated machine yield machines resemble animals currently found earth allows machines specialized behaviors environments differ model organism additionally optimization process serve controlled repeatable study evolutionary developmental behavioral theory however generalization design automation include optimization robot neural controllers body plans proven problematic recent successes demonstrated potential effective optimization control policies agents fixed morphologies lesser extent optimization morphologies body plans agents minimal fixed control policies two seen limited success inability perform robot scale experienced noted informally many researchers involved robot optimization yet published rarely thus lack publication presumably field lacks incentives publication negative results rather lack negative results unpublished works joachimczak provide anecdotal example premature convergence robot brain body plan fig cheney analyze phenomenon premature convergence embodied machines suggest traditional evolutionary algorithms hindered setting primarily ability perform continued optimization morphology robot hypothesized premature convergence may due effect embodied cognition individual body plan brain incentive specialize behaviors complement one another specialization makes improvements either subsystem difficult without complementary changes highly unlikely event given current algorithms thus results embodied agent fragile respect design perturbations background attempts solve problem robot evolution frequently traced back work sims work introduced use evolutionary algorithms produce behaviors morphologies simultaneously despite advance work represented evolved robots tended composed small number components figures show mean segments per robot typically controlled neurons suggested current computational power vastly increased scale complexity robots evolved using sims similar methods yet surveys evolved robots fail exhibit significant increase size complexity wide range hypotheses lack scalability proposed focused lack efficient evolutionary search algorithms genetic encodings others pointed lack incentives complexity simple tasks environments yet attempts evolve robots using methods designed overcome challenges yet obviously surpass sims work terms complexity scale work investigates different hypotheses first suggested considers way agent brain body plan interact optimization process iii ethods interdependency body brain investigate notion specialization brain body plan one another evolution creates fragile system easily amenable change specialization creates local optima search space premature convergence suboptimal designs paper explore direct solution problem fragile coupled systems explicitly readapting one subsystem body plan brain evolutionary perturbation proposed method differs traditional evolutionary algorithm evaluates fitness newly proposed variation immediately readaptation uses valuation fitness determine long term potential variation thought experiment consider hypothetical robot partially optimized body plan example quadrupedal form partially optimized controller example legs swing forward back sagittal plane suppose controller morphology evolution step forward robot contracts muscle near hip enough force swing one legs forward enough force land front body successfully take step consider variation morphology proposed evolutionary algorithm new robot possesses longer legs controller changed machine evolved rapid locomotion longer legs able take longer strides would beneficial variation successful original design however evaluation new robot original controller applies amount force leg failing move thus frustrating robot ability walk coordinated fashion current evolutionary methods would treat robot recipient detrimental mutation remove population variations display immediate detrimental fitness impact rejected regardless long term potential evolutionary algorithm converged local optima search space however newly proposed morphology would resulted robot outperformed predecessor coupled controller suited morphology determine newly proposed morphology superior suppressing mutations body plan allowing readaptation controller properly coordinate behavior herein lies foundation proposed algorithm readapt controller new proposed morphology fit predecessor controller readaptation obvious method modeling controller readaptation would protect lineage recently experienced mutation body plan allowing undergo several generations evolutionary change restricted control subsystem member lineage achieves higher fitness ancestor time period descendant retained otherwise new morphological variant dies however unclear set time period protection priori surely amount time controller takes readapt new morphology depends many specific features complexity genetic encoding desired behavior current ability level robot changes optimization time determining correct value parameter would require full parameter sweep various values readaptation time new combination brain body environment goal simply optimize robot searching value unique optimization scenario intractable proposed method morphological innovation protection response unintuitive nature optimal value readaptation length proposed approach free parameter descendants robots experience morphological mutations retained population number generations elapsed since mutation occurred tracked referred morphological age robot two individuals found population latter robot exhibits better performance desired task experienced fewer generations since morphological mutation former robot latter robot said dominate former robot optimization age fitness former robot removed population effect latter robot exhibited ability recover ancestor morphological mutation improve upon others population concept tracking age using optimization objective borrowed major difference age refers length time subsystem agent morphology remained unvaried rather original definition total time since random individual introduced population procedure effect protecting new morphologies poorly adapted controllers thus henceforth refer procedure morphological innovation protection form diversity maintenance though reduced selection pressure newly mutated morphologies various methods exist encouraging diversity fitness sharing crowding random restart parallel hillclimbers novelty speciation however age chosen simplicity implementation parallels learning biology helps avoid cost extended control morphologies since optimization allows fitness comparisons new child morphologies equal readaptation time even yet fully readapted rather making comparisons predefined amount readaptation evolutionary algorithm optimization performed evolutionary algorithm trials follow strategy parents mutants population size trials last generations crossover considered work mutation chance creating variation either morphology controller given robot ratios morphology controller mutations considered none showed significant effect resolving premature convergence resulting fitness preliminary trials without innovation protection genetic encoding soft robot morphologies consistent prior work studying robot morphologies controllers choose soft robots model system due complexity deformable morphologies distributed controllers soft robot morphologies encoded compositional pattern producing network cppn cppn encoding produces cell fate voxel robot type neural network takes cell geometric location cartesian coordinates radial polar coordinate outputs variety morphogens work one determine whether cell present location one determine whether present voxel muscle passive tissue cell since nearby voxels tend similar coordinate inputs also tend produce similar outputs network creating continuous muscle tissue patches cppns also produce complex geometric patterns activation functions node take variety functions sigmoid sine absolute value negative absolute value square negative square square root negative square root functions tend produce regular patterns features across coordinate inputs example absolute value node input would produce symmetry sine node input would produce repetition network optimized produce high performing morphologies iterating various proposed perturbations include addition removal node edge network well mutation weight edge activation function node soft robot resolution cppn genetic encoding continuous function mapping location cell cell fate may discretized phenotype resolution creating number voxels morphology unique controller voxel practice resolution limited computational resources elements computationally expensive simulate default treatment discretization occurs space robots use phenotypes created resolution note distance values noted absolute number voxels higher fitness values tend produced phenotypes higher resolution controllers genetic encoding unique controller optimized muscle cell robot morphology voxel controller two parameters optimized outputs separate cppn inputs one phase offset individual cell muscle oscillations global sinusoidal oscillator acts central pattern generator second frequency global clock since cppns currently enable global parameters done averaging local values cell produce single global value controllers output value time step corresponds linear change dimension muscle cell original length original volume defining robot behavior passive tissue cells remain original size though also deform based intrinsic compliance encoding simple straightforward ability produce complex behaviors multiple patches muscle groups sync real valued phase offset also ability produce gradually varying sweeps phase offset resulting propagating waves excitation across large muscle groups furthermore optimization global frequency able produce oscillation speeds fine tuned properties individual morphologies optimizing maximize resonance soft tissues appendages encoding morphology controller robot two separate cppn networks emphasizes false dichotomy robot brains body plans however explicit separation allows make changes specific isolated either brain body necessary proposed algorithm controller readaptation requires iterating controller changes without affecting morphology physics simulation evaluation morphology controller given robot specified fitness locomotion distance robot determined constructing simulating robot voxcad physics simulator simulations last actuation cycles may variable amount time depending length globally optimized frequency though method normalizing number steps taken leads fair comparison normalizing amount time per simulation morphological innovation protection newly proposed method set morphological age zero new child result morphological mutation current individual population means individual large value age objective individual must result large number successive controller mutations without change robot body plan setup thus allows simple comparison method individuals similar amounts controller adaptation current morphologies dominated individual would fitness ability morphology paired controller adapted diversity maintenance mechanism encourages exploration new peaks rugged landscape plans implicit assumption unique morphologies correspond peaks landscape method age resets corresponding morphological mutations existing individuals differs prior technique inserting completely random individuals allows improvement fitness levels individuals time case traditional optimization individuals drawn distribution fitness values random genotypes regardless created however case morphological innovation protection individuals random inherit many properties parents meaning show higher fitness values time demonstrate empirically performed linear regression individuals fig containing morphological innovation protection showed significant increase fitness time generation generation confirms major difference proposed technique standard approach diversity maintenance introduction random individuals statistical analysis treatments performed independent trials random seeds consistent treatments plots show mean values averaged across fit individual trials condition shaded areas representing bootstrapped confidence intervals average generated wilcoxon test esults effect morphological innovation protection fitness task locomotion ability standard task evolutionary robotics first optimize robots using traditional method greedy fitness evaluations selection criteria immediate locomotion ability determines survival population candidate morphologies innovation protection setup traditional method produces robots average fitness voxels bootstrapped confidence interval voxels additionally optimized robots task environment setup time using morphological innovation protection selection method individuals equal lesser amounts controller adaptation current morphologies treatment produces significantly effective robots mean distance travelled voxels voxels increase fitness shows morphological innovation protection effective way optimizing robots yet conclusively demonstrate intuition correct work demonstrated asymmetric difficulty optimizing morphology robot compared optimizing controller drew hypotheses morphology encapsulated controller acting translator language cognitive functions outside environment experiment help support intuition controller must readapt new morphological encoding also introduces confounding effects added population diversity afforded protection added dimensionality search space protection age moving search singleobjective optimization problem tease apart influence two confounds present treatment controllers robot undergo equivalent protection morphologies experiment treatment individuals others whose morphology equal lesser amounts readaptation newly mutated controllers deemed controller innovation protection condition provides potential advantages search added diversity temporary reductions selection pressure yet rely idea broken morphological treatment robots locomote voxels average voxels fails show significant improvement singleobjective case protection performs significantly worse protection morphological innovations fitness trajectories evolutionary time fig demonstrate typical logarithmic fitness improvement first generations show stagnation traditional optimization procedure without innovation protection mean fitness value treatment without protection shows significant improvement turnover morphologies unique robot body plans fig fitness impact distance travelled voxels optimization time generations various types plan protection mechanisms values plotted represent mean value independent trials bootstrapped confidence intervals denoted colorized regions generation average fitness values voxels respectively contrary treatment morphological innovation protection shows significant improvements time generation voxels generation rapid improvement controller innovation protection protection cases first generations contradict hypotheses fragile morphological language coupling morphology controller takes time become established would introduce fragility system effect morphological innovation protection population stagnation perhaps telling average locomotion ability end optimization time examination optimization process within individual run figs represent typical runs help give intuition optimization process figures randomly colored line represents unique morphology plotted locomotion ability optimization time note continued improvement performance fit individual optimization time case morphological innovation protection fig consistent trend seen average fig seen case without innovation protection fig best individual found generation generation fitness reached final value consistent observation see fit individual fig changes rapidly continually turning trial morphological innovation protection color figure represents unique morphology notice wide variety different morphologies hold title average runs morphological innovation protection unique morphologies point optimization runs without protection show significantly less question temporary reduced selection pressure via dimension morphological innovation protection may help improve overall fitness continued optimization may best demonstrated pop box fig see current best morphology teal morphology unable improve time see fitness value flatlining parent morphology child new proposed variation morphology highlighted red original fitness value morphology falls parent individual empirically shows worse performance parent thus would considered viable solution traditional evolutionary method however since new morphology controller well adapted controller specialized previous morphology teal morphological innovation protection expect new robot immediately outperform parent keeps individual consideration one could hold long term potential show immediate promise indeed see number controller optimization iterations later occurring equal amounts parent child intermediate period child morphology red overtakes parent morphology teal achieving higher fitness demonstrating indeed hold better long term potential parent despite immediate drop fitness fitness parent outperformed child less time controller morphology assume parent unlikely promising robot body plan long run thus remove population see trend overtakes children start worse fitness parents eventually outperform continuing throughout run blue child overtakes red parent green overtakes blue pop fig see morphological overtakes significantly often runs morphological innovation protection average overtakes first generations without innovation protection overtakes trials controller innovation protection morphological overtakes average interestingly number morphological overtakes morphological innovation protection treatment significantly different number controller overtakes controller innovation protection treatment child robot new controller variation readapts morphology catch back parent controller happens times average combined finding fig morphological innovation protection outperforms two treatments suggests greater potential relative importance morphological overtakes controller overtakes reinforces asymmetry morphologies controllers optimization perspective fig single optimization trial featuring morphological innovation protection unique morphology represented random color pop figure highlights example overtake new child morphology red initially performs worse parent morphology teal outperform previous morphology successive control optimization parent child fig fitness time single optimization trial initial conditions fig without innovation protection unique color represents unique morphology prevalence colored dots filling space underneath best performing individual represent new morphologies initially performed worse current best individual thus rejected thrown population opposed individuals protected eventually led overtakes fig pronounced fig effects morphological innovation protection progression morphologies evolutionary time potential convergence across initial conditions visual inspection actual morphologies evolutionary time supports proposed method improved optimization efficiency ability escape local optima fig shows current best individual various points evolutionary time top runs treatment morphological innovation protection note fitness values robots increase time left right indicated color robot also note final morphology robots runs result identical morphologies despite starting range starting morphologies finding convergent morphology hundreds thousands generations optimization process contrast sustained turnover morphologies shown fig shows snapshots best runs treatment without innovation protection notice colors robots tend show little change evolutionary process mirroring stagnation shown fig convergence final morphologies present well gross morphologies found variants full cube appendages found early search often provided set random initial morphologies treatment gross morphological changes tend absent generation full generations differences figs suggests traditional method without morphological innovation protection tends converge prematurely morphologies early evolutionary search morphological innovation protection may allow search escape local optima converge global optima convergence morphologies across varied initial conditions even explore question scale apply morphological innovation protection evolution robots higher resolution morphologies voxels rather lower resolution morphologies voxels employed previous experiments increased number voxel cells make robot allows greater expressiveness finer details morphology however also presents challenge algorithm number cells increases effect changing single voxel minimum morphological variation reduced extreme concept readapting controllers since last morphological change less straightforward increasingly small changes modify minor details morphology without affecting overall function help address problem morphology changes introduce parameter represent minimum percentage voxels must varied order qualify gross morphological change important note parameter specific soft robot implementation employed work thus optimal setting parameter great importance generalization outside soft robot encoding general concept threshold minimum morphological change universal concept could applied robot instantiation necessary case robots morphologies find resetting morphological age individual mutation changes voxels produces optimal results value found via parameter sweep minimum fig progression morphologies evolutionary time morphological innovation protection rows represent top performing runs generation columns represent snapshots morphology various points optimization process note runs converge upon morphology front back legged robot despite starting varying initial conditions color morphologies represent fitness values voxels changed also investigated minimum change threshold controller innovation protection found benefit falls threshold increases showing optimal performance threshold ignore threshold minimum controller change setting basic morphological innovation protection threshold travels significantly farther voxels case protection voxels case controller innovation protection voxels trials employing threshold morphological changes achieve higher average fitness case morphological innovation protection without threshold robots traveled voxels average difference significant level side note difficult trials robots need optimize neural network controller depicted visually due space constraints trials morphological change threshold significantly outperform treatments travel voxels trials protection controller protection morphological innovation protection threshold travel voxels respectively fig progression morphologies evolutionary time top performing runs innovation protection color legend column snapshots times identical fig visualization soft robot morphologies time trials morphological change threshold depicted fig compared case protection fig ability evolution converge morphology across many independent trials despite starting different initial conditions suggests particular soft robot implementation inclusion thresholded morphological innovation protection able escape local optima around starting conditions find global optima search space case without protection search stagnates quickly appears unable escape local optima near initial conditions fig interestingly low resolution soft robot implementation figs benefit inclusion threshold presumably discrete construction robots mean mutations large enough create meaningful morphological change iscussion results demonstrate new method entire robot evolution brain scalable terms continued optimization longer periods time better resulting fitness traditional evolutionary methods method morphological innovation protection helps prevent premature convergence many local fig progression morphologies evolutionary time setting morphologies evolution morphological innovation protection threshold rows represent top independent trials columns represent progression evolutionary time color represent fitness values robot locomotion speed warmer values depicting fit individuals note convergence runs final morphology optima appear present rugged search space robot morphologies controllers hypothesis fragile brain body plan caused specialization one subcomponent consistent findings work also reveals asymmetry brain body plan protecting innovations morphology leads effective optimization protecting innovations controller benefits temporarily reduced selection pressure provided morphological innovation protection suggests potential immediate fitness impact morphological mutation always correlated thus require form diversity maintenance help evolution rate proposed solutions based potential rather immediate fitness impact shown using morphological innovation protection purpose help reduce premature convergence search space stagnation suboptimal values believe first example design automation algorithm robotics considers interdependence neural controllers body plans arising psychological theory embodied cognition uses intuition propose method escape local optima fig progression morphologies evolutionary time top performing runs innovation protection color legend column snapshots times identical fig fitness landscape despite significant improvement ability simultaneously optimize brain body plan embodied robots much work still done firstly proposed method applied one class robot class may actually represent simplest form distributed sensing actuation information processing cellular soft robot paradigm designed possess helps blur line physical interactions morphology environment information processing controller case centralized controllers robots composed rigid components topological rather parametric changes cognitive architecture would required control readaptation morphological mutations add remove physical components limbs future work explore effect morphological innovation protection paradigm potential morphological changes drastically change function robot thus readaptation morphologies play even critical role optimization experiments genotype encoding soft robots modularized one part genome dictated shape material properties robot separate part encoded actuations form volumetric deformations voxels behavior robots even complex phenotypes one envision different splits constitutes morphology control qualifies morphological change proposed algorithm would interest investigate effect various splitting points dichotomy efficacy morphological innovation protection furthermore idea distinct split body brain false dichotomy information processing physical processes happen throughout agent rather various mutation operators affect brain robot desire agents grown develop result rich crosstalk feedback loops processes responsible creating components creating complex adaptive systems like see biological organisms proposed method relates informs biological analogs entirely clear rough analogies drawn example observation brains learn adapt much faster rate bodies grow fits paradigm even slower change gross morphological features evolutionary time way readaptation controller strategies varying body plans built neuroplasticity brain one could also imagine periods evolutionary time entire species relatively little morphological selection pressure environmental shock suddenly reapplies pressure also possible periods single individual lifetime selection pressure varies example human infants may selected highly locomotion speed parents tend physically carry protect early life believe methods introduced restricted particular domain algorithm simple implement requiring age counter check variations brain body mutation optionally criterion minimal gross morphological change thus believe widely applicable future work test supposition due recent interest neural network topology weights also note work agent controller embodied within morphology closely related neural network weights embodied within topology future work show whether method proposed show similar optimization gains design neural network topologies well onclusion demonstrate example robot design automation algorithm considers interdependence neural controllers body plans due theory embodied cognition optimization process use intuition temporarily reduce selection pressure newly mutated robot morphologies thus allowing agents readapt controllers better escape local optima fitness landscape shown technique deemed morphological innovation protection produces evolutionary optimization delays premature convergence stagnation results efficient evolved robots showcase ability technique escape local optima search space demonstrating convergence similar morphology across many independent trials randomly initial conditions hope technique surpassed future developmental process feedback loops body brain propose algorithm short term improvement current techniques morphology control virtual creatures vii acknowledgment thank nasa space technology research fellowship army research office contract support steve strogatz advice kathryn miller copy editing uvm morphology evolution cognition lab feedback eferences auerbach bongard environmental influence evolution morphological complexity machines plos comput biol baluska levin head cognition throughout biological systems frontiers psychology beyer schwefel evolution comprehensive introduction natural computing bongard morphological change machines accelerates evolution robust behavior proc national academy sciences bongard evolutionary robotics comm acm bongard pfeifer evolving complete agents using artificial ontogeny machines new species pages springer japan cheney bongard lipson evolving soft robots tight spaces proceedings annual conference genetic evolutionary computation pages acm cheney bongard sunspiral lipson difficulty morphology control evolved virtual creatures alife fifteenth international conference synthesis simulation living systems volume pages mit press cheney clune lipson evolved electrophysiological soft robots alife fourteenth conference synthesis simulation living systems volume pages cheney maccurdy clune lipson unshackling evolution evolving soft robots multiple materials powerful generative encoding proc genetic evol comp pages acm chestnutt lau cheung kuffner hodgins kanade footstep planning honda asimo humanoid robotics automation icra proceedings ieee international conference pages ieee collins ruina tedrake wisse efficient bipedal robots based walkers science cully clune tarapore mouret robots adapt like animals nature fanelli negative results disappearing disciplines countries scientometrics fernando banarse blundell zwols rusu pritzel wierstra pathnet evolution channels gradient descent super neural networks arxiv preprint geijtenbeek pronost interactive character animation using simulated physics review computer graphics forum volume pages wiley online library geijtenbeek van panne van der stappen flexible locomotion bipedal creatures acm transactions graphics tog gordon whitley serial parallel genetic algorithms function optimizers icga pages hiller lipson dynamic simulation soft multimaterial objects soft robotics hornby alps population structure reducing problem premature convergence proceedings annual conference genetic evolutionary computation pages acm hornby pollack using lsystems generative encoding proceedings annual conference genetic evolutionary computation pages morgan kaufmann publishers joachimczak suzuki arita artificial metamorphosis evolutionary design transforming robots artificial life langton artificial life publishing company redwood city lehman stanley evolving diversity virtual creatures novelty search local comp proc genetic evol comp conf pages acm lenski ofria collier adami genome complexity robustness genetic interactions digital organisms nature lenski ofria pennock adami evolutionary origin complex features nature levine finn darrell abbeel training deep visuomotor policies journal machine learning research miikkulainen liang meyerson rawal fink francon raju navruzyan duffy hodjat evolving deep neural networks arxiv preprint mouret clune illuminating search spaces mapping elites arxiv preprint pfeifer bongard body shapes way think new view intelligence mit press pfeifer lungarella iida embodiment biologically inspired robotics science raibert blankespoor nelson playter team bigdog quadruped robot proceedings world congress volume pages proceedings seoul korea schmidt lipson pareto optimization genetic programming theory practice viii pages springer sims evolving virtual creatures proceedings annual conference computer graphics interactive techniques pages acm stanley compositional pattern producing networks novel abstraction development genetic programming evolvable machines stanley miikkulainen evolving neural networks augmenting topologies evolutionary computation wilcoxon wilcox rapid approximate statistical procedures lederle laboratories wood first takeoff biologically inspired robotic insect ieee transactions robotics
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criteria universality quantum gates adam katarzyna center theoretical physics pas warsaw poland may consider problem deciding set quantum gates universal provide compact form criteria leading simple algorithm allows deciding universality given set gates number steps moreover criteria indicate type gates added turn universal set universal quantum gates play important role quantum computing quantum optics ability effectively manufacture gates operating many modes using example optical networks couple modes light natural motivation consider universality problems qubits also higher dimensional systems qudits see also fermionic linear optics quantum metrology quantum computing qudits universal set gates consists gates together additional gate map separable states onto separable states see recent results context universal hamiltonians set gates however generated using finite number gates say gates universal gate built arbitrary precision using gates known almost sets qudit gates universal sets given cardinality measure zero characterised vanishing finite number polynomials gates entries conjugates surprisingly however polynomials known hard find operationally simple criteria decide gates universality special cases optical gates recently studied approach providing algorithm deciding universality given set quantum gates implemented quantum automata proposed see also algorithms deciding finitely generated group infinite main obstruction problems considered lack classification finite subgroups nevertheless show paper one still provide reasonable conditions universality gates without knowledge efficiency universal sets typically measured number gates needed approximate gates given precision theorem states universal sets roughly efficient precisely number gates needed approximate gate bounded logc may depend recently bit flurry karnas area single qubit gates showing using number theoretic results conjectures one construct universal sets approach presented contributions unified author pointed connection new results seminal work distributing points sphere uses results concerning optimal spectral gap averaging operator moreover authors showed existence spectral gap implies symmetric universal sets single qudit gates symmetric mean set although still problems solve area seems progress would require development advanced methods pure mathematics rather quantum information developments include verification spectral gap conjecture currently known true additional assumption gates algebraic entries paper present approach allows decide universality checking spectra gates solving linear equations whose coefficients polynomial entries gates complex conjugates moreover method indicates type gates added make universal paper organised follows start presenting basic facts concerning adjoint representation adjoint representations assigns every matrix matrix adu give explicit formula adu necessary condition universality lemmas formulated using matrices adu adu boils checking dimension kernel matrix given next assume necessary condition universality satisfied provide sufficient conditions infinite thus dense precisely contains least one element whose distance nonzero less infinite combining basic results number theory arrive main results theorem state universal contains least one matrix whose spectrum belong finite list exceptional spectra also provide algorithm allows deciding universality given set gates also contains matrices exceptional spectra finite number steps discuss correctness algorithm provide instructive examples necessary condition universality let begin introducing basic notation used paper explaining adjoint representation set gates called universal set generated elements uim uij dense closure fact always lie group group say generates let denote lie algebra recall iff antihermitian traceless matrix moreover lie algebra real vector space equipped nondegenerate positive inner product defined trxy define adu one easily checks adu linear operator acting also invertible adu adu preserves inner product adu tru trxy therefore adu orthogonal transformation acting dimensional vector space upon choice orthonormal basis basis statisfies transformation adu expressed basis matrix belonging adtu adu det adu entries matrix adu real defined identity adu adu thus give adu note also way obtain homomorphism known adjoint representation set real matrices let denote set matrices commuting matrices adjoint representation absolutely irreducible real representation therefore extended version schur lemma matrix commutes matrices adsu adu proportional identity matrix words adsu example adjoint representation particularly nice form matrix written form cos sin pauli matrices satisfies similarly matrix form sin xij eij eji eij matrix whose non vanishing entry one easily verifies adjoint representation given adu calculation matrices adu done using formula upon choice orthonormal basis basis given example matrices multiplied imaginary unit higher one construct orthonormal basis analogous way general considerations found show group either infinite connected infinite consists connected components component dimension manifold finite note cases group infinite number elements thus first provide criteria distinguish case cases end use adjoint representation let ads adu note thus generates matrix commutes ads commutes adsu therefore universal ads adsu turns see converse true one additional assumption namely infinite lemma set special unitary matrices assume infinite ads proof lemma based structure theory semisimple lie groups found make additional remarks regarding calculation ads let vec vectorisation matrix vector obtained stacking columns matrix top one another one easily calculates adu adu adu vec complex conjugate transpose let adu adun lemma ads kernel emphasise role adjoint representation crucial lemma particular infinite subgroups example provide subgroup next characterise space ads let recall composition two unitary matrices unitary matrix determined cos cos cos sin sin sin cos sin cos sin sin sin moreover two unitary matrices commute iff similarly two orthogonal matrices commute one even multiple making use facts show fact noncommuting space larger one equal odd integer infinite next describe conditions infinite group commutator defined note equivalent distance elements measured using norm tru two elements following relation distances identity distance group commutator identity let ball radius centred elements det need assume let turns noncommuting elements belonging generate infinite subgroups lemma assume infinite one steps proof lemma uses relations show sequence converges integer see full discussion next describe end note tru tru trace determined spectrum desired condition expressed terms eigenvalues given mod direct calculations lead let next assume belong one show always exists integer belongs given let smallest integer satisfying condition prove modified version dirichlet approximation theorem use find upper bound nsu maxu way every nsu thus taking powers nsu move every element assume next necessary condition universality satisfied ads lemma one easily deduce assumption infinite intersection dense shown necessary condition universality places significant constrains structure also case finite group namely subset thus finite elements belong discussion summarised lemma let assume ads least one matrix belongs know every element put taking powers bounded nsu hence finite introducing must equivalent introducing condition phrased terms specra matrices definition assume spectrum called exceptional consists nth roots nsu set exceptional spectra finite set illustrate ideas find nsu list exceptional spectra note spectrum given therefore determined one angle angle corresponding exceptional spectrum called exceptional angle moreover centre consists two matrices start recalling dirichlet approximation theorem fact dirichlet given real number positive integer exist integers differs let spectral angle fact given integers multiplying inequality obtain note simplifies sin sin thus given search satisfying arcsin arcsin thus arcsin arcsin upper bound nsu attained arcsin see figure hence nsu exceptional spectra determined roots order equivalently primitive roots unity order precisely given fig condition gray segments determined sin white segments sin gcd number exceptional angles calculated using euler totient function equal higher dimensional groups discuss number nsu grows exponentially main result theorem assume adun least one matrix nonexceptional spectrum iii algorithm deciding universality case matrices exceptional spectra requires algorithm next present algorithm allows deciding universality given set gates finite number steps algorithm step check ads done checking dimension kernel matrix constructed entries matrices adun thus linear algebra problem answer stop set universal yes set step step check matrix belongs nsu done using formula answer yes universal answer set step define new set adding words length products elements length new equal old one group finite otherwise step major advantage approach fact make decisions steps finite time also clear randomly chosen matrices algorithm terminates probability step direct consequence fact exceptional spectra form finite set let next discuss correctness algorithm assume passes positively necessary condition universality step group finite algorithm terminates step finite hand direct consequence lemma infinite algorithm must terminate step finite one also argue words exceptional spectra form dense subset thus dense must contain words finite length spectra words terminate algorithm step moreover following fact assume dense length word gives termination universality algorithm length words length form covers arbitrary small proof assume words length built elements form arbitrary small let element whose distance identity exactly see figure definition must least one word length contained ball radius centred ball contained hence gives termination universality algorithm step result follows exceptional angles product exceptional otherwise algorithm terminates step either pair finitely many find step gives termination algorithm detailed discussion results connection finite subgroups found short words turns algorithm terminates step step fact algorithm checking universality terminates moreover set universal algorithm terminates main conclusion fact one decide universality subset looking words length examples remaining part paper demonstrate approach calculating examples chosen particularly elucidate importance conditions given theorem example let irrational multiple example infinite order since commute infinite abelian fact however adu adu hence example commutes adu interestingly however fig proof fact formulation fact related results contained order demonstrate efficient algorithm determine maximal gives termination step step respectively simplicity consider form end enough consider case understand structure group note hence normaliser thus group consists two connected components first one given group one elements form adjoint representation able identify infinite disconnected subgroups whereas defining representation moreover know exactly fix set example add one matrix neither parallel orthogonal spectrum whose neither parallel orthogonal case fact adh adt need check infinite distinguish three possibilities first assume exceptional theorem algorithm deciding universality terminates step next consider exceptional angles fig two component group generated example let hadamard gate phase gate arbitrary phase using notation goal check case generated group finite cyclic group order order case odd fact adh larger hence fact finite dicyclic group order whose generators fixing universality case requires example adding matrix babai proceedings third annual acm siam symposium discrete algorithms orlando acm new york babai beals rockmore proc international symposium symbolic algebraic computation issac acm press bocharov phys rev linear algebra appl gcd look product using formula calculate cos compare cos exceptional angles find never agree hence exceptional thus theorem get algorithm deciding universality terminates step left gcd exactly four angles calculations shows exceptional gcd moreover taking products results finite subgroup consisting elements exceptional spectra known binary octahedral group algorithm deciding universality terminates step fixing accomplished example adding one gate arbitrary see example algorithm requires words length terminate acknowledgment would like thank anonymous referees suggestions led improvements paper work supported national science centre poland grant sonata bis bouland aaronson phys rev bourgain gamburd eur math soc bourgain gamburd invent issue dieck representations compact lie groups new york bromberg phys rev lett chen mathematics quantum computation boca raton chapman press childs quantum info comput curtis reiner representation theory finite groups associative algebras interscience publishers john wiley sons derksen symb comput detinko flannery journal symbolic computation deutsch proc roy soc lond hardy wright introduction theory numbers oxford clarendon press field proc amer math soc freedman math res lett hall lie groups lie algebras representations elementary introduction springer gtm harrow math phys kliuchnikov kliuchnikov ieee transactions computers kuranishi nagoya math lloyd phys rev lett lubotzky comm pur appl math nielsen chuang quantum computation quantum information cambridge university press oszmaniec phys rev vol oszmaniec phys rev politi science reck phys rev lett sarnak letter scott aaronson andy pollington theorem sawicki quantum info comput sawicki karnas universality single qudit gates selinger quant inf comp zeier math phys zeier math phys phys rev
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iterators recursors interaction nets ian jorge sousa miguel mackie jsp jmvilaca oct lix cnrs umr polytechnique palaiseau cedex france departamento cctc universidade minho braga portugal abstract propose method encoding iterators recursion operators general using interaction nets ins two main applications method used obtain visual notation functional programs used extend existing translations ins languages recursive types introduction use visual notations functional programs long active research topic whose main goal notation used define functional programs visually animate execution paper propose graphical system functional programming based ins system offers adequate solution classic problems visual notations including treatment functions recursion based use recursion operators system implements semantics straightforward correspondence functional programs graphical objects approaches visual programming simply propose notation programs program evaluation animated representing visually intermediate programs result executing reduction steps initial program using operational semantics underlying functional language approach different use formalism semantics advantages using ins visual programming programs data represented graphical formalism programs animated without leaving interaction formalism external constructors recursive definitions expressed naturally interaction rules involving agents reintroduced side interaction net formalism offer satisfactory semantic interpretation behaviour functional symbols moreover many interaction net systems defined functional reading missing clear correspondence functional definitions interaction paper establish correspondence agents obviously functional interaction rules functions defined recursion operators encoding encodings use interaction system two different symbols exist application one syntactic symbol introduced translation corresponding agents principal ports facing root exists term depicted triangles second symbol used computation simplify figures corresponding agents depicted circles equally labelled principal ports face net represents applied function make possible interaction agents translation ttp encodes terms system rtp generates nets containing active special sym bol used evaluation token agent traverses net transforming evaluation rules involving tailored occurrences specific evaluation strategy rtp consists following rules comprises evaluation rules involving computation rule involving management copying erasing rules omitted start reduction symbol must connected root port term let denote net following correctness result holds iff ttp ttp evaluation relation defined standard evaluation rules language used paper extended natural numbers booleans lists iterators types bnl defined following syntax terms range set variables iterbool suc iternat nil cons iterlist encoding bnl extend bnl translation interaction system rtp novelty encoding aspect rather approach recursion first consider data structures implementation interaction rule token agent constructor symbol stop evaluation bnl define system rbnl consists symbols nil arity suc arity cons arity recursive program encoded interaction system specifically generated major novelty approach interaction system extended introducing fixed set symbols instead new symbol introduced occurrence recursion operator interaction rule different constructor dedicated interaction system generated term system constructed recursive function defined nil suc cons bool iterbool itbool itv ritbool nat iternat itnat itd itnat list iterlist itlist itd itlist ritlist consists following interaction rules others similar iterator symbols introduced pairs first symbol used syntactic agents second computation agents similarly bnl program translated net defined system rtp rbnl rtp defined section given bnl program iterbool iternat iterlist net given follows implementations terms translated syntax trees syntactic iterator agents turned computation counterparts token agents first key aspect approach interaction rules computation iterator agents internalise iterator parameters instance net iterlist cons reduces iterlist second key aspect new symbol auxiliary ports correspondence free variables iterator term end section correctness result proofs found proposition correctness closed bnl term canonical form conclusions future work presented approach encoding ins functional programs defined recursion operators given full details application approach implementation language results convenient visual notation language approach easily extended richer sets recursive types recursion operators also new strategies novel characteristics encoding interaction system generated dynamically program internalisation parameters recursion operator interaction rules respect previous work encoding recursion interaction nets fixpoint operators studied elsewhere interaction net implementations shown elsewhere binding recursion operator pcf implemented setting prototype system visual functional programming developed integrated tool inblobs interaction net programming tool consists evaluator interaction nets together visual editor compiler module translates programs nets latter module allows users type functional program visualize follow evaluation visually step step current compiler module automatically generates systems additionally visual editing mode available allows users construct nets corresponding functional programs current implementation way convert visual programs back textual ones translation however representative work area concentrated designing efficient translations samples let one translation typically constructed introducing application symbol principal port connected root port treatment iterators adapted setting removing evaluator tokens introducing iterator agents principal port immediately facing argument iterated function closed term correctness result easily established let closed term iternat iternat suc iternat references almeida mackie pinto encoding iterators interaction nets available http almeida pinto nets functional languages giesl editor proceedings international workshop reduction strategies rewriting programming wrs volume electronic notes theoretical computer science pages almeida pinto tool programming interaction nets proceedings eighth international workshop programming rule appear elsevier entcs asperti guerrini optimal implementation functional programming languages volume cambridge tracts theoretical computer science cambridge university press gonthier abadi geometry optimal lambda reduction proceedings acm symposium principles programming languages popl pages acm press mackie geometry interaction machine proceedings acm symposium principles programming languages popl pages acm press january mackie yale yet another lambda evaluator based interaction nets proceedings international conference functional programming icfp pages acm press mackie efficient interaction nets van oostrom editor proceedings international conference rewriting techniques applications rta volume lecture notes computer science pages june sinot interaction nets urzyczyn editor tlca volume lecture notes computer science pages springer inblobs webpage http
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generalization without systematicity compositional skills recurrent networks brenden lake marco baroni feb abstract humans understand produce new utterances effortlessly thanks compositional skills person learns meaning new verb dax immediately understand meaning dax twice sing paper introduce scan domain consisting set simple compositional navigation commands paired corresponding action sequences test generalization capabilities variety recurrent neural networks rnns trained scan methods find rnns make successful generalizations differences training test commands small apply strategies solve task however generalization requires systematic compositional skills dax example rnns fail spectacularly conclude experiment neural machine translation suggesting lack systematicity might partially responsible neural networks notorious training data thirst introduction human language thought characterized systematic compositionality algebraic capacity understand produce potentially infinite number novel combinations known components chomsky montague example person knows meaning usage words twice learns new verb dax immediately understand produce instructions dax twice dax type compositionality central human ability make strong generalizations dept psychology center data science new york university facebook artificial intelligence research correspondence brenden lake brenden marco baroni mbaroni limited data lake set influential controversial papers jerry fodor researchers argued neural networks plausible models mind associative devices capture systematic compositionality fodor pylyshyn marcus fodor lepore marcus calvo symons last years neural network research made astounding progress practical domains success crucially depends generalization perhaps strikingly recurrent neural networks currently dominate machine translation bojar since overwhelming majority sentences even word sequences language occur even large corpus baroni points strong generalization abilities still commonly observed neural networks extremely sample inefficient requiring large training sets suggests may lack algebraic compositionality humans exploit might sensitive broad patterns lots accumulated statistics lake paper introduce grounded navigation environment learner must translate commands given limited form natural language sequence actions problem naturally framed task due simplicity ideal study systematic generalization novel examples controlled setup thus use test wide range modern recurrent network architectures terms compositional abilities results suggest standard architectures lstms attention bahdanau generalize well novel examples feature mixture constructions observed training however models catastrophically affected systematic differences training test sentences sort would trivial agent equipped algebraic mind marcus generalization without systematicity scan tasks call data set scan simplified version commai navigation tasks mikolov learner goal translate commands presented simplified natural language sequence actions since command unambiguously associated single action sequence scan unlike original commai tasks straightforwardly treated supervised semantic parsing task dong lapata jia liang herzig berant input vocabulary given set words used commands output set actions available learner several examples scan presented fig formally scan consists commands generated grammar presented appendix corresponding sequence actions produced according semantic interpretation function see appendix intuitively scan grammar licenses commands denoting primitive actions jump denoted jump fig walk denoted walk lturn denoted turn left refer primitive also accepts set modifiers conjunctions compositionally build expressions referring action sequences left right modifiers take commands denoting undirected primitive actions input return commands denoting directed counterparts jump left fig opposite modifier produces action sequence turns agent backward specified direction executing target action jump opposite left around makes agent execute action step turning around specified direction jump around right fig modifiers trigger repetition command take scope combine two action sequences although scan examples fig focus jump primitive instance jump replaced either walk run look generate yet commands many combinations possible licensed grammar scan grammar lacking recursion generates finite large set unambiguous commands precise commands decoded compositionally applying corresponding interpretation function means discovers right interpretation function learner understand commands seen training example learner might observed primitive jump command training scan available https introducing primitive turning actions lturn rturn considerably simplifies interpretation function compared capturing orientation specifying arguments movement actions jump jump learned meaning twice around left verbs able decode complex command jump around left jump twice models setup approach scan successful framework two recurrent networks work together learn mapping input sequences output sequences sutskever fig illustrates application approach scan example first recurrent network encoder receives input sequence forming lowdimensional representation entire command second representation passed recurrent network decoder generates output sequence decoder output compared ground truth backpropagation algorithm used update parameters encoder decoder note although encoder decoder share network structure number layers hidden units otherwise share details regarding rnn provided appendix using framework tested range standard recurrent neural network models literature simple recurrent networks srns elman long memory networks lstms hochreiter schmidhuber gated recurrent units grus chung recurrent networks attention become increasingly popular last years thus also tested network without attentional mechanism using model bahdanau see appendix details finally make evaluations systematic possible hyperparameter search conducted varied number layers number hidden units per layer amount dropout applied recurrent layers word embeddings varying hyperparameters leads different network architectures run experiment replicated times different random reporting results focus architecture determined extensive hyperparameter search winning architecture lstm hidden units per layer attention dropout applied level although detailed analyses follow focus particular model recently convolutional networks reached comparable superior performance machine translation gehring investigate future work small number runs complete thus every network runs generalization without systematicity jump jump left jump around right turn left twice jump thrice jump opposite left walk thrice jump opposite left walk around left jump lturn jump rturn jump rturn jump rturn jump rturn jump lturn lturn jump jump jump lturn lturn jump walk walk walk lturn walk lturn walk lturn walk lturn walk lturn lturn jump figure examples scan commands left corresponding action sequences right jump jump twice walk eos sos jump jump walk jump eos walk figure framework applied scan symbols eos sos denote respectively encoder left ends first eos symbol decoder right begins sos experiments following experiments recurrent networks trained large set commands scan tasks establish background knowledge outlined training networks evaluated new commands designed test generalization beyond background set systematic compositional ways evaluating new commands networks must make generalizations produce appropriate action sequence based solely extrapolation background training experiment generalizing random subset commands chitecture experiment individually also reported analyzed networks trained following specifications training consisted trials presenting sequence updating networks adam optimization algorithm used default parameters including learning rate kingma welling gradients norm larger clipped finally decoder requires previous step output next step input computed two different ways training half time network outputs passed back next step half time groundtruth outputs passed back next step teacher forcing williams zipser networks implemented pytorch based standard training accuracy network key experiments least experiment specifically note experiments number distinct training commands well randomly sampled replacement reach target size code used publicly available link http experiment scan tasks randomly split training set test set training set provides broad coverage task space test set examines networks decompose recombine commands training set instance network asked perform new command jump opposite right walk around right thrice generalization test set although conjunction whole novel parts training set features many examples parts contexts jump opposite right turn opposite right jump right twice walk around right thrice bold appear times training set succeed network needs generalize recombining pieces existing commands interpret new ones overall networks highly successful generalization network experiment achieved correct test set accuracy values averaged five training runs topperforming architecture lstm attention layers hidden units dropout network achieved correct interestingly every architecture successful classic srns performed poorly best srn achieved less correct test time performance training set equally low however srns learned commands much better achieving correct average test set range across srn generalization without systematicity value times experiment generalizing commands demanding longer action sequences figure generalization training random subset scan tasks network trained varying proportions set distinct tasks generalization measured new tasks bar shows mean training runs corresponding sem architectures lstms grus attention essential since many highest performing architectures use indicated main split quite generous providing commands training time total distinct examples strong combinatorial coverage next network varying numbers distinct examples actual number training presentations kept constant results shown fig commands shown training examples network performs poorly correct coverage performance improves correct test set coverage performance correct results show networks generalize random subsets tasks relatively sparse coverage compositional command space well line success architectures machine translation test sentences likely never encountered training still even sparser coverage differences training test instances dramatic let example consider set commands without conjunction walk around thrice run jump opposite left twice commands sort occur test set training coverage split either components conjunction also occur corresponding training set average occurrences even split one test command also occur training split frequency occurrence commands training set average study next systematic form generalization models must bootstrap commands requiring longer action sequences seen training set contains commands requiring sequences actions whereas test set includes remaining commands requiring action sequences lengths split example test time network must execute command jump around left twice walk opposite right thrice requiring sequence actions although elements used command observed training network never asked produce sequence length ever seen around twice command conjoined opposite thrice command although observe components conjoined others thus must productively generalize familiar verbs modifiers conjunctions generate longer action sequences fair task system correctly translating input commands know walk around jump function conjunction immediately able walk around jump even never performed action sequence length test turns challenging models best result average runs achieved gru attention one hidden layer dropout interestingly model considerably less capacity best setup model achieves accuracy fig top shows partial success almost entirely explained generalization shortest action sequence lengths test set although might expect even humans able generalize long action sequences sharp drop extrapolating actions striking bottom panel fig shows accuracy test set organized command length word tokens model gets right longest commands tokens training set longest action sequences invariably associated commands containing tokens thus model correctly generalizing cases similar training instances focus action sequence length rather command length since former exhibits variance longest commands words given conjunction two directed primitives modified twice jump around left twice run opposite right hand relatively short command jump around left thrice demands actions accuracy new commands generalization without systematicity strong effect action sequence length average accuracy ranging commands requiring actions commands requiring actions experiment generalizing composition across primitive commands accuracy new commands action sequence length command length figure generalization commands action sequence lengths seen training top accuracy distribution action sequence length bottom accuracy distribution command length lengths attested test set shown cases bars show means runs model sem finally studied whether difficulty long sequences mitigated proper length provided oracle evaluation difficulty relatively straightforward issue decoder terminating early provide unrealistic fix difficulty symptomatic deeper problems generalization change small effect oracle network performance improved correct notable insufficient master long sequences model showed substantial improvement although improved networks far perfect still exhibited key difficulties long sequences output actions even top model attempt decoder terminate action sequence eos ignored second strongest action chosen sequence proper length produced next test closest dax thought experiment presented introduction training phase model exposed primitive command denoting certain basic action jump model also exposed primitive composed commands actions run run twice walk walk opposite left run twice test time model execute composed commands action saw primitive context jump twice jump opposite left run twice according classic thought experiments fodor colleagues easy know meaning run jump run twice also understand jump twice means run two variants experiment generalizing turn left jump respectively since turn right distributionally identical turn left sense occurs exactly composed commands walk run look distributionally identical jump redundant test commands moreover ensure networks highly familiar target primitive command jump turn left overrepresented training roughly training presentations obtain strikingly different results turn left jump turn left many models generalize well composed commands best performance achieved gru network attention one layer hidden units dropout accuracy overallbest model achieved accuracy hand jump models almost completely incapable generalize composed commands best performance accuracy lstm attention one layer hidden units dropout model reached accuracy case turn left although models exposed primitive command training see action denotes lturn many times used accomplish many directed actions example training example walk left jump left interpretation lturn walk lturn jump apparently seeing action sequences containing lturn suffices model understand composed commands turn left probably model receives direct evidence without performance consistently worse report generalization without systematicity table nearest training commands sample commands respective cosines jump trained isolation run trained compositionally italics mark low similarities cosine run jump look walk run walk walk run turn right run thrice run run twice run look right twice walk twice turn right turn right run twice look twice run twice look opposite right thrice run twice run right twice run twice look opposite right twice walk twice run twice lturn used context hand action denoted jump jump occurs primitive command training model generalize minimal context new composed ones take closer look results focusing run model representative run model observe even successful turn left case model errors surprising one would expect errors randomly distributed perhaps pertain longest commands action sequences instead errors made model conjunctions one components simple turn left cases turn left thrice cases particularly striking network produced correct mapping turn left training well turn left thrice test time gets many conjunctions right ironically including turn left thrice turn left turn left thrice turn left conclude even network apparently learned systematic composition almost perfectly got way hard conceive someone understood meaning turn left jump right turn left twice network gets right jump right turn left one examples network missed jump experiment network could correctly decode two composite cases starting execution primitive jump conjoined different action jump run opposite right jump walk around left thrice instructive look representations network induced various commands latter experiment table reports nearest neighbours sample commands command similarity measured cosine final encoder hidden state vectors computed respect commands present training set run provided example primitive command model exposed full composed paradigm training one would expect run close jump twice walk walk run walk walk opposite right walk look right walk walk right walk primitive commands look walk well short conjoined commands contain primitive run one conjuncts observe similar pattern jump representation induced experiment instead since jump different training distribution primitive commands model capture similarity shown low cosines nearest commands since fails establish link basic commands model generalize modifier application jump although run twice similar conjunctions primitive tasks composed twice jump twice isolated representational space far nearest neighbours look arbitrary tested systematicity purest form model exposed jump isolation asked bootstrap compositional paradigm based behaviour primitive commands walk look run although suspect humans would problems setup arguably opaque computational model could lack evidence jumping sort action walking suppose give network evidence jumping composes like walking showing composed jump command training network able generalize full composed paradigm question answered figure new primitive command compositions training make presentations even shown different composed commands jump training time network generalize composed commands correct weak generalization starts appearing network presented composed tasks training significant generalization still far perfect shows training set contains especially distinct composed commands respectively conclude network failing generalize simply original setup evidence jump generalization without systematicity figure generalization adding primitive jump compositional jump commands network trained different numbers composed jump commands generalization measured new composed jump commands bar shows mean runs varying training commands along corresponding sem behave like commands hand runs composed examples confirm found experiment network display powerful generalization abilities simply conform behaviour would expect systematically compositional consequence require positive examples emerge experiment compositionality machine translation final experiment findings broadly applicable limitations recurrent networks regards systematic compositionality extend beyond scan problems machine translation first trained standard code short words sentence pairs begin english phrases contractions randomly split training testing informal hyperparameter search led pick lstm attention layers hidden units dropout hyperparameters training procedure used scan tasks section network reached respectable bleu test score steps second examine compositionality introduction new word trained fresh network adding repetitions sentence daxy suis daxiste training data bleu score original test set dropped less point tested network embedding daxy following constructions daxy daxiste daxy est daxiste daxy suis pas daxiste daxy pas daxiste daxy est pas daxiste daxy suis daxiste daxy daxiste daxy est daxiste training model saw constructions occurring distinct predicates average limiting counts perfect matches excluding still model could get one translations right daxy comparison adjective tired occurred different constructions training corpus network accuracy testing constructions daxy one also occurred tired training set although machine translation problem preliminary result suggests models similarly struggle systematic compositionality larger data sets adding new word vocabulary ways people clearly discussion thirty years since inception systematicity debate many tested ability neural networks solve tasks requiring compositional generalization mixed results christiansen chater marcus phillips chang van der velde botvinick plaut wong wang bowers botvinick plaut brakel frank frank frank bowman however best knowledge first study testing systematicity modern models results confirm mixed picture one hand experiment turn left results experiment show standard recurrent models reach high zeroshot accuracy relatively training examples would like stress important positive result showing controlled experiments models make powerful generalizations indeed interesting direction future work understand precisely generalization mechanisms subtend networks success experiments human language plenty generalization patterns easily accounted algebraic compositionality see goldberg hand networks fail spectacularly link training testing data dependent ability extract systematic rules results change instead repeating daxy times insert times occurrences sentence training data get translations right generalization without systematicity seen trivial confirmation basic principle statistical machine learning training test data come distribution results also point important difference humans current models generalize since doubt human learners generalize unseen data data governed rules learned crucially training data experiments provide enough evidence learn composition rules affording correct generalizations experiment training data contain examples modifiers connectives needed test time producing longer action sequences experiment usage modifiers connectives illustrated training time application many combinations different primitive commands test time network apply new command encountered isolation training thus believe fundamental component current models missing ability extract systematic rules training data model abstract away surface statistical patterns operate rule space extract rules translate translate translate translate twice translate translate meaning new command translate jump learned training time acts variable rules applied learning needed test time represented abstract way training test distributions quite similar even differ terms shallower statistics word frequency encourage models extract rules data rather exploiting shallower pattern recognition mechanisms think several exclusive avenues explored first approach thrun pratt risi finn network exposed number different learning environments regulated similar rules objective function requiring successful generalization new environments might encourage learners discover shared general rules another promising approach add structure neural networks taking inspiration recent neural program induction modular network models reed freitas johnson could endow rnns set ideally learned functions interpreting individual modifiers connectives primitives job rnn would learn apply compose functions appropriate interpreting command similarly differentiable stacks tapes memory joulin mikolov graves could equip models memory structures enabling separate storage variables turn might encourage abstract rule learning solutions copying mechanisms special ways initialize embeddings novel words might help solve scan tasks specifically unlikely help general problems remains seen course proposed approaches offer truly general solution nonetheless see suggestions directions worth pursuing perhaps simultaneously complementary ways goal achieving systematicity scan beyond given astounding successes models challenging tasks machine translation one might argue failure generalize systematic composition indicates neural networks poor models aspects human cognition little practical import however systematicity extremely efficient way generalize person learns new english adjective daxy immediately produce understand infinity sentences containing scan experiments machine translation experiment experiment suggest ability still beyond grasp neural networks likely contributing striking need large training sets results give hope neural networks capable systematic compositionality could greatly benefit machine translation language modeling applications acknowledgments thank kruszewski adam liska tomas mikolov kristina gulordava gemma boleda michael auli matt botvinick sam bowman jeff dean jonas gehring david grangier angeliki lazaridou gary marcus commai team audiences facebook dialogue summit paris syntax semantics colloquium feedback advice scan tasks based navigation tasks available https references bahdanau dzmitry cho kyunghyun bengio yoshua neural machine translation jointly learning align translate proceedings iclr conference track san diego published online http main baroni marco distributions text anke merja eds corpus linguistics international handbook volume mouton gruyter berlin germany bojar chatterjee rajen federmann christian generalization without systematicity graham yvette haddow barry huck matthias jimeno yepes antonio koehn philipp logacheva varvara monz christof negri matteo neveol aurelie neves mariana popel martin post matt rubino raphael scarton carolina specia lucia turchi marco verspoor karin zampieri marcos findings conference machine translation proceedings first conference machine translation berlin germany botvinick matthew plaut david memory serial order recurrent neural network model psychological review dong lapata mirella language logical form neural attention proceedings acl berlin germany elman jeffrey finding structure time cognitive science finn chelsea abbeel pieter levine sergey modelagnostic fast adaptation deep networks proceedings icml sydney australia fodor jerry lepore ernest compositionality papers oxford university press oxford botvinick matthew plaut david empirical computational support representations serial order reply bowers damian davis psychological review fodor jerry pylyshyn zenon connectionism cognitive architecture critical analysis cognition bowers jeffrey damian markus david colin fundamental limitation conjunctive codes learned pdp models cognition comment botvinick plaut psychological review frank stefan getting real systematicity calvo paco symons john eds architecture cognition rethinking fodor pylyshyn systematicity challenge mit press cambridge bowman samuel manning christopher potts christopher composition neural networks without architectures arxiv preprint frank stefan haselager willem van rooij iris connectionist semantic systematicity cognition brakel frank stefan strong systematicity sentence processing simple recurrent networks proceedings cogsci amsterdam netherlands calvo paco symons john architecture cognition rethinking fodor pylyshyn systematicity challenge mit press cambridge chang frankilin symbolically speaking connectionist model sentence production cognitive science chomsky noam syntactic structures mouton berlin germany christiansen morten chater nick generalization connectionist language learning mind language chung junyoung gulcehre caglar cho kyunghyun bengio yoshua empirical evaluation gated recurrent neural networks sequence modeling proceedings nips deep learning representation learning workshop montreal canada published online http gehring jonas auli michael grangier david yarats denis dauphin yann convolutional sequence sequence learning https goldberg adele constructions work nature generalization language oxford university press oxford graves alex wayne greg reynolds malcolm harley tim danihelka ivo agnieszka colmenarejo sergio gomez grefenstette edward ramalho tiago agapiou john badia hermann karl moritz zwols yori ostrovski georg cain adam king helen summerfield christopher blunsom phil kavukcuoglu koray hassabis demis hybrid computing using neural network dynamic external memory nature herzig jonathan berant jonathan neural semantic parsing multiple proceedings acl volume short papers vancouver canada hochreiter schmidhuber long memory neural computation generalization without systematicity ronghang andreas jacob rohrbach marcus darrell trevor saenko kate learning reason module networks visual question answering proceedings iccv venice italy jia robin liang percy data recombination neural semantic parsing proceedings acl berlin germany johnson justin hariharan bharath van der maaten laurens hoffman judy zitnick lawrence girshick ross inferring executing programs visual reasoning international conference computer vision joulin armand mikolov tomas inferring algorithmic patterns recurrent nets proceedings nips montreal canada kingma diederik welling max efficient gradientbased inference transformations bayes nets neural nets international conference machine learning icml lake brenden ullman tomer tenenbaum joshua gershman samuel building machines learn think like people behavorial brain sciences press marcus gary rethinking eliminative connectionism cognitive psychology marcus gary algebraic mind integrating connectionism cognitive science mit press cambridge mikolov tomas joulin armand baroni marco roadmap towards machine intelligence arxiv preprint url http montague richard universal grammar theoria phillips steven feedforward recurrent networks systematic analysis implications connectionist cognitive architecture connection science reed scott freitas nando neural programmerinterpreters proceedings iclr san juan puerto rico published online http main risi sebastian vanderbleek sandy hughes charles stanley kenneth novelty search escapes deceptive trap learning learn proceedings gecco montreal canada sutskever ilya vinyals oriol quoc sequence sequence learning neural networks proceedings nips montreal canada thrun sebastian pratt lorien learning learn kluwer dordrecht van der velde frank van der voort van der kleij gwendid kamps marc lack combinatorial productivity language processing simple recurrent networks connection science williams ronald zipser david learning algorithm continually running fully recurrent neural networks neural computation wong francis wang william generalisation towards combinatorial productivity language acquisition simple recurrent networks proceedings kimas waltham yonghui schuster mike chen zhifeng quoc norouzi mohammad macherey wolfgang krikun maxim cao yuan gao qin macherey klaus klingner jeff shah apurva johnson melvin liu xiaobing kaiser lukasz gouws stephan kato yoshikiyo kudo taku kazawa hideto stevens keith kurian george patil nishant wang wei young cliff smith jason riesa jason rudnick alex vinyals oriol corrado greg hughes macduff dean jeffrey google neural machine translation system bridging gap human machine translation http
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apr alternate modules subsymplectic april abstract paper alternate module finite abelian group application alternate zero diagonal shall prove alternate module subsymplectic lagrangian cardinal exists abelian group order submodule standard symplectic module keywords alternate modules symplectic modules lagrangians finite abelian groups mathematics subject classification contents definitions statement result lagrangians alternate modules alternate module subsymplectic preliminaries proof fundamental lemma proof theorem remarks irma institut recherche university strasbourg rue descartes strasbourg france guerin definitions statement result stating result give standard definitions along properties regarding alternate modules concept defined wall tignol amitsur among results establish might directly read references also let abelian group given application say bilinear module bilinear seen definition underlying group bilinear module associated bilinear form sequel bilinear modules assumed finite let bilinear module say alternate module alternate module since alternate bilinearity since alternate whence sum saying alternate module particular let alternate module say orthogonal since alternate element orthogonal relation orthogonality symmetric bilinear form let alternate module subset define orthogonal subset elements orthogonal one directly check subgroup let alternate module subgroup bilinear module induced also alternate module called induced submodule let two alternate modules define orthogonal sum modules straightforward verification alternate module verifying induced submodule conversly alternate module two subgroups written orthogonal sum two induced modules let finite module define dual group mor result direct consequence classification finite abelian groups states isomorphic group remark isomorphism canonical let alternate module define dual application associated kernel definition kernel ker dual application words set elements orthogonal element let alternate module say symplectic module kernel trivial application alternate module associate symplectic module setting mod mod fact well defined symplectic module clear symplectic module always denoted called symplectic module associated important example symplectic module following example let abelian group define symplectic module sequel symplectic module always denoted underlying bilinear form form proof clearly alternate module let implies trivial morphism furthermore mor since isomorphic mor although isomorphism canonical implies therefore trivial symplectic module let alternate module subset say isotropic furthermore subgroup called lagrangian general see proposition cardinal lagrangian depends shall denote cardinal lagrangian example subgroups lagrangians module let alternate modules say isometric exists group isomorphism verifying isometric relation clearly equivalence relation come important definition paper let alternate module say subsymplectic exists abelian group order included standard symplectic module result going prove theorem alternate module subsymplectic whereas classification symplectic modules easily delt seems hopeless thing alternate modules general statement alternate modules subsymplectic appears closest possible result classification alternate modules another paper use result classify conjugacy classes centralizers irreducible subgroups second section shall characterize study lagrangians alternate modules third section prove theorem last section make remarks proof alternate modules general lagrangians alternate modules results section proven papers shall give references begin elementary propoosition proposition let symplectic module subgroup induced submodule symplectic module induced submodule also symplectic module furthermore proof see proposition lemma whereas alternate modules hard classify following property implies symplectic module isometric one constructed example particular set symplectic modules quite rigid corollary let symplectic module exists isomorphic proof see lemma theorem construction seen analogous symplectic base base finite abelian group dual base direct consequence corollary two symplectic modules isometric isomorphic groups somehow surprising symplectic base exists finite modules similar fashion bilinear forms spaces whereas spaces endowed alternate form necessarily symplectic always sum kernel form complementary case alternate modules general indeed one consider following let denote base define alternate bilinear form associated matrix kernel direct factor particular sum module kernel proof straightforward computation one check isomorphic remark furthermore element order verify particular direct factor little example shows kernel necessarily direct factor furthermore one check written orthogonal sum strictly smaller submodules sense irreducible suggests classification alternate modules sense would like one sequel complicated another corollary proposition corollary let symplectic module lagrangian cardinal proof definition applying first point proposition get paper tignol amitsur interested lagrangian symplectic modules apply division algebra need understand lagrangians alternate modules recall results lagrangians case proposition let alternate module lagrangians exactly maximal elements inclusion within set isotropic subsets words subset lagrangian isotropic subset containing proof let isotropic subset assume lagrangian isotropic subset containing since orthogonal whence since lagrangian whence since assumption thus maximal among isotropic subsets assume lagrangian since isotropic exists belong define clearly furthermore isotropic orthogonal assumption whence isotropic subspace whence maximal among isotropic subsets let alternate module set isotropic subspaces empty contains trivial group finite since finite therefore exists maximal element inclusion whence alternate module admits lagrangian furthermore isotropic subset always maximal isotropic subset containing therefore isotropic subset contained lagrangian particular exists lagrangian containing applying done hai next proposition generalization corollary proposition let alternate module lagrangian cardinal proof define associated symplectic module projection onto clearly application leads bijective correspondance isotropic subgroups containing isotropic subgroups since maximal elements among isotropic subsets isotropic subgroups containing follows induces bijective correspondance lagrangians lagrangians sending particular lagrangian form lagrangian corollary see cardinal lagrangians constant alternate modules however isomorphism class lagrangians may vary see also example let alternate module defined denote subgroups lagrangians isomorphic isomorphic particular isomorphic proof hence cardinal grangian using proposition directly check isotropic furthermore order therefore isomorphic whence isomorphic particular isotropic cardinal lagrangian maximality lagrangians proposition lagrangian finally clearly isomorphic likewise isotropic cardinal maximality lagrangians proposition lagrangian general much convenient work abelian finite abelian groups remark proposition let alternate module isometric orthogonal sum subsymplectic subsymplectic proof begin showing clearly contains direct sum finalely true likewise whence follows since hypothesis exists abelian gorup order nai whence included alternate module since equation follows definition subsymplectic result corollary let alternate module denote prime dividing unique dividing induced submodule subsymplectic subsymplectic proof well known result distinct primes dividing group isomorphic product furthermore spi spj order dividing power power follows order trivial particular induced modules spi spj orthogonal follows spr corollary direct consequence decomposition proposition classical properties show theorem alternate module subsymplectic basically idea make induction cardinal kernel following lemma major step proof fundamental lemma let prime number alternate module symplectic exists alternate module submodule sequel say extension modules via inclusion inclusion alternate modules inclusion modules extension constant lagrangians first subsection gathers preliminary results second subection proof fundamental lemma subsections fixed prime number alternate module assumed underlying group third subsection prove theorem preliminaries next lemma interesting generalizes second point proposition lemma let alternate module exists submodule symplectic module isometric proof let alternate module submodule symplectic also submodule since symplectic trivial follows subgroup definition orthogonal follows submodule let canonical projection since symplectic trivial isomorphic using first point proposition clearly follows finally since submodule modules cardinal isometric next lemma gives sufficient condition extension modules constant lagrangians lemma let prime number alternate module subgroup index denote induced submodule included proof let element since cardinal element uniquely written define since since result finite subgroup exponent dividing whence either trivial unique cyclic subgroup order trivial orthogonal since orthogonal generated follows orthogonal whence assumption contradiction follows cyclic cardinal let let order since hand since follows since order divides since end implies result shown finally let lagrangian isotropic proposition exists lagrangian containing using lagrange theorem divides using proposition divides follows positive integer equal one prime number particular last lemma preleminaries allows construct extension modules fairly simple way lemma let prime number alternate module group assume exist elements hei assume furthermore order strictly lesser order exists alternate module order times order inclusion inclusion submodules proof let order times order clearly mapping defines injective morphism groups order define bilinear form suffices define generating set define also define finally remark group pdivisible element admits root define one element verifying remark order thus defined divides times order assumption divides order also divides order times order follows divides order whence following equations allow define group morphism clearly alternate module let induced module via whence since alternate forms particular shown submodule via tools prove fundamental lemma proof fundamental lemma recall first statement fundamental lemma fundamental lemma let prime number alternate module symplectic exists alternate module submodule proof prove lemma strong induction cardinal module let lemma true alternate module let alternate module symplectic cardinal two different cases consider case included classification abelian decompose group product cyclic subgroups denote order may assume divides let decomposition may write since exists least one coefficient among divisible denote one index divisible define group let generator define alternate form clearly alternate form furthermore inclusion given mapped clearly leads inclusion modules finally since orthogonal since divide hence found element kernel applying lemma hence contains cardinality lagrangians strictly greater case constructed extension constant lagrangians without using induction hypothesis case included write hei denote order order may assume divides divides assume exists order equal order since order divides order bilinearity follows divides since also divides definition two elements whose order equal order result submodule generated isometric particular submodule symplectic lemma follows let induced submodule induction hypothesis exists module submodule denote clearly submodule remark since symplectic whence construction case also construct extension constant lagrangians assume order strictly divides order condition applications lemma define order times order inclusion defined bilinear form recall second case denote canonical projection onto since follows particular since clearly element follows know isomorphic corollary denote base corresponding dual base write since follows least one divisible exchanging may assume divisible let order since order furthermore definition remark definition order since divisible order whence trivial remark since particular element applying lemma fundamental lemma proven shall see theorem follows easy induction cardinal kernel alternate module proof theorem recalling result want prove following theorem theorem alternate module subsymplectic proof corollary suffices prove alternate modules prime number let show following induction let alternate module subsymplectic ispsymplectic therefore corollary isometric follows subsymplectic definition alternate module symplectic kernel trivial whence fundamental lemma exists alternate module submodule since follows apply induction hypothesis exists abelian group order since submodule finally included module follows subsymplectic definition remarks proof constructive indeed explicitely gives construction induction symplectic module containing basically order define algorithm computing one needs algorithm given computes kernel diagonalization associated symplectic module remark given alternate module might exist cardinal included let abelian group endowed trivial bilinear form follows let two inclusions two inclusions inclusions submodules endowed natural structure symplectic form defined example furthermore clear extensions constant lagrangians however stated introduction paper theorem used another paper order give classification centralizer irreducible subgroups interesting thing would classification alternate modules let alternate module say indecomposable whenever orthogonal sum since alternate module clearly sum indecomposable ones induction want classify alternate modules need characterize indecomposable ones leads following question question let indecomposable alternate module follow follow computed many examples conjecture seems verified however still proof acknowledgements computations performed examples leading result performed using gap system gap would like thank thesis advisor olivier guichard whose suggestions helpful would also like thank collinet encouraged fully investigate topic tignol pointing helpful references finally would like thank fellow student mohamad maassarani carefully listening proofs helpful comments references amitsur tignol kummer subfields division algebras israel journal mathematics vol nos amitsur tignol symplectic modules israel journal mathematics vol gap gap group gap groups algorithms programming version http tignol wadsworth value functions simple algebras associated graded rings springer monographs mathematics wall quadratic forms finite groups related topics topology vol
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jun lucretia type system objects languages viviana marcin aleksy dipartimento informatica dell university may abstract scripting languages javascript python gain popularity due flexibility still growing code bases written languages call methods make possible automatically control properties programs ensure stability running time propose type system called lucretia makes possible control object structure languages reflection subject reduction soundness type system respect semantics language proved introduction scripting languages javascript python perl ruby became popular due succinctness program expressions flexibility idioms languages optimise programmer time rather machine time effective small programs constructed advantages languages help short programs developed detrimental case big applications created short code clear advantages small programs constructed provide less information person looks clues particualr piece code one frequent activities see http work partly supported prin disco grant work partly supported polish government grant introduction software maintenance tasks see moreover strong invariants programmer rely case statically typed languages longer valid type particular value change instruction mostly uncontrolled way function call program still systems handle complex critical tasks swedish premium pension system deployed continuously maintained therefore desirable study methods help programmers understanding code keeping consistent therefore propose type system handles dynamic features scripting languages help understanding existing code type system propose presented style many type annotations must present guarantee correct typechecking particular require function declarations well labelled instructions decorated types type system strong reconstruction types probably undecidable however system designed way primarily used top type inference algorithm admits wide range type inference heuristics complete provide correct type annotations wide range programs addition would like encourage programmers add type annotations programs since annotations serve important documentation code invariants assumed developers program construction phase inherent feature scripting languages running time type particular variable may change course program execution problem solved extent introduction single assignment form local variables still applied object fields natural semantics programs fields change efforts circumvent principle probably result complicated solution therefore statement statically describes evolution running time type variable type name must reflect journey running time type throughout control flow graf program would inconvenient repeat structure whole control flow graf variable program therefore makes sense describe type visible variable statically available program point approach follow paper present type system inspired works systems however present typing judgements slightly different manner one effect described pair constraints express update performed particular instruction sense pair together typed expression viewed triple variant hoare logic constraints use express type information object approximations actual types matching constraints similar ones model language semantics paper structured follows language model scripting languages presented section type system presented section properties demonstrated section end present related work section conclude section model language semantics syntax expressions work presented fig figure presents raw syntax programs also syntax evaluation contexts represent ifnormation particular point program semantical reduction rules elaborate moreover present full syntax expressions may show evolution expression reduced reflected fact permit locations occur expressions intent directly visible programs source form definition types type information associated expression language consider combined two items one representation actual type second constraint expression approximates shape type components generated help following grammar type constant constt type variable set types set elements generated set typesb set elements generated set typesr set elements generated set constr set elements generated set objects set partial functions identifiers values set stores set finite partial functions set locations set objects set evaluation contexts set sequences basic evaluation contexts presented figure set expressions set expressions given figure semantics defined relation relates triples let partial function write denote function dom evaluation context sequence write model language semantics identifiers labels locations constants function value values objects fields list function expression expressions identifiers identifiers func let opn else ifhasattr else break new basic evaluation let texts opn else break stores figure syntax model language semantics ehlet ehlet let ehlet hvi ehe ehop ehop ehop hvi ehop ehopn hvk ehop ehop opn ehfunc ehfunc ehfunc hvi ehfunc ehfunc hvk ehe else ehif else ehif htruei else ehe ehif ehif hfalsei else ehe ehifhasattr else ehe dom ehifhasattr else ehe dom hei hbreak hvi occur label hvi ehvi new setref deref ehnewi ehli fresh ehvi dom figure semantic rules type system relation holds triples write relation ehei precise semantical rules given figure model inheritance multiple inheritance scripting languages assume viewed notational shortcut direct presentation objects therefore model features class models method updates classes python constructs language language propose features object creation new field reference dot notation field modification addition object introspetion operation available ifhasattr else construct flow information controlled let expression addition creation local variable makes possible execute sequence traditional split control flow depending computed condition realised else exceptional flow realised labelled instructions combined break statements last loops organised recursively defined functions notably use types labelled instructions function declarations types fact omitted since play part respective reduction rules possible errors may result execution operation value operation prepared take present system appropriate definition check types values supplied arguments type system typing judgements form environment mapping variables mapping locations type variables set constraints form intended meaning judgment evaluating environment types locations match mapping store satisfying leads value type store satisfying record update information record updated represented constraint variable rule record update comes two variants depending whether constraint mentions field updated type system know record contains field updated forget old value hence type ignored store update reflects new value otherwise constraint amended information new value record access access field provided constraint record guarantees field definite type access type definite type constant type variable function type disjunction definite types defined inductively follows conditional instruction basic way split processing depending value use conditional instruction instruction typed system following way bool else typing branches need typable type however constraints principle different therefore need rule weaken constraint result set need operation merges type constraint two branches defined following way type system combination two sets constraints need incorporated type derivation done using following weakening rule object structure introspection dynamic languages split processing depending condition defined terms actual values also depending types expressions therefore introduce construct performs appropriate check provide typing rule handles ifhasattr ifhasattr else typing rule must update typing information available branches instruction type information one branch takes granted attribute present need pass information attribute missing branch since case actual value field must expressed type description alternative tik tik assume base types implemented distinguished field makes possible check base type ifhasattr else variable location access information type variable recorded type environment use variable referred expression type system similarly information locations stored location environment exploit analogous way function definition call whenever want type function definition must rely type annotation associated function add type decoration functions since used recursive way inference type circumstances must rely kind fixpoint computation difficult task assume type given typechecking procedure either hand help programmer kind automatic type inference algorithm type obtain function definition type explicitly given function however check addition body function indeed obeys declared type therefore assume formal parameters types expressed precondition function type obey constraints noted constraint set precondition expect resulting type equal one postcondition type constraints match constraints postcondition worth noting constraints precondition postcondition take account formal parameters also global variables visible function body notably type function body typing interfere typing function definition reflects dynamic character language particular running time type global variable may differ function definition point expected one typing impossible particular moment however situation may different execution site therefore defer check compatibility global variables function call site func fdecl case real application function expression made arguments check application indeed lead type error case therefore typing rule application must check type information global variables call site actual parameters call elaborated accordance type information available functional type checked relation type system definition rules operate constraint sets follows reduce problem one record types relation defined follows proposition relation reflexive transitive proposition constraint sets execution function body causes changes values held global variables changes may give rise changes types must reflected type information must updated accordingly therefore need operation update definition type information update two sets constraints define inductively set constraints words constraints updated follows dom important observe update preserves least type information present constraint make update lemma constraint sets type system present rule function application rule elaborates first expressions actual parameters updates constraints side effects taken account weakens result match constraints input part function type resulting set constraint set constraints last argument updated constraint information contained result type function fapp control flow break instructions language propose includes nonlocal control flow instructions may used simulate exceptions jumps loops recursive calls case labelled instruction resembles call anonymous function parameters remember type information want achieve result execution note prospective break instructions reside inside recursive function call therefore assume type give explicitely result finitary collecting typing information break statements inside label case encounter break instruction check expression value given result indeed obeys expected result type labelled instruction surrounding break break break let expression language statement instruction sequencing operation therefore compute constraints first step second one last combine together logical cut rule forgets middle formula let let properties type system object creation object creation rule introduce information new object created expressed fact set constraints extended information object known type system fields fresh new new disjunction also add rule weaken assumptions expression primarily necessary obtain subject reduction property case instruction reduced one branches type system infers information one branches information one must added artificially properties type system soundness subject reduction need take store account hence need express fact store instance satisfies certain constraints store mapping locations types set constraints form definition constant type function values unc fun unc location obj properties type system particular every object lemma soundness constraint sets proof sufficient prove proceed first induction derivation nontrivial rules obj marginally nontrivial case derived via disjunction rule case derive inserting premise final obj rule original derivation need prove done induction definition lemma contains defined iii properties type system proof induction derivation last rule obj thesis follows immediately hand last rule definition one following holds dom dom holds thesis follows directly induction hypothesis case assumption well moreover either induction hypothesis defined appropriate thus rule completes proof lemma value completeness value properties type system proof induction derivation constant function rules thesis follows directly hand last rule used induction hypothesis hence lemma progress either value proof expressions may get stuck record access case lack field function application case arity mismatch reducibility record access follows lemma lemma value stability value moreover proof induction derivation thesis follows directly axioms function definition disjunction rule use induction hypothesis rule thesis follows proposition since value rules used derivation lemma weakening proof routine induction derivation lemma subject reduction reduction preserves typings properties type system ehei ehe exist moreover thesis holds proof induction derivation consider several cases depending last rule used case new new case since lemma field type hence lemma case else bool else hei reduces either value cases induction hypothesis otherwise properties type system typing rule lemma else case func func func simplicity consider case func func hfunc consider occurrence variable axiom type derivation stability lemma hence replacing occurrences appropriate type derivations get lemma side condition application hence transitivity lemma hence proved related work structural rules last rule used induction hypothesis thesis holds hence reapplying disjunction rule also last rule used induction hypothesis get thesis applying rule theorem soundness either ehei infinite reduction path reduces value exist value ehei proof progress lemma reduction stop value required properties final value state follow lemmas induction lreduction length related work starting point research paper guha type system scripting languages presented type system address types objects infers typing information concerning base types including special type references take account object simple conclusions future work way provide type system addition one guha infer meaningful typing information objects addition paper guha relies runtime tags present semantics dynamicly typed languages javascript checks runtime type tags viewed asserts check particular value expected type primitive however one disadvantage namely reflect split control flow makes possible use particular type still scripting languages operators check actual running time type value typeof function operator javascript hasattr getattr python consider natural rely operators instead tagchecks typing framework works fashion principle typing information present scenario realistic expectations scripting languages programmers taken account real effort bring static information program development must consider explicated realised scpecific way thorn scripting language led interesting type architecture conclusions future work type system presented paper gives expressible framework typing objects dynamic languages python javascript follows general view systems running time program property must expressed dynamical change express change means hoarelike triples describe structure relevant objects expression executed execution type system presented fashion similar necessary typing information must provided programmer work helpful heuristics infer type information make annotational effort programmer minimal way system types integrated real program development scripting languages references viviana bono michele bugliesi matching lambda calculus objects theoretical computer science references bard bloom john field nathaniel nystrom johan gregor richards rok jan vitek tobias wrigstad thorn robust concurrent extensible scripting jvm proceedings acm sigplan conference object oriented programming systems languages applications pages new york usa acm david gifford john lucassen integrating functional imperative programming proceedings acm conference lisp functional programming lfp pages new york usa acm arjun guha claudiu saftoiu shriram krishnamurthi typing local control state using flow analysis proceedings european conference programming languages systems part joint european conferences theory practice software esop pages berlin heidelberg andrew brad myers michael coblenz htet htet aung exploratory study developers seek relate collect relevant information software maintenance tasks ieee transactions software engineering december daniel marino todd millstein generic system proceedings international workshop types language design implementation tldi pages new york usa acm john ousterhout scripting programming century computer march lutz prechelt empirical comparison seven programming languages computer october william sasso cognitive processes program comprehension empirical analysis context software reengineering journal systems software september tiobe software tiobe programming community index april web page http april references stephenson fallback application built six months earns prime role fallback application built six months earns prime role http june tobias wrigstad patrick eugster john field nate nystrom jan vitek software hardening research agenda proceedings workshop script program evolution stop pages new york usa acm tobias wrigstad francesco zappa nardelli sylvain lebresne johan jan vitek integrating typed untyped code scripting language sigplan notices january tobias wrigstad francesco zappa nardelli sylvain lebresne johan jan vitek integrating typed untyped code scripting language proceedings annual acm symposium principles programming languages pages new york usa acm tobias wrigstad nate nystrom jan vitek editors stop proceedings workshop script program evolution new york usa acm
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nov language probabilistically oblivious computation david darais university maryland chang liu university california berkeley ian sweet university maryland michael hicks university maryland oblivious computation one free direct indirect information leaks due observable timing memory access patterns paper presents obliv core language whose type system enforces obliviousness prior work oblivious computation focused deterministic programs obliv new consideration programs implement probabilistic algorithms involved cryptography obliv employs substructural type system novel notion probability region ensure information leaked via distribution visible events use regions motivated source unsoundness discovered type system oblivm language implementing state art oblivious algorithms data structures prove obliv type system enforces obliviousness show nevertheless powerful enough check oblivious data structures stacks queues even oblivious rams introduction cloud computing allows clients conveniently outsource computation must trust cloud providers exploit mishandle sensitive information remove provider trusted computing base work industry research strived realize secure abstract machine comprising execution engine protected memory adversary see sensitive data operated observe data rest memory abstract machine realized encrypting data memory performing computations using cryptographic mechanisms secure computation yao secure processors hoekstra suh thekkath unfortunately secure abstract machine defend adversary observe memory access patterns islam maas zhuang instruction timing brumley boneh kocher among side channels information cloud computing adversary cloud provider physical access machines observe memory bus countermeasure unscrupulous provider augment secure processor store code data oblivious ram oram maas suh first proposed goldreich ostrovsky goldreich goldreich ostrovsky oram obfuscates mapping addresses data encrypting addresses along data replacing ram oram solves much security problem incurs substantial slowdown practical situations liu maas add overhead polylogarithmic size memory recent work explored methods reducing cost programming oram liu developed family type systems check partial use oram alongside normal encrypted ram results loss security addresses secret data could indirectly reveal sensitive information must data stored oram optimization provide asymptotic performance improvements wang working draft july david darais chang liu ian sweet michael hicks explored build oblivious data structures odss queues stacks standard counterparts implemented top oram technique involves specializing ideas oram algorithms particular data structures resulting asymptotic performance gains common cases followup work liu devised programming language called implementing oblivious data structures liu earlier work treats oram black box oblivm one program oram algorithms well odss key feature oblivm careful treatment random numbers heart oram ods algorithms goal oblivm programs secure formal argument made paper present obliv core language oblivious computation inspired oblivm obliv extends core imperative language equipped information type system sabelfeld myers primitives generating using uniformly distributed random numbers prove obliv type system guarantees probabilistic memory trace obliviousness mto possible distribution execution traces independent values secret variables property generalizes deterministic mto property enforced liu consider use randomness carrying work discovered oblivm type system unsound important contribution obliv address problem without overly restricting complicating language obliv type system aims ensure probabilistic correlation forms secrets publicly revealed random choices oblivious computations frequently case random choice made store particular block oram eventually choice made visible adversary another block later looked visibility long revealed value communicate information secret obliv ensures guaranteeing number always still uniformly distributed obliv type system presented section ensures uniform revelations part treating randomly generated numbers meaning freely copied prohibition prevents revealing number twice second revelation uniform unfortunately strong practical algorithms make copies random numbers obliv type system thus allows random numbers coerced numbers long secret version never revealed moreover obliv requires numbers allowed choice random numbers since might violate latter uniformity example able coerce random number secret look secret decide reveal random number one type system prevents problem using new mechanism call probability regions track transitive dependence numbers one region another probability regions missing oblivm source unsoundness section outlines proof obliv enjoys mto obliv program well typed observable behavior oblivious details proof appendix obliv type system oblivm implemented type checker obliv section presents version tree oram shi oram implementation language type checks also show section oblivious stacks kind oblivious data structure wang type checked implemented top oram far aware implementations constitute automated proofs data structures indeed oblivious believe obliv promising step forward generalizing work came section discusses related work http language probabilistically oblivious computation let add mem int int array int int let len length let rec iterate int pair int int unit len else wrong code first let mem mem pair else iterate pair correct code let mem let addr pair let mux addr pair mem iterate iterate fig adding association array obliviously aka add trivial oram background section presents threat model background deterministic oblivious execution next section motivates sketches novel type system enforcing probabilistic oblivious execution threat model assume powerful adversary make observations program execution particular use generalization program counter security model molnar adversary knows program executed observe execution well contents patterns memory accesses secret memory contents may encrypted public memory memory addresses still visible relevant instantiation consider untrusted cloud provider using secure processor like sgx hoekstra memory directly observed secret memory encrypted using key kept processor pattern accesses timing information system features instruction cache misses provide information another instantiation secure computation mpc using secret shares goldreich two parties simultaneously execute program thus know program program counter certain values entered one party kept hidden using secret sharing techniques also handle weaker adversaries observe memory make timing measurements observe memory oblivious execution goal ensure memory trace obliviousness mto kind noninterference property goguen meseguer sabelfeld myers property states despite able observe address instructions data fetched public value adversary able infer anything manipulated secret values see inference could occur consider program figure syntax program takes array pairs secret integers mem secret address secret value adds david darais chang liu ian sweet michael hicks pair free slot array value address commented code lines illustrate implementation code works line extracting current array element line checking address updating element otherwise line continuing consider next array element notice designated int means public int index array revealed adversary observe memory accesses program executes mem idx adversary observe address read knowing starting address array mem adversary compute idx watching address trace adversary learn something array contents particular sequence reads followed write implies slots free slot writes means free slots adversary also watch program counter notice true false branch conditional taken revealing information code lines problems mux line key part takes three arguments boolean condition whether current slot free also whether pair address mux evaluates pair comprised mux second third arguments order otherwise pair contents reversed line updates mem current slot element pair recurses consider next element passing second element pair code oblivious reads writes every slot array always executes exactly statements matter contents array index value arguments observing code execution reveals nothing secrets manipulates code actually add operation trivial oblivious ram oram implementation goldreich goldreich ostrovsky oram provides api like array makes sure address trace reveal relationship index value read oram write oram operations index kept secret later consider sophisticated oram algorithms use trivial oram building block obliviousness typing liu developed type system ensures programs mto types extended indicate values allocated per example data public secret also reside oram liu operational semantics reduces expression memory heap expression emitting trace event trace events include fetched instruction addresses public values addresses public secret values read written addresses values data stored oram modeled black box basically treated array whose accesses opaque adversary similar modeling encrypted messages dolev yao model mto property means running two memories agree public produce exact memory trace along output heaps results formally imply operator denotes enforcement probabilistic obliviousness liu type system enforces obliviousness deterministic programs use oram however work oblivious data structures wang shown probabilistic uses oram far uses visible memory traces need identical computing secrets rather must identically distributed possible memory traces equally likely matter secrets used indeed property oram language probabilistically oblivious computation let stackop oram stack ispush bool let rid let read oram rid let mux ispush let mux ispush rid rid write oram let rid mux ispush rid rid fig deterministic oblivious stack built using full recursive oram algorithms open black box actually providing section motivates probabilistic programming orams sketches type system enforces mto programs section shows type system actually powerful enough implement oram algorithms directly need treated black boxes motivation naive oblivious stack goal oblivious stack hide data operations pushes pops taking total number operations revealed incorrect approach would implement stack usual using secret encrypted array would hide contents array indexes array would visible adversary discussed section would reveal indexes indicate pushes decreasing ones indicate pops hide index store stack oram instead array merge code push pop operations shown stackop function figure observation reveal operation taking place code stack consists oram secret numbers reference storing index root function stackop takes stack indicating whether operation push pop value push returns value code reads value root index line next line copies value operation pop else puts push line determines index write perform line index one root index push current root index write line puts given value next slot case push writes back value current root pop finally line adjusts root index line returns result either popped value pushed value push code works perfectly well would typecheck liu system probabilistic version stack using oram would particular require extra space current size stack whereas version requires extra space size oram see next recursive oram understand source digress consider features oram algorithms particular consider oram shi stefanov wang two parts structure storing actual data blocks position map maps logical data block indexes position tags indicate block position tree simplest instantiation oram size position map size hidden adversary could stored memory deployment oram position tags mask relationship logical index david darais chang liu ian sweet michael hicks location corresponding block tree blocks read written around data structure new locations recorded position map note position map need hidden chip rather much stored recursively oram reducing space overhead client tree part contains data blocks sometimes called oram noram short oram read write operations implemented sequence two noram operations called readandremove short add noram idx int tag int add noram idx int tag int block unit type noram designates tree portion oram data structure stores data blocks type operation int argument desired block logical index hidden adversary int argument position tag extracted map returned data block add operation similar except position tag secret adds block tree rather removing full oram read index requires looking tag position map performing read remove value adding back value updating position map fresh tag write requires remove old value add write new value updating position map depending details noram algorithm add may perform many reads writes tree simplicity think call noram pos producing single abstract event noram pos event indicates operation noram performed may one position tag visible correlates actual memory addresses accessed retrieve requested block add operation behavior depend position tag simply generates event add noram fact position tag visible adversary means traces involving operations might dependence particular secret values due innocuous random variation hence require general obliviousness discussed running time add log position map requires storage access time stored client hidden adversary stored recursively oram position map imposes storage client log access time standard deployment full recursive oram necessary position map accesses visible adversary deployments turns obliv powerful enough implement orams scratch present details section optimized oblivious stack returning question implement oblivious stack consider always access stack via head using root index thus code figure full oram internally ever uses one slot position map thus better using noram directly stack manage position tag root short implement oblivious stack triple comprising noram index root element position tag latter two act kind pointer noram block stored noram contains data position tag next block stack language probabilistically oblivious computation type ostack rnd noram int ref rnd ref let stackop noram ostack ispush bool let rid pos let rid pos ispush let noram reveal rnd let pos let pos rnd let add noram use pos rid pos else let pos noram rid reveal pos let rnd let add noram rid pos rid pos fig probabilistic oblivious stack built using oram use mux simplicity code implementing stack following design given figure note code branches ispush variable make easier read actual implementation must use muxs conditionally execute statement branches ensure line extracts current root index position tag lines handle push operation line dummy read noram saw oblivious add figure using index results dummy block returned position tag argument unimportant case line constructs new block push consists given data paired current root position tag pos thus creating pointer block generate fresh position tag pos new root block add block noram random numbers generated rnd type rnd coercion reveal ascribes random number type int per line use gives type int line explain constructs next subsection new root index old one plus one root tag dummy block passed returned line lines handle pop real work extracting block corresponds root index position tag generate dummy block add oram updated root index old one minus one position tag returned fetched block returned version oblivious stack performs better version figure space overhead due added pointers root within oram size current stack size oram running time still random numbers generated lines eventually revealed operations lines many possible events sequence secret inputs code safe possibility visible trace reveals secret information need ensure traces identically distributed formally traces probability evaluating produces trace notice structure branches roughly parallel makes converting use muxes straightforward david darais chang liu ian sweet michael hicks probability evaluating also produces property call probabilistic trace obliviousness ensured underlying noram implementation need ensure programs use preserve turns property holds stack type system described next obliv obliviousness typing probabilistic programs main contribution paper obliv core language ensuring obliviousness programs use randomness obliv type system establishes programs like one figure oblivious employing two mechanisms treatment random values probability regions track dependences random values could leak information value revealed random numbers produced executing rnd given type rnd probability region explained shortly values type like secret values invisible adversary converted int numbers use coercion converted int numbers reveal coercion example figure line creates random value pos converts secret number passes add operation line sets position tag root line line root position tag converted public number passed argument revealing adversary obliv type system enforces random number made public treating values type rnd duplicated assignments consumed operations use example variable pos consumed line passed use consumed assigned line subsequent reference line would illegal similarly line variable pos read root tree type rnd consumed line given reveal important consider example variable type int let rnd rnd let mux output reveal output reveal bad lines code safe generate two random numbers invisible adversary store one depending whether secret zero revealing line safe regardless whether contains contents fact uniformly distributed means whatever revealed nothing learned however revealing line revealed safe seeing two zeroes row likely happens zero violates obliviousness similar problem would arise oblivious stack changed line figure let pos pos would set position tag new root old root tag pos revealed result leak information secret ispush variable next operation whether push previously performed pop revelation type system catches mistake assigning pos line consumes reading line would disallowed probability regions unfortunately reasons needed expressiveness use convert random numbers secret ones computing distribution revealed values see consider program language probabilistically oblivious computation let rnd rnd let use let mux let mux output reveal bad line generates two random numbers line creates secret copy consume line sets otherwise reverse true problem uniformly distributed likely contain zero number result outputting line dangerous seeing zero likely thus says likely zero notice violated random number revealed publicly type system addresses problem novel construct call probability regions normal random number types ascribed region region represents set random numbers generated program points whose rnd instructions annotated elided region name far normally write rnd regions allow type system reason random numbers probabilistically independent particular regions follow lattice ordering say two regions independent meet otherwise regions may depend one another meaning revelation random number one region may compromise uniformity distribution number could result leak see example region number region since derived two independent lack independence problematic conditioning return means output longer uniform hand guard mux region independent regions branches uniformity output threatened normal numbers never random numbers region independent random number notational convenience write secrecy label int int means region looking example line perfectly safe assuming type int region independent region allowing safe code critical implementing oblivious algorithms formal language section presents obliv syntax sampling semantics type system next section presents main metatheoretic result typing ensures memory trace obliviousness syntax figure shows syntax obliv syntax terms kind form simplify semantics expressions comprise let binding conditionals atomic expressions comprise various computational forms discussed shortly pico expressions comprise variables literals tagged security label literals either bits either natural numbers security labels either public secret semantically literals labeled invisible adversary labeled think literals encrypted returning atomic expressions takes complement bit represents arbitrary arithmetic operation asnat converts bits nats asbit reverse expression randomly generates bit type call values type coin annotation coin probability region defer discussion section coin david darais chang liu ian sweet michael hicks label region parameter aop parameter var lit pico atom asnat asbit use reveal mux arrayn expr let let else security label probability region bits natural numbers arithmetic ops lexical variables literals pico expressions atomic expressions bit complement arithmetic ops conversion conversion coin convert bit reveal public atomic conditional tuple creation array creation array read atomic array compound expressions local variable binding local tuple binding conditional fig syntax obliv semantically secret limited use coerce one public bit via reveal secret bit via use expression mux unconditionally evaluates returns pair given order evaluates opposite order evaluates operation critical obliviousness operation atomic contrast normal conditionals else evaluate either depending never instruction trace communicates branch taken components tuples constructed accessed via let finally last three atomic expression forms constant array creation read update respectively designed obliv capture key features used programming oblivious data structures notably numbers tuples arrays random numbers omits obviously useful features avoid unimportant complexity many missing features added without implementation example easily encode random numbers tuple bits ready conversions nats allocate arrays support lambda terms bounded recursion using standard ideas imagine obliv programs post inlining allowing opposed bounded recursion language probabilistically oblivious computation technically delicate also doable discuss implementation issues including support richer types section sampling semantics figure shows operational semantics obliv programs main judgment states steps probability form rational number contains expression environment store special halt form indicates terminated computation expression replaced value value labeled literal address pair values environment maps variables values store maps array addresses sequences values bottom gives rules main judgment basically standard notice variables put environment rather substituted target term rules inherit probability annotation subsidiary judgment states atomic expression steps value output store probability turn rules invoke pjpk evaluate pico looked environment literals returned atomic expression rules probability except rules flip return probability rules preceding computing bits numbers straightforward assume given semantics function operations rules mux evaluate branches values pair order values depends bit guard evaluates array allocation stores sequence given values fresh address store array read extracts value given index notice must public array write returns existing value updating appropriate position sequence present store array update return old value critical coexisting type system treatment random numbers described attempting access array bounds note represent information leak array indexes public traces adversary observations execution trace sequence whereby sequence proof judgment halted parts trace adversary see according function obs function takes sequence produces corresponding sequence bulleted omit function space reasons given appendix figure operation straightforward bullet original except secret numbers random replaced capture idea adversary see full instruction trace structure expressions visible address trace addresses values public literals secret numbers hidden type system given next able ensure even adversary see trace learn nothing program secret values typing figure shows type system obliv judgment states type environment expression type yields residual type environment types environments figure bits numbers respectively type label probability region respectively mentioned earlier characterizes coin region implicitly label types array characterize arrays pairs respectively environments map variables either types inaccessibility tags latter used support discussed shortly david darais chang liu ian sweet michael hicks env var store addr halt addr pico env val pjxk pjxk compo compi aop asno asni asb flipo flipi use rev muxi muxo tup arr read write pjpk pjpk asnat pjpk asnat pjpk asbit pjpk flip flip use reveal mux mux array pjp fresh atom lett halt let hlet ifi hif else ifo hif else fig sampling semantics obliv hlet language probabilistically oblivious computation type array tenv var type type kind type kind array fig type kind language obliv high level type system enforcing three properties type errors occur runtime operating array number secrets inferred information probabilistic correlation forms secrets publicly revealed random numbers properties enforced using standard techniques example program operate array number rule aop types expression numbers arrays moreover operation correctly captures secret information annotating result type security label join labels recall likewise rules ensure array indexes conditionals leak information improperly property main novelty work need arises obliv support random numbers used enforce secrecy later seen adversary key invariant coin type always following properties distribution independent values distribution stable meaning probability possible bit value type system allows creating manipulating eliminating values type ways preserve properties particular property maintained treating values prevents duplication property maintained tracking probability regions values combined used computations enforce variable used program type removed output environment figure kinding metafunction assigns type either kind freely duplicatable arrays nats bits always universal always pair considered either components rule varu figure types variables output environment input environment rule vara types variable marking output environment rule rule problematic example section rules rules use reveal permit converting types bits via use reveal coercions respectively converts bits make inaccessible second converts bitp make inaccessible essence type system enforcing random number made adversaryvisible secret copies never revealed allowed arrays permitted contain universal values latter case safe read array element directly since former would become immediately inaccessible hence rule read requires array element type kind rule write applies arrays david darais chang liu ian sweet michael hicks varu vara litb litn asn rev use bits mux muxflip mux array arr write let lett asbit use aop reveal read comp tup asb asnat muxbit flip array array array let let else fig typing judgment obliv language probabilistically oblivious computation returned value old one read written one serves immediate replacement guaranteeing store remains finally note variables could made inaccessible branches conditional types branch initial context joins output contents variable made inaccessible one branch inaccessible joined environment probability regions consider role probability regions type system probability regions partially ordered set purpose regions track dynamic probabilistic dependencies values regions must distributive lattice least element distributive lattice must support join meet operations meets must distribute joins notate meet two call region independence distributivity regions bottom necessary conjunction region independence coincide region independence join rule flip types coin region annotation static name coin generated program point region must represents independence random values coin dependent created coin independent distribution value independent region stable meaning possible bit value equal probability given interpretation type able typecheck value type also type given rule comp rules literals litb litn track security label literal given probability region signify probabilistically independent random values rules use reveal mentioned crucially preserve probability region input output type type system track potential dependence values rules asn asb convert bit nat types also preserving region per rule aop result type arithmetic operation given join operand labels join operand regions particular joining regions indicates whatever correlates may also correlate result operation safe correlation result binary arithmetic operation operates directly values distributions values critical feature type system handling mux expressions treated rules muxbit muxflip bit types respectively mux rule bits treats mux ternary bit operation takes join security labels probability regions argument mux rule types designed avoid passing probabilistic dependence guard labels output pair types acceptable probability region guard independent regions arguments revelation values stability independence example following program let let let let use let mux output reveal david darais chang liu ian sweet michael hicks according muxflip rule type region probabilistically independent revelation line stability hand rule muxflip prevents second problematic example section page example mux line disallowed region guard independent region argument remaining type rules handle tuples arrays let binding normal conditionals remark important features first notice read write require array index label important assume adversary able observe array accesses also avoids leaks due programs accessing array bounds rule similarly requires guard label since execution trace reveals branch taken second let lett remove bound variables output environment write split context part binds binds rest latter part returned dropping binding probabilistic memory trace obliviousness section describes proof memory trace obliviousness obliv begin presenting model distributions probabilities conditional probabilities probabilistic dependence next give denotational semantics based model prove coincides sampling semantics section equipped model distributions denotational semantics state precisely memory trace obliviousness describe proof approach proceed informal terms describe key semantic invariants hold denotational semantics key invariants prove strengthened memory trace obliviousness property strengthened property implies simpler property initially stated details proof including precise mathematical properties given appendix distributions use discrete model distributions universe entropy model distribution elements universe notated number available coin distributions represented mapping bitvector length element domain supports modeling result terminating computation arbitrary number coins model support nonterminating computations especially may number coins example suppose want represent distribution count coin come thus function takes vector coin returns natural number sums occurrences vector probability measure distributions counting occurrences dividing example probability would model distributions particularly useful expressing measure conditional probability conditional probabilities distributions counting simultaneous occurrences dividing occurrences conditional probabilities therefore derived notion opposed axiomatic notion typically case modeling distributions universe informative omit write notate joint probabilities notate point distributions language probabilistically oblivious computation key approach careful bookkeeping set coin upon information observable adversary depends notate probabilistic dependency prove two random variables depend disjoint sets coin probabilistically independent written means joint probability factors often notate probability conditioned probability matter values coin notate probabilistic dependency conditioned set coin modulo probabilistic independence modulo call two distributions equivalent notated probability measures coincide elements call equivalent modulo notated probability measures coincide elements conditioned another key invariant proof regard distribution stability notated stable stability property outcome distribution equally likely call distribution stable modulo notated stable stable conditioned denotational semantics facilitate proof memory trace obliviousness obliv denotational semantics distributions prove corresponds sampling semantics previously section denotational semantics two parts evaluation semantics expressions notated trace semantics expressions notated evaluation semantics denotes expression partial function distributions environments stores distributions values environments stores recall expression environment store execution trace sequence trace semantics denotes expression partial function distributions environments stores distributions traces exp store store exp store semantics partial due possibility runtime type errors however type safety proved later guarantees denotational semantics total expressions theorem sampling denotation correspondence sampling semantics denotational semantics correspond halt memory trace obliviousness design obliv well typed programs property memory trace obliviousness mto state prove mto respect adversary able observe memory instruction access pattern execution mto theorem establishes adversary learns nothing observations david darais chang liu ian sweet michael hicks theorem memory trace obliviousness mto given well typed obliv expression two value environments agree public values publicly observable traces executing using yields equivalent distributions jek obs jek obs obs obs public observations value environment store using obs see section notate lifting distributions obs proof outline section outlines key ideas invariants required proof mto provide mathematical detail key properties appendix property prove type safety shows partial denotation functions actually total assumption term well typed theorem type safety given obliv expression distributions value environments value stores evaluation semantics ejek always result exists ejek prove type safety trace semantics consequence type safety evaluation semantics next step proof give semantics regions appear types relate semantics regions evaluation semantics achieve region semantics notated applied produces map regions concrete set coin occurred region annotation evaluation call map region model notated key invariant region model disjoint regions map disjoint sets achieve two parts first dependency semantics notated maps case base types set coin set dependencies resulting value conditioning public revelations given set coin dependencies desired relationship regions values whenever expression annotated region evaluates value dependency set predicted dependency semantics included set coin associated region model make steps precise theorems used proof theorem dependency soundness given obliv expression distributions value environments value stores models dependencies environment store respectively conditioned set publicly revealed bits result dependency semantics results evaluation semantics conditioned equal larger set publicly revealed bits language probabilistically oblivious computation theorem region soundness given obliv expression distributions value environments value stores region model conditioned set publicly revealed bits result region semantics results dependency semantics conditioned equal larger set publicly revealed bits primary purpose region semantics dependency semantics soundness comproperties establish terms typed disjoint regions turn two key invariants puted values probabilistically independent justify memory trace obliviousness presence reveal expressions partitioning stability say evaluation contexts two bit distribution values type flip occurring positions evaluation context guaranteed independent conditioned publicly revealed information prove theorem evaluation semantics preserves well partitioning conis ditioned public information theorem given obliv expression distributions value environments value stores every contained flip value flip values conditioned set publicly revealed bits result evaluation semantics equal larger set publicly revealed bits key part proof reveal expressions revelation assumed values well partitioned immediately reveal expressions values remain conditioned larger set publicly revealed bits publicly revealed added subsequent execution publicly revealed bit may end correlating values result public expression however partitioning quotiented publicly revealed bits correlation ends benign mechanism type system enforces semantic property treatment values type flip flip values duplicated always one copy coin context therefore value independent values stability also property evaluation contexts two bit distribution values type flip evaluation context either hguaranteed stable distrii butions conditioned publicly revealed information stable prove theorem evaluation semantics preserves stability conditioned public information theorem stability given obliv expression distributions value environments value stores every contained flip value stable conditioned set publicly revealed bits result evaluation semantics stable conditioned equal larger set publicly revealed bits proof stability uses proof partitioning lemma key part proof reveal expressions revelation assumed values stable immediately revelation must shown coin remain stable consequence guarantees revealed coin correlate coin therefore adding revealed bit set public information perturb remaining stable coin context mechanism type system enforces semantic property stability typing rules bit complement mux bit complement safe stability prove david darais chang liu ian sweet michael hicks mux rule stable conditions region independence turn imply underlying distributions values probabilistically independent public revelation two language forms manipulate coin typing rules guarantee resulting coin remain stable order prove desired memory trace obliviousness property state stronger property implies desired property supports direct proof induction term stronger trace obliviousness property includes assumptions lemmas lemma applied case proof induction show details theorem strong memory trace obliviousness smto given obliv expression satisfying properties lemmas applied two sets distribution value environments stores distributed equally publicly observable values conditioned publicly revealed bits obs obs obs observable traces applied either set obs contexts yield identical distributions conditioned publicly revealed bits jek obs jek obs proof prove theorem induction syntax terms applying theorem developed thus far example let expressions proof goal form jlet obs jlet obs trace semantics goal becomes obs jek obs obs obs jek obs obs ejak product concatenation operations respectively lifted distributions goal broken using product chain rule distributions states using notation case three elements case two elements tuple independent factors without conditioning per independence using product chain rule order show two distributions goal equivalent show component independent elements threef element equivalent pairwise tuple obs obs recursive occurrence trace equivalent element obs obs pairwise conditioned establishing independence elements show obs obs obs obs obs jek obs obs obs jek obs obs goals consequences partitioning stability theorems goal based observation new contexts depend uniquely equal larger set publicly revealed bits contexts remain stable therefore condition potentially larger set publicly revealed bits language probabilistically oblivious computation demonstrate equivalence distributions module jek obs jek obs point inductive hypothesis applies proof let complete proofs let tuples public statements follow structure analogous proving smto recover proof mto instantiation smto empty empty set publicly revealed bits store case study implemented type checker extension obliv adds functions bounded recursion various arithmetic operations used check two classical probabilistically oblivious algorithms oram noram orams mentioned section used noram building block implement oblivious stack section discuss implementation noram tree oram latter makes use former follows write nat secret number region nat public number nat secret region also write type rnd random natural number region implementation encode using tuples flip oram oram noram foundation orams norams provide performance enable oblivious data structures noram organizes data blocks complete tree depth log capacity oram node tree bucket essentially trivial oram size call bucket size trivial oram basically array reads writes require iterating element array showed trivial oram add operation section noram data block bucket associated randomly generated position tag log random number corresponding one leaf tree key invariant noram data block always resides bucket somewhere path root leaf corresponding position tag particular following signature type noram bucket array type bucket block array type block bit nat nat rnd rnd noram array buckets represent complete tree style heap data structure node index parents left child right child correspond nodes index respectively bucket array blocks last element actual data ideally noram able store data arbitrary type since language yet support polymorphism data type rnd rnd independent choose type illustrate values stored noram set implementation tree oram next actual interpreter duplicate code type wish store three components block secret region bit indicating whether block dummy index block position tag block noram supports two primitives add introduced greater generality section signatures implementation david darais chang liu ian sweet michael hicks noram idx nat tag nat data rnd rnd add noram idx nat tag nat data rnd rnd unit due noram invariant position tag corresponding block known implemented linear scan path corresponding tag block given index log buckets accessed blocks therefore routine implemented log time oram constructions circuit oram wang parameterized constant renders overall time complexity log much faster linear scanning trivial oram code let noram noram idx nat tag nat rnd rnd let length noram let length noram let rec bucket idx nat nat read rnd rnd read else read current block let isdummy data bucket true check current block index matches queried one let toswap bit isdummy idx let data read mux toswap read data let isdummy mux toswap false isdummy toswap false equivalent writting data back otherwise read stores found block data field passed next iteration let bucket isdummy data bucket idx read let rec level nat read nat let base nat pow level compute first index bucket array depth level base length noram read else let nat base tag base bucket path access let bucket noram let read readbucket bucket idx read level read let read read implementation involves two nested loops line implements outer loop iterates buckets path root leaf corresponding tag line iterates blocks bucket essentially trivial oram read things notice first noram array arrays since array addresses read noram directly line hand buckets contain blocks data part must replace read block dummy one shown lines line toswap computed determine whether swap block whether index matches queried one line muxes toswap picks either block data argument read pairs type rnd rnd toswap region pairs independent region rule muxflip ensures returned value still store data noram would require region code type check language probabilistically oblivious computation add routine noram also simple adds data bucket corresponding root node tree bucket index array similar routine omit implementation avoid root bucket due repeated adds oram employs additional eviction routine evict blocks buckets noram nodes closer leaf buckets routine also maintain key invariant data block always reside path corresponding position tag oram implementations choices eviction strategies one simple eviction strategy picks random nodes level tree reads single block bucket writes block one level either left right dummy block written opposite direction make operation oblivious recursive oram use noram need way get position tag reads writes recursive oram employs additional position map solve issue map maps index capacity oram randomly generated position tag given add routines noram tree oram routine access get position tag invoke noram routine retrieve data block mentioned section one way implement position map use regular array stored hidden memory secure processor deployment oram however possible deployments adversary observe access pattern map avoid problem use another smaller oram store particular oram blocks block contain elements parameter oram therefore also contains another position map stored oram capacity construction continue recursively position map bottom small enough constructed trivial oram therefore logc norams constructed store position maps thus overall runtime recursive oram logc times runtime noram following implement oram full oram thus type oram given type oram noram array type array type bit nat rnd rnd short oram sequence norams sequence contains actual data remainder serve position map eventually terminate trivial oram trivial oram simply array blocks contain isdummy index data recall trivial orams use position tags implementations data pair position tags main idea think position map essentially array size stored array array access idx essentially access idx mod implement call function takes additional public level argument indicate point list norams start work initially called level let rec oram oram idx nat level nat rnd rnd let norams oram let levels nat length norams present readable version code give full picture section david darais chang liu ian sweet michael hicks level levels trivial oram case returning block rnd rnd else let rnd rnd oram idx level let tag mux idx mod rnd let tag mux idx mod tag let oram idx level norams level idx reveal tag let oram oram idx nat rnd rnd oram idx line checks whether hit base case recursion case lookup idx returning back rnd rnd pair code shown otherwise enter recursive case line essentially reads lines obliviously read idx mod tag replacing freshly generated tag satisfy requirement finally line writes updated block back using analogous routine level finally line reveals retrieved position tag index idx passed routine noram level corresponds actual data oram returned client routine similar show recursively adds corresponding bits position tag array norams level recursion snippet like following let rnd let use consume let rnd rnd oram idx level let tag mux idx mod replaces new tag let tag mux idx mod tag let oram idx level add norams level idx data adds tree oram lines generate new tag make secret copy new tag stored recursive similar replace found tag garbage value appropriate level position map line finally used store data appropriate level noram limitations astute reader may noticed code snippet add type check particular argument type nat add requires type position tags noram level stored data noram level regions put region require single noram metadata data reside regions solve problem level use opposite pair regions one solves type error compromise noram independence requirement unfortunately means oram type made bit awkward essentially type oram noram noram array basically operate two levels time order satisfy type checker basic logic addition require type idx nat probability region rather region lattice idx passed unrolled levels one requires unfortunate limitation implement noram also implement oblivious stack see consider operation implemented language probabilistically oblivious computation data type nat rnd required oblivious stack component stack element value second position tag stack next element issue stack element position tag region position tags underlying noram violates independence requirement mux line however violation false positive tags happen region actually tag generation conditioned value tag never stored block indeed using general noram signature presented section obliv type system stack figure secure checking implementation stack noram likely requires generative treatment regions use existential types features may simplify solution recursion problem well pursuing ongoing work related work lampson pointed various covert side channels information leakage program execution lampson defending leakage challenging previous works attempted thwart leakage various angles processor architectures mitigate leakage timing kocher liu power consumption kocher fletcher liu maas ren program analysis techniques formally ensure program bounded leakage instruction traces molnar timing channels agat molnar russo zhang memory traces liu algorithmic techniques transform programs algorithms counterparts introducing mild costs works mitigating timing channel leakage askarov barthe zhang preventing leakage blanton eppstein goldreich goldreich ostrovsky goodrich shi stefanov wang zahur evans often comprehensive approach combining areas advances fact several aforementioned works indeed combine subset algorithms architecture programming language techniques fletcher liu ren zhang core language generalizes line prior works timing channel security agat program counter security molnar obliviousness liu also provides distinct core language captures essence obliviousness without treating key use black box thus express algorithmic results formally reason security implementations thus building bridge algorithmic programming language techniques oblivm liu language programming oblivious algorithms intended run secure multiparty computations yao type system also employs types ensure random numbers used however provides mechanism disallow constructing distributed random number random numbers generated distinguished attacker uniformly distributed random numbers revealed therefore type system oblivm guarantee obliviousness obliv use probability regions enforces random numbers uniformly random thus eliminates david darais chang liu ian sweet michael hicks channel information leakage moreover prove mechanism others obliv prove pmto work belongs large category work aims statically enforce noninterference typing sabelfeld myers volpano probabilistic memory trace obliviousness property bears resemblance probabilistic notions noninterference much prior work ngo russo sabelfeld sabelfeld sands smith concerned random choices made thread scheduler could cause distribution visible events due values secrets source nondeterminism external scheduler rather program case smith consider random numbers may likelihood certain outcomes mostly concerned termination channels programming model rich secret random number never permitted made public ability main source complexity obliv crucial supporting oblivious algorithms prior work aims quantify information released possibly randomized program rybalchenko clark according measures work verifying correctness private algorithms barthe zhang kifer essentially aims bound possible leakage contrast enforce information leaks due program execution conclusions paper presented obliv core language suitable expressing computations whose execution oblivious powerful adversary observe execution trace instructions memory accesses see private values unlike prior formalisms obliv used express probabilistic algorithms whose security depends crucially use randomness obliv tracks use randomly generated numbers via substructural type system employs novel concept called probability regions latter used track random number probabilistic dependence random numbers proved together mechanisms ensure random number revelation visible trace perturb distribution possible events make secrets likely demonstrated obliv type system powerful enough accept sophisticated algorithms including forms oblivious rams oblivious data structures data structures reach prior type systems implementations represent automated proofs algorithms secure currently pursuing two threads ongoing work first working powerful notion probability region grained accept oblivious data structure algorithms end end second working expand general expressiveness language example support containers requires form parametric polymorphism support deployments oram requires polymorphism implementable dependent types ultimately hope integrate ideas oblivm liu compiler oblivious algorithms thereby ensure programs indeed secure acknowledgments thank aseem rastogi kesha heitala comments earlier drafts paper elaine shi helpful suggestions comments work underway material based upon work supported national science foundation grant nos language probabilistically oblivious computation darpa contracts opinions conclusions recommendations expressed material author necessarily views national science foundation references johan agat transforming timing leaks popl aslan askarov danfeng zhang andrew myers predictive mitigation timing channels ccs gilles barthe boris federico olmedo santiago zanella probabilistic relational reasoning privacy acm trans program lang syst gilles barthe tamara rezk alejandro russo andrei sabelfeld security multithreaded programs compilation acm transactions information system security tissec marina blanton aaron steele mehrdad alisagari graph algorithms secure computation outsourcing asia ccs david brumley dan boneh remote timing attacks practical usenix security dolev yao security public key protocols proceedings annual symposium foundations computer science sfcs david eppstein michael goodrich roberto tamassia geometric algorithms geographic data gis christopher fletcher ling ren xiangyao marten van dijk omer khan srinivas devadas suppressing oblivious ram timing channel making information leakage program hpca goguen meseguer security policy security models ieee goldreich towards theory software protection simulation oblivious rams stoc goldreich micali wigderson play mental game stoc oded goldreich rafail ostrovsky software protection simulation oblivious rams acm michael goodrich olga ohrimenko roberto tamassia graph drawing model algorithms corr matt hoekstra intel sgx dummies intel sgx design objectives https mohammad islam mehmet kuzu murat kantarcioglu access pattern disclosure searchable encryption attack mitigation network distributed system security symposium ndss paul kocher ruby lee gary mcgraw anand raghunathan security new dimension embedded system design proceedings annual design automation conference dac srivaths paul kocher timing attacks implementations rsa dss systems crypto boris andrey rybalchenko automation quantitative analysis formal methods dynamical systems butler lampson note problem commun acm chang liu austin harris martin maas michael hicks mohit tiwari elaine shi ghostrider hardwaresoftware system memory trace oblivious computation asplos chang liu michael hicks elaine shi memory trace oblivious program execution csf chang liu yan huang elaine shi jonathan katz michael hicks automating secure computation ieee chang liu xiao shaun wang kartik nayak yan huang elaine shi oblivm programming framework secure computation ieee isaac liu jan reineke david broman michael zimmer edward lee pret microarchitecture implementation repeatable timing competitive performance iccd martin maas eric love emil stefanov mohit tiwari elaine shi kriste asanovic john kubiatowicz dawn song phantom practical oblivious computation secure processor ccs david molnar matt piotrowski david schultz david wagner program counter security model automatic detection removal side channel attacks icisc chunyan david clark abstraction quantifying information probabilistic semantics workshop quantitative aspects programming languages qapl tri minh ngo stoelinga marieke huisman programs journal computer security ling ren xiangyao christopher fletcher marten van dijk srinivas devadas design space exploration optimization path oblivious ram secure processors isca david darais chang liu ian sweet michael hicks alejandro russo john hughes david naumann andrei sabelfeld closing internal timing channels transformation annual asian computing science conference asian alejandro russo andrei sabelfeld securing interaction threads scheduler sabelfeld myers security ieee commun andrei sabelfeld david sands probabilistic noninterference programs elaine shi hubert chan emil stefanov mingfei oblivious ram log cost asiacrypt smith probabilistic noninterference weak probabilistic bisimulation smith rafael secure information flow random assignment encryption workshop formal methods security fmse smith rafael fast probabilistic simulation nontermination secure information flow plas emil stefanov marten van dijk elaine shi christopher fletcher ling ren xiangyao srinivas devadas path oram extremely simple oblivious ram protocol ccs edward suh dwaine clarke blaise gassend marten van dijk srinivas devadas aegis architecture processing ics david lie chandramohan thekkath mark mitchell patrick lincoln dan boneh john mitchell mark horowitz architectural support copy tamper resistant software sigops oper syst rev dennis volpano cynthia irvine smith sound type system secure flow analysis comput secur xiao wang hubert chan elaine shi circuit oram tightness lower bound ccs xiao shaun wang kartik nayak chang liu hubert chan elaine shi emil stefanov yan huang oblivious data structures ccs andrew yao generate exchange secrets focs samee zahur david evans circuit structures improving security privacy tools danfeng zhang aslan askarov andrew myers predictive mitigation timing channels interactive systems ccs danfeng zhang aslan askarov andrew myers control mitigation timing channels pldi danfeng zhang daniel kifer lightdp towards automating privacy proofs popl danfeng zhang yao wang edward suh andrew myers hardware design language security asplos xiaotong zhuang tao zhang santosh pande hide infrastructure protecting information leakage address bus sigarch comput archit news language probabilistically oblivious computation appendix david darais chang liu ian sweet michael hicks idxn dep notation elements parameterized set sequence elements length sequence elements length distributions elements coin index sequences length probabilistic dependency set cond mux cond cond mux cond sel idx stable sel sel fig distributions language probabilistically oblivious computation addr vbr env var env addr vbr vbr env env env store store store store cond mux cond cond cond mux join cond cond cond cond cond cond join join join join join store read addr store store addr read join hread join fig domains david darais chang liu ian sweet michael hicks env env atom env store store exp env expr env store store pico env pico atom ejxk pico ejxk pico ejpk ejpk pico atom ejp cond ejpk atom atom pico pico pico atom ejasnat ejpk asnat pico atom asbit ejpk ejasbit env store atom ejflip sel pico pico ejuse atom ejreveal atom hmux ejmux atom pico pico hhb atom pico pico pico pico pico fig denotational evaluation semantics language probabilistically oblivious computation ejarrayn vbn atom ejpn pico fresh hread atom pico pico atom pico pico pico ejak ejak exp atom ejlet ejek exp exp ejak atom ejek ejlet exp exp hhb ejak atom ejif else pico vbr ejpk exp eje exp eje exp cond cond cond fig denotational evaluation semantics david darais chang liu ian sweet michael hicks loc source code locations env var store addr trace loc env store store env env nvr env store store store expr env store race jal env store ejak atom env store jlet ejak jek env store jlet atom hhb ejak jek jif else cond atom ejpk pico fig denotational trace semantics language probabilistically oblivious computation pico penv var pico pstore addr penv penv pstore dep pstore dep atom env store atom penv pstore dep pstore dep exp env store exp pico pico djxk djlk djpk djpk pico atom djp djpk pico pico atom pico atom djpk djasnat pico atom djpk djasbit atom atom pico djuse atom djreveal atom djmux hhw atom pico pico pico djmux hhw atom pico pico pico pico atom fig dependency semantics david darais chang liu ian sweet michael hicks pico pico atom djarrayn djp fresh atom ejp pico atom pico pico djak djak exp atom djek djlet exp exp ejak djak atom atom djek djlet exp exp ejak hhw djak atom atom djpk djif else exp pico dje exp dje exp fig dependency semantics language probabilistically oblivious computation renv region dep renv renv atom env store atom renv renv exp env store exp atom atom rjak rjak exp exp atom rjek rjak rjlet exp atom ejak exp exp atom rjlet rjek rjak atom ejak atom rjif else rje rje exp exp fig region semantics exp david darais chang liu ian sweet michael hicks bit arr arr flip nat tup array dist nat bit arr tup tup array base flip vbr vbr vbr fig value vbr language probabilistically oblivious computation fig env store david darais chang liu ian sweet michael hicks loc cloc rloc btype tcxt trxt env store ret bit nat flip loc lit cxt store cxt ret rxt rloc lit env rxt store rxt type bit type type loc btype env cloc lit cxt rxt env store env store tenv tstore type tenv tstore type type nat type type type type flip type tcxt env tcxt tcxt cloc btype type tcxt store trxt trxt type ret trxt rloc btype trxt type env trxt type store fig locations language probabilistically oblivious computation flip flip flip flip flip cxt tcxt cxt tcxt trxt rxt rxt trxt fig partitioning stable flip flip flip cxt tcxt cxt tcxt fig stability david darais chang liu ian sweet michael hicks loc lit env source code locations var store addr halt trace lit lit lit obs obs lit label lit obs env var addr store obs lit obs obs env obs hobs obs obs env obs env env store obs obs store obs store store obs obs env env obs obs obs env env obs halt halt obs obs obs obs trace obs obs obs env obs env env store store obs store trace obs obs obs obs obs obs obs env obs obs store obs obs fig adversary observations language probabilistically oblivious computation theorems theorem type safety ejak well typed evaluation ejak theorem dependency soundness obs env obs djak store obs obs store theorem region soundness djak rjak theorem djak djak theorem stability david darais chang liu ian sweet michael hicks theorem strong memory trace obliviousness smto obs env obs env obs env obs obs jek obs trace obs store store jek obs trace store corollary memory trace obliviousness mto env env obs obs jek obs jek obs trace trace lemmas regarding distribution model lemma cond probability measure semantics cond proof cond cond cond cond cond language probabilistically oblivious computation lemma cond stability assume cond proof given goal show cond cond cond probability measure semantics lemma conditional independence distribution equivalence common factor proper probability measure lemma lemma probabilistic independence assume lemma cond independence assume cond model probability measures allows proving following lemmas david darais chang liu ian sweet michael hicks lemma proper probability measure proof lemma joint probability symmetry proof immediate lemma chain rule proof corollary alt chain rule consequence lemma language probabilistically oblivious computation lemma bayes rule proof corollary alt bayes rule consequence lemma david darais chang liu ian sweet michael hicks lemma total probability proof
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sep control variate approach improving efficiency ensemble monte carlo tarik borogovac francis alexander pirooz vakili electrical division computer engineering department boston university saint mary boston usa los alamos national laboratory los alamos usa systems engineering mechanical engineering department boston university saint mary brookline usa abstract paper present new approach control variates improving computational efficiency ensemble monte carlo present approach using simulation paths nonlinear stochastic equation core idea extract information one nominal model parameters use information gain estimation efficiency neighboring parameters idea basis general strategy called database monte carlo dbmc improving efficiency monte carlo paper describe strategy implemented using variance reduction technique control variates show initial setup cost extracting information incurred approach lead significant gains computational efficiency initial setup cost justified projects require large number estimations performed constraints key words monte carlo variance reduction control variates pacs portions work carried los alamos national laboratory auspices national nuclear security administration department energy tarik borogovac pirooz vakili supported part national science foundation grants corresponding author email address tarikb tarik borogovac preprint submitted elsevier january introduction purpose paper present novel approach efficient estimation via monte carlo method approach broadly applicable present main ideas narrow focus ensemble monte carlo estimation based stochastically independent trajectories system illustrate use simulation nonlinear processes monte carlo particularly general powerful numerical method compared available alternatives nonlinear processes general models used among others statistical mechanics data assimilation climate weather ocean modeling financial modeling quantitative biology hence developing efficient methods may significantly impact wide range applications known weakness slow rate convergence assume random quantity defined paths process let denote standard deviation convergence rate estimating expected value number independent paths process general canonical rate convergence improved upon hence since inception method number variance reduction techniques devised reduce see early account recent discussions techniques lead estimators form weighted average samples techniques prescribe recipe selecting samples set weights arrive prescriptions one must rely existence specific problem features ability user method discover effectively exploit features lack generality significantly limited applicability techniques point departure new strategy called database monte carlo dbmc address shortcoming devise generic techniques generically applied techniques bring additional information bear estimation problem however mentioned information problem specific relies exploiting special features problem hand contrast clarified paper dbmc adds generic computational exploration phase estimation problem relies gathering information one nominal model parameter achieve estimation efficiency neighboring parameters advantage approach generality wide applicability quite easy plement wrap existing ensemble codes hand computational exploration phase dbmc approach may require extensive simulations computationally costly therefore initial setup cost needs justification setup cost may justified projects involve estimations many model parameters projects computational constraint first type project setup cost may lead efficiency gain subsequent estimation large enough number subsequent estimations easily justified projects constraint setup cost passive cost lead estimates significantly higher quality lower statistical error higher quality many projects justifies setup cost paper limit presenting implementation technique control variates dbmc setting see discussion techniques technique compared technique importance sampling less utilized computational physics requires identifying number random variables called control variates say correlated known means correlation implies carry information technique way utilizing information included controls known means help estimation mean variable dbmc setting assume depends model parameter use neighborhood departure classical technique use high quality estimates rather precise values arrive controlled estimator argue paper elsewhere departure allows substantially broader choices control variates makes technique significantly flexible effective dbmc method shares similar intent histogram reweighing method markov chain monte carlo literature different setting implementation broader applicability example rely boltzmann distribution exp structure given generality potential applications among others ensemble weather prediction hydrological source location climate ocean optimal control stochastic simulations biological systems remainder paper organized follows section discuss preliminaries including details example numerical study timedependent tdgl equation well method control variates estimation mean outcomes tdgl equation range temperatures interest especially considering large difference behavior coexistence curve section describe dbmc methodology motivation general context section discusses implementation results dbmc applied estimation quantities generated tdgl equation results numerical study conclude section preliminaries present aspects approach numerical results context ginzburg landau tdgl model worth noting model chosen illustrative purposes make use specific features ginzburg landau use canonical equation kinetics stochastic tdgl equation two spatial dimensions written represents local order parameter magnetization point time denotes transpose noise mean zero covariance choose potential constant function temperature high corresponds low temperature use discrete form using forward stochastic integrator stencil laplacian denoted simulation time step lattice spacing independent identically distributed standard normal random variables spacetime point follows applies discretization schemes well estimation problem cover broad range estimation problems consider estimation quantities related specific point quantities global tire lattice particular time quantities depend entire time evolution system specifically consider following representative quantities point magnetization total magnetization specific time total magnetization problem estimating expected value one quantities represented vector random numbers representing single complete path dynamics temperature related parameter random sample quantity interest magnetization single sample path denotes expectation note knowing noise parameter completely determines path sample quantity interest control variate technique give brief review classical control variate technique variance reduction see let assume random variables called control variates correlated assume means known let defined controlled estimator estimator uses information included samples controls degree deviation known means estimator bring closer unknown mean key idea alternatively viewed fitted value linearly regressed variables words includes part variation explained unbiased estimator vectors coefficient vector minimizes variance covariance matrix vector covariances used variance given therefore hence precisely theoretical degree variance reduction controlled estimator used estimate opposed crude estimator called variance reduction ratio vrr statistic control variates note upper limit degree achievable variance reduction since potentially close controls highly correlated estimation variable words technique potentially effective leading orders magnitude variance reduction practice general known exactly need estimated samples typically estimated samples used construct controlled estimator practice adds bias small sample sizes thus makes effective decrease estimator mean squared error precisely equal variance reduction ratio bias converges zero faster standard error thus expending computational resources generating separate pilot samples estimating considered justifiable insightful detailed discussion technique see challenges using technique critical task using technique finding effective controls controls selected rest procedure fairly routine effective control say needs satisfy two requirements simplify discussion consider scalar control needs correlated needs available user known main barrier finding effective controls second requirement namely requirement known mean modification technique called biased control variate bcv reduces burden requirement allowing good approximation evaluated analytically bcv lowers requirement barrier expands range available choices effective controls nonetheless limits potential scope implicitly assuming analytic path arriving approximate value describe next section dbmc approach turn second requirement computational task words use statistical estimation obtain good estimate therefore barrier completely removed range choices controls dramatically expanded relevant question becomes whether computational investment estimating pays enough dividends make investment worthwhile dbmc control variate starting point dbmc approach observation many parametric estimation settings including example considered paper quantities highly correlated random input used generate close suggests using control variates estimating close identified potentially effective controls sufficient information known needs evaluated brings second feature dbmc method corresponds initial computational information stage stage corresponds statistical estimation details given dbmc algorithm dbmc approach consists setup stage estimation stage setup stage dbmc setup phase involves generating large number input random vectors obtaining high quality estimates let similar observation basis histogram methods simulation single state point characterized ising model choice temperature magnetic field one gain information properties point also neighboring region page large denote large set random inputs set represents database given database averages controls precisely calculated schematic stage given figure generate according distribution inputs simulate path evaluate value control find jdb average ith control datebase jdb fig dbmc setup stage estimation stage estimate close select small sample say size uniformly database sample resimulate equation using obtain samples values controls available database using evaluate controlled estimate schematic version steps given figure select uniformly database simulate path evaluate estimation variable find controlled estimator jbcv jdb fig dbmc estimation stage implementation choices two general schemes implementation approach corresponding described requires storing simulation inputs outputs database later resampling utilize resampling storage data beyond recording calculated control means implementations feasible first preferable cases second may preferred cases elaborate implementation database random inputs either directly stored enough information input seeds number generator stored able regenerate precisely paths corresponding generally simulated parallel elements random vector progressively generated random input say value controls stored implementation setup stage completed values stored high quality estimates means values jdb estimation phase random input vectors generated anew paths simulated using new random inputs path calculated finally using values controlled estimator evaluated statistical properties computational efficiency promise approach following anchoring estimation via high quality estimates possible obtain high quality estimates locations parameter space far fewer samples actual statistical properties resulting estimators computational efficiency generating reflect choices made implementing given problem example much computation invested exploration phase points parameter space explored two important questions need investigation choices generally involve problem dependent tradeoffs leave future studies instead analysis follows meant provide general qualitative understanding statistical properties computational efficiency tradeoffs involved discussion general possible consistent numerical study described section implementation choices made utilizing basic familiarity problem discussion see statistical properties give analysis implementation words assume database analysis implementation shows similar estimator statistical properties simplify discussion consider single control say let var var assume database input variables generated let random variables corresponding generated uniformly replacement database let denote means variances variables conditioned database controlled estimator exactly classical estimator results classical apply example scalar unbiased estimator known optimal prescribed classical take random variables defined database measure variance reduction due using controlled estimator use controlled estimator estimator assume optimal used define assume general therefore biased estimator bias introduced sampling database opposed probabilistic assessment bias reduce increasing size database specifically bias obtain approximate probability confidence interval ignore low order bias results typical procedure estimating optimal confused resampling bias discussed section quantile standard normal distribution words high probability bias order assume large bias sufficiently small disregarded focus key measure computational gain using controlled estimator estimate computational efficiency generating large database pointed earlier corresponds initial setup cost let computational cost generating sample cost involves generating simulating path evaluating reasonable assumption many problems cost cost generating database obtaining averages controls approximately let denote variance reduction ratio ratio variance uncontrolled sample controlled sample statistical error controlled estimator based samples approximately samples uncontrolled estimator thus ratios computational costs two estimators arrive statistical accuracy therefore serve measure benefit dbmc approach setup cost dbmc approach justified two types applications first type applications require solving many instances estimation problem many total number instances sufficiently large variance reduction achieved average instances large fixed cost dwarfed total computational savings many estimations second type applications setup cost viewed cost enabling significant efficiency gains critical task estimation typically cost delay estimation higher merely computational justifying even much larger computational effort numerical results numerical results section intended give qualitative illustration efficiency gains achieved using dbmc approach specifically estimate variance reduction achieved regular crude sampling estimating three quantities interest point magnetization total magnetization specific time total magnetization range parameter choices size database number samples used estimation range parameter values controls simply illustration purposes however expect numerical results qualitatively quite representative simulate tdgl dynamics lattice lattice spacing fixed path evolve system total time steps sufficient system exhibit behavior specific temperature region critical point system parameter range interest extends sides critical point build database simulate paths evaluate point magnetization total magnetization specific time total magnetization two nominal values quantity interest consider three control variate estimators first two estimators use single controls corresponding respectively chose anchor estimators two nominal values located opposite sides phase transition line third estimator uses controls simultaneously use samples crude estimators estimate variance estimators following simulation approach see use independent macro simulations consisting independent micro simulations obtain variance estimates macro simulation average resulting values obtain overall variance estimate report ratios variance estimates sampling vrr results total magnetization problem given table corresponding graph given figure graph point magnetization problem given fig results total magnetization time problem quite similar excluded table variance reduction ratios estimators applied integral magnetization several values based results draw following conclusions controlled estimators produce dramatic variance reduction parameter variance reduction ratios time integral magnetization vrr fig variancep reduction ratios estimators integral magnetization range values log scale values close nominal parameters substantial variance reduction values moderately close nominal estimation problems adding second control consistently improves performance cases leading substantial reduction variance compared single controls course incorporating information points sides critical temperature expected give better coverage either single control estimators however better either single control estimators even regions suggests control provides relevant information estimation problem opposite region vrr values total magnetization somewhat larger point total magnetization specific time expect true generally path integrals compared values specific time instances variance reduction ratios point magnetization vrr fig variance reduction ratios estimators point magnetization range values log scale conclusions paper described new strategy database monte carlo dbmc improving computational efficiency ensemble monte carlo specific nonlinear dynamics showed approach lead significant efficiency gains range estimation problems selection controls illustration purposes work required better understand options available computational tradeoffs involved end current research focused derivation specific guidelines selection effective control variates implementation dbmc strategy conjunction variance reduction techniques example stratification importance sampling iii application method specific domains example estimation problems geophysical fluids biochemical systems references binder heermann monte carlo simulation statistical physics introduction springer evensen data assimilation ensemble kalman filter springer glasserman monte carlo methods financial engineering new york inc wilkinson stochastic modelling systems biology crc press hammersley handscomb monte carlo methods john wiley asmussen glynn stochastic simulation algorithms analysis springer vakili zhao borogovac database monte carlo new strategy efficient simulation tech boston university college engineering borogovac vakili database monte carlo approach effective control variates tech boston university college engineering ferrenberg swendsen new monte carlo technique studying phase transitions phys rev lett gulbahce alexander johnson statistical mechanics histories cluster monte carlo algorithm phys rev ferreira toral hybrid monte carlo method systems phys rev robert casella monte carlo statistical methods springer science business media schmeiser taaffe wang biased estimation iie transactions schmeiser chapter simulation experiments heyman sobel eds handbooks stochastic models elsevier
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threshold selection multivariate data phyllis wan richard davis mar department statistics columbia university amsterdam avenue new york phyllis rdavis abstract regular variation often used starting point modeling multivariate data random vector regularly varying radial part regularly varying asymptotically independent angular part goes infinity conditional limiting distribution given large characterizes tail dependence random vector hence estimation primary goal applications typical strategy look angular components data radial parts exceed threshold large class methods proposed model angular distribution exceedances choice threshold scarcely discussed literature paper describe procedure choosing threshold formally testing independence using measure dependence called distance covariance generalize limit theorem distance covariance unique setting propose algorithm selects threshold algorithm incorporates subsampling scheme also applicable weakly dependent data moreover avoids heavy computation calculation distance covariance typical limitation measure performance method illustrated simulated real data keywords phrases distance covariance data multivariate regular variation threshold selection introduction multivariate data principal objective often study dependence tail distribution achieve goal assumption multivariate regular variation typically used starting point random vector said multivariate regularly varying polar coordinates kxk norm satisfy conditions univariate regularly varying slowly varying function infinity converges weakly measure referred index regular variation called angular distribution characterizes limiting tail dependence equivalent definitions regular variation resnick one convenient purposes given observations corresponding polar coordinates straightforward procedure estimating look angular components data radii greater large threshold studies one takes large empirical quantile extensive research choosing threshold distribution regularly varying limit condition little research devoted ensuring threshold large enough independence reasonable limit condition end haan ronde fit parametric extreme value distribution model marginal examined parameter stability plot coordinate plot looked joint tail empirical measure way equivalent examining extremal behavior resnick suggested automatic threshold selection plot observed thresholds sometimes systematically underestimated study threshold based inference parametric processes jeon smith suggested choosing threshold minimizing mse estimated parameters paper propose algorithm selects threshold modeling motivation implied property given become independent given sequence candidate ver file date march wan davis threshold levels test degree dependence truncated data level dependence measure use distance covariance introduced measure ability account various types dependence applicable data higher dimensions resulting test statistics given form compared across levels subsampling scheme enables extract accurate information test statistics overloading computational burden remainder paper organized follows first provide theoretical backgrounds multivariate regular variation section distance covariance theoretical properties introduced section applying dependence measure conditioning setting propose test statistic prove relevant theoretical results section proposed algorithm threshold selection presented section illustrated simulated real examples section paper concludes discussion multivariate regular variation problem one way approach multivariate data notion multivariate regular variation detailed review see example chapter resnick let random variable defined cone define polar coordinate transformation kxk denotes norm regularly varying exists probability measure unit sphere function denotes vague convergence measure defined chosen inf convergence implies denotes weak convergence words given large conditional distribution independent limit view restrict measure throughout remainder paper angular measure characterizes tail dependence structure concentrated components asymptotically independent tail case known asymptotic independence mass lying subspace extreme observation direction implicates positive probability extreme observation direction case known asymptotic dependence hence estimation observations important problem often primary goal multivariate modeling following convergence implied suggests estimating using angular data whose radial parts satisfy large motivation behind method seek virtually independent given candidate threshold sequence formally test independence index set use pearson correlation dependence measure unsuitable case two reasons first correlation applicable univariate random variables whereas lies sphere dimension second correlation describes linear relationship two random variables thus zero correlation sufficient condition independence instead use powerful dependence measure distance covariance introduced next section threshold selection multivariate data distance covariance section briefly review definition properties distance covariance detailed descriptions proofs found davis let two random vectors distance covariance defined denote joint marginal characteristic functions suitable measure order ensure one following conditions assumed satisfied throughout paper davis finite measure infinite measure one advantage distance covariance say pearson covariance positive lebesgue density independent another attractive property dependence measure readily applies random vectors different dimensions estimate observations define empirical distance covariance respective empirical characteristic functions assume symmetric origin conditions exists also computable form coshs davis popular choice first mentioned feuerverger extensively studied defined lemma choice gives moreover choice distance covariance invariant relative scale orthogonal transformations note order integral exist required utilize described weight measure simulations data analyses section applied log transformation ensure moment condition satisfied detailed davis sequence stationary ergodic wan davis independent condition centered gaussian field hand dependent limit implying diverges naturally one devise test independence using statistic null hypothesis independence rejected level upper practice distribution intractable typically approximated bootstrap hence main drawback using distance covariance computation burden brings large sample size computation single distance covariance statistic requires operations finding values via resampling requires much additional computation method however overcomes problem subsampling data described section theoretical results let iid observations multivariate regularly varying distribution satisfying polar coordinate transformations given threshold measure dependence conditional empirical distance covariance set conditional empirical characteristic function eisrj corresponding empirical conditional marginal characteristic functions section establish limiting results adapted conditional distance covariance ease notation let theoretical empirical probability exceedance let theoretical conditional joint marginal characteristic functions threshold selection multivariate data recall become asymptotically independent converge respectively denote characteristic functions corresponding limit distributions exp isr lim exp lim following results theorem let iid observations generated multivariate regularly varying index let conditional empirical distance covariance angular radial component defined assume npn weight measure satisfies addition satisfies npn centered gaussian process covariance function cov defined remark case regularly varying index similar results hold replace log moments exist proof theorem delayed appendix following remark discuss certain sufficient conditions assumption remark assume measures respectively symmetric origin section davis condition equivalent npn cos let denote conditional joint distribution given respective conditional marginals expressed npn wan davis npn npn iid copies one way verify assume like condition distribution example assume signed measure finite borel set unit sphere scalar function components asymptotically independent equivalent second order condition multivariate regular variation resnick choose sequence npn npn npn npn case finite measures bounded satisfied since integrand written npn special case met provided chosen measures infinite verified specific cases illustrated following example example let follow bivariate logistic distribution cdf exp asymptotically independent components shown regularly varying index using threshold selection multivariate data coordinate transform pdf consider case infinite weight measure given derive condition sequence conditions theorem hold first observe denotes generic constant whose value may change line line throughout paper last inequality comes facts letting max log wan davis first term bounded terms bounded way since infinite first moment apply distance correlation log integral bounded npn log log log log log converges zero therefore chosen theorem holds result theorem generalized iid regularly varying time series setting present next theorem multivariate stationary time series set regularly varying measure property borel set see example page davis mikosch follows easily assume assume following conditions sequence threshold verified various time series models davis mikosch assume exists sequence lim lim sup kxj iii npn threshold selection multivariate data theorem let multivariate regularly varying time series tail index coefficients assume conditions weight measure sequence thresholds theorem hold condition holds centered gaussian process particular proof theorem given appendix note limiting distributions theorem theorem intractable practice quantiles distributions calculated using resampling methods iid case done straightforwardly weakly dependent case one needs apply block bootstrap stationary bootstrap obtain desired result see davis following section present threshold selection framework subsampling scheme require independence observations threshold selection section propose procedure select threshold estimating spectral measure observations let first consider case specific threshold given specifies empirical distance covariance conditional assumption theorem number observations practice limit distribution intractable one resort bootstrapping consider hypothesis testing framework independent given independent given define testing versus follows sufficiently small consider decreasing sequence candidate thresholds sequence pvk corresponding threshold obtained goal find smallest threshold conditional reasonably considered independent note pvk independent since computed set data conventional multiple testing procedures bonferroni correction problematic implement dependent counter limitations propose intuitive direct method based subsampling idea outlined follows fixed level choose subsample size conditional empirical cdf subsample compute distance covariance compute assumption conditional empirical distribution product conditional marginals take large number subsamples size calculate value subsample pvk empirical value relative process starting initial subsample wan davis repeated times produces estimates pvk pvk independent conditional original sample averaged sequence levels produce sequence independent choice threshold independent dependent otherwise based examination path mean note following two observations independent given pvk pvk iid approximately center around dependent given pvk well closer studying sequence call mean path choose threshold smallest around method situation cusum algorithm page detects changes mean sequence looking partial sums algorithm use spline fitting method based cusum approach called wild binary segmentation wbs proposed fryzlewicz wbs procedure uses cusum statistics subsamples fits piecewise constant spline setting may choose knot spline fitted value comfortably several advantages using subsampling scheme first recall path pvk obtained whole data set complicated serial structure varies greatly realization contrast mean subsampling conditionally independent center around small variance total sample size number subsample large turns helps present justifiable estimation threshold second calculation distance covariance extremely slow moderate sample size using smaller sample sizes subsamples computational burden greatly reduced addition procedure amenable parallel computing reducing computation time even third subsampling makes possible accommodate stationary dependent data waiving stringent independent assumption idea looking mean path inspired mallik used mean multiple independent tests detect change points population means data illustration section demonstrate threshold selection method simulated real data examples practice set sequence thresholds corresponding upper quantiles sequence quantile levels subsample size threshold set designed subsample fraction eligible data points choice boils choice reflect following considerations large enough ensure good resolution levels sufficient small subsamples contain much overlap observations iii larger requires heavier computation distance correlation examples total sample size ranges find suitable choice number subsamples set large computation capacity allows examples take examples choose weight function distance covariance number replications used calculate ensure moment conditions met distance correlation applied log radial part examples simulated data known threshold illustrate methodology simulate observations distribution known threshold become independent threshold selection multivariate data let absolute value degrees freedom independent random variables beta set upper independent given let simulated observations generate iid observations distribution figures show data cartesian polar coordinates goal recover tail angular distribution choosing appropriate threshold sequence candidate thresholds selected empirical upper quantiles corresponding equidistant points apply procedure described section data mean calculated using random subsamples size observations figure shows mean path wbs algorithm set threshold largest thresholds quantile level fitted spline stays threshold levels chosen good agreement true independence level empirical cdfs truncated corresponding chosen thresholds shown figure see true tail angular cdf accurately recovered simulated logistic data simulate data bivariate logistic distribution bivariate regularly varying recall example follows bivariate logistic distribution cdf example set generate iid observations distribution similar previous example threshold corresponding upper quantile chosen equidistant points mean calculated using random subsamples size observations figures show scatterplots data used transform data polar coordinates algorithms suggests using data estimate angular distribution estimated cdf angular distribution shown theoretical limiting cdf derived figure even though independent threshold procedures produce good estimates limiting distribution real data example look following exchange rate returns relative dollar deutsche mark dem british pound gbp canadian dollar cad swiss franc chf time spans data total days observations examine pairs estimate angular density tail pair figures present scatter plots data marginals observations standardized using rank transformation proposed joe log rank chosen equidistant points mean calculated using random subsamples size observations note may reasonable view observations iid subsampling scheme still applied choose threshold independence course selection rules used example conservative approach would choosing threshold largest fitted spline stays radius angle estimated truth cdf change point fitted spline wan davis upper quantile theta fig example scatterplot scatterplot scale scatterplot mean path black triangles fitted wbs spline blue line chosen threshold quantile red vertical line estimated cdf using threshold chosen compared truth black dotted mean paths shown figures threshold levels selected three pairs respectively figures show shape estimated angular densities pairs expected tails two central european exchange rates dem chf highly dependent contrast cad chf almost independent simulated varying data example generate data model regularly varying let random variable standard pareto distribution let independent random variables set log integer log integer positive integer verify estimated truth cdf angle change point fitted spline radius threshold selection multivariate data upper quantile theta fig example scatterplot scatterplot scale scatterplot mean path black triangles fitted wbs spline blue line chosen threshold quantile red vertical line estimated cdf using threshold chosen compared theoretical limiting cdf black dotted hence convergence regularly varying let iid observations distribution figures show data cartesian polar coordinates apply threshold selection algorithm data threshold upper quantile levels chosen equidistant points mean calculated using random subsamples size observations shown figure model radial part regularly varying dependent given expect mean well observed threshold selected algorithm suggests technique potentially used detect misspecified models regular variation assumption especially scenario observed dependence suspected discussion paper propose threshold selection procedure multivariate regular variation approximately independent beyond threshold problem set multivariate setting utilize distance covariance measure dependence algorithm wan davis gbp dem density density cad density chf chf chf theta theta theta fig example analysis paired exchange rate returns respect usd scatter plots standardized paired exchange rate returns estimated angular densities using estimated thresholds chosen essentially change point detection method based generated subsampling schemes hence may generalized problem settings potentially incorporates dependence measures though proposed automatic selection threshold based fitted mean path would like emphasize like hill plot viewed visual tool rather optimal selection criterion final threshold based automatic procedure conjunction visual inspection path note choice norm polar coordinate transformation may result significant differences choice thresholds indicates rate convergence limit spectral density especially evident near asymptotic independence case mass angular distribution concentrates axes illustration simulated iid observations bivariate logistic distribution cdf given apply polar coordinate transformation respect note case satisfy triangular inequality however shown holds limiting angular distribution exists bivariate logistic distribution compare threshold selection results figure note cases threshold levels chosen upper respectively case possible select threshold dependence levels shown significant indeed seen figure compare histogram given kxkp large across three levels truncations together theoretical limiting density curve limiting angular density poorly approximated truncated data levels two norms truncated threshold selection multivariate data change point fitted spline upper quantile change point fitted spline upper quantile change point fitted spline upper quantile fig example analysis paired exchange rate returns respect usd mean paths black triangles fitted wbs splines blue lines chosen threshold quantiles red vertical line ang wan davis uppe quan examp sca erp sca erp sca sca erp mean ack ang wbs chosen hresho quan red ver observat ons accord ected thresho prov decent approx mat ons true dens angu component one poss exp anat thresho concave hence observat ons agona much eas ass fied extremes near resu est mator angu dens uses observat ons near agona may fact ose enough cho norm nterest top subject ongo research acknowledgement fore exchange rate data obta ned oanda qrmtoo wou thank bodh sattva sen pfu scuss ons wou thank tor referees many construct ghtfu comments references cke chura convergence ter parameter stochast processes app cat ons ann statist ngs probabi ity measure ence dav kosch extremogram corre ogram extreme events bernou threshold selection multivariate data change point fitted spline upper quantile change point fitted spline upper quantile change point fitted spline upper quantile fig simulated logistic data sample size threshold selection algorithm applied mean paths black triangles fitted wbs splines blue lines chosen threshold quantiles red vertical line davis mikosch cribben towards estimating extremal serial dependence via bootstrapped extremogram econometrics davis matsui mikosch wan applications distance covariance time series haan ronde sea wind multivariate extremes work extremes doukhan mixing properties examples new york feuerverger consistent test bivariate dependence internat statis fryzlewicz wild binary segmentation multiple detection ann jeon smith dependence structure spatial extremes using threshold approach joe smith weissman bivariate threshold methods extremes jrss mallik sen banerjee michailidis threshold estimation based framework regression settings biometrika page continuous inspection schemes biometrika resnick hidden regular variation second order regular variation asymptotic independence wan davis theta theta theta theta theta theta density density density density density density density density density theta theta theta fig simulated logistic data sample size histogram truncated levels tremes resnick phenomena probabilistic statistical modeling new york multivariate extremes models constant conditional correlations empir rizzo bakirov measuring testing dependence correlation distances ann appendix proof theorem note definition empirical distance covariance integrand expressed isrj isrj itt isrj eit isrj itt threshold selection multivariate data writing ujn eisrj vjn eit ujn vjn ujn vkn since eujn evjn eujn vjn convenient mean correct summands obtain ujn vjn ujn vkn note averages iid random variables first prove second part theorem first part theorem follows easily similar fashion proof theorem order show suffices establish implied notice npn hence equivalent prove npn npn npn show convergence proposition npn holds provided npn follows similar fashion proposition wan davis proposition assume satisfies npn npn centered gaussian process covariance function proof proposition first show npn implied finite distributional convergence npn tightness write ujn vjn npn yjn yjn iid random variables mean fixed note var hand apply central limit theorem triangular arrays checking lyapounov condition see billingsley yjn npn npn var var follows easily fixed npn distribution obtained using device covariance function verified calculations show tightness npn note isrj isrj isrj threshold selection multivariate data eit without loss generality show tightness npn npn npn follows argument first introduce notation following bickel wichura fix let subset form ease notation suppress dependence define increment stochastic process sufficient condition theorem bickel wichura tightness following statement holds corresponding npn npn implied follows npn npn isrj npn eis eitk eitk var eis eitk eis itk eitk eitk taylor series argument eix wan davis hence npn since bounded supn regular variation assumption proves tightness define bounded set using continuous mapping theorem npn hand npn npn ujn vjn ujn vjn vjn eujn vjn vjn eisrj eit eisrj srj srj srj srj rrn rrn rrn rrn eit rrn therefore lim lim sup npn lim lim sup npn threshold selection multivariate data lim lim sup rthe dominated convergence theorem combined shows convergence npn hence completes proof proposition proof theorem remains show similar proof proposition show npn centered gaussian process hence npn npn argument follows similarly continuous mapping theorem bounding tail integrals proof theorem similar proof theorem suffices show convergence follows trivially general results proof theorem hence suffices show recall irn let rrn rrn corresponding marginal measures hence fixed eist using argument vjn follows dominated convergence concludes proof appendix proof theorem following notation steps proof theorem appendix suffices prove following convergences mixing case npn wan davis npn prove propositions respectively proof follows similar fashion proofs propositions rely following lemma throughout proof make use results stationary coefficient see section theorem doukhan lemma let multivariate stationary time series regularly varying mixing coefficient sequence set let bounded functions vanish outside unit open ball sets discontinuity measure zero set efi assume condition holds npn covariance matrix particular npn proof lemma provided proofs propositions proposition assume condition holds proof npn apply lemma kxk result follows proposition assume condition holds satisfies respectively npn centered gaussian process threshold selection multivariate data proof let first establish convergence npn fixed take eiskxk eiskxk eiskxk eiskxk lemma npn npn covariance structure derived implies npn complex normal process covariance matrix relation matrix distributional convergence generalized using device omit calculation covariance structure tightness condition functional convergence follows arguments proof proposition bickel wichura equality replaced variance calculation sum components using inequality condition verified argument completes proof proposition lemma proof follows theorem davis mikosch outline sketch proof detail parts differ proof vague convergence var iii cov without loss generality suppress let first consider marginal convergence dependency set ytn var cov also following two results lim lim sup lim lim sup wan davis lim lim sup kxj kxj cln lim lim sup condition lim cov lim lim lim lim condition apply technique blocks used davis mikosch let sizes big small blocks respectively let ikn kmn jkn index sets big small blocks respectively set ikn big blocks first observations removed simplicity set assume number big blocks npn case generalized without additional difficulties denote ytn ytn npn ikn npn npn jkn let npn iid copies prove convergence show following npn suffices npn npn limiting distribution npn npn npn jkn npn npn npn statement holds npn follows argument equation davis mikosch threshold selection multivariate data condition suffices show npn jkn var npn note npn jkn var var npn npn lim sup lim sup var yjn var lim sup lim sup var lim lim sup lim lim sup last step follows dominated convergence term ysn ytn cov cov cov note size big blocks ikn distance consecutive small blocks jkn last limit follows finish proof need establish central limit theorem note iid calculate variance recall size big block small block removed var var yjn var yjn cov var yjn cov wan davis lim lim var yjn lim lim cov also lim lim sup lim lim sup lim lim therefore lim var lim lim lim defined show infinite sum converges suffices show follows condition lim sup kxj lim inf kxj leads contradiction apply central limit theorem verify lindeberg condition yjn npn var npn npn completes proof convergence npn threshold selection multivariate data joint convergence follows line argument together device particular cov npn completes proof lemma remark lemma general result independent interest result generalized functions defined compact support case condition modified lim lim sup kxj support also seen proof lemma conditions relaxed
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