kmeans The Advantages of Careful Seeding David Arthur Sergei Vassilvitskii Abstract The kmeans method is a widely used clustering technique that seeks to minimize the average squared distance between - PDF document

Presentation on theme: "kmeans The Advantages of Careful Seeding David Arthur Sergei Vassilvitskii Abstract The kmeans method is a widely used clustering technique that seeks to minimize the average squared distance between"— Presentation transcript

k-means++:TheAdvantagesofCarefulSeedingDavidArthurSergeiVassilvitskiiyAbstractThek-meansmethodisawidelyusedclusteringtechniquethatseekstominimizetheaveragesquareddistancebetweenpointsinthesamecluster.Althoughitoersnoaccuracyguarantees,itssimplicityandspeedareveryappealinginpractice.Byaugmentingk-meanswithaverysimple,ran-domizedseedingtechnique,weobtainanalgorithmthatis(logk)-competitivewiththeoptimalclustering.Prelim-inaryexperimentsshowthatouraugmentationimprovesboththespeedandtheaccuracyofk-means,oftenquitedramatically.1IntroductionClusteringisoneoftheclassicproblemsinmachinelearningandcomputationalgeometry.Inthepopulark-meansformulation,oneisgivenanintegerkandasetofndatapointsinRd.Thegoalistochoosekcenterssoastominimize,thesumofthesquareddistancesbetweeneachpointanditsclosestcenter.SolvingthisproblemexactlyisNP-hard,evenwithjusttwoclusters[10],buttwenty-veyearsago,Lloyd[20]proposedalocalsearchsolutionthatisstillverywidelyusedtoday(seeforexample[1,11,15]).Indeed,arecentsurveyofdataminingtechniquesstatesthatit\isbyfarthemostpopularclusteringalgorithmusedinscienticandindustrialapplications"[5].Usuallyreferredtosimplyask-means,Lloyd'salgorithmbeginswithkarbitrarycenters,typicallychosenuniformlyatrandomfromthedatapoints.Eachpointisthenassignedtothenearestcenter,andeachcenterisrecomputedasthecenterofmassofallpointsassignedtoit.Thesetwosteps(assignmentandcentercalculation)arerepeateduntiltheprocessstabilizes.Onecancheckthatthetotalerrorismonotoni-callydecreasing,whichensuresthatnoclusteringisre-peatedduringthecourseofthealgorithm.Sincethereareatmostknpossibleclusterings,theprocesswillal-waysterminate.Inpractice,veryfewiterationsareusu-allyrequired,whichmakesthealgorithmmuchfaster StanfordUniversity,SupportedinpartbyNDSEGFellow-ship,NSFGrantITR-0331640,andgrantsfromMedia-XandSNRC.yStanfordUniversity,SupportedinpartbyNSFGrantITR-0331640,andgrantsfromMedia-XandSNRC.thanmostofitscompetitors.Unfortunately,theempiricalspeedandsimplicityofthek-meansalgorithmcomeatthepriceofaccuracy.Therearemanynaturalexamplesforwhichthealgo-rithmgeneratesarbitrarilybadclusterings(i.e., OPTisunboundedevenwhennandkarexed).Furthermore,theseexamplesdonotrelyonanadversarialplacementofthestartingcenters,andtheratiocanbeunboundedwithhighprobabilityevenwiththestandardrandom-izedseedingtechnique.Inthispaper,weproposeawayofinitializingk-meansbychoosingrandomstartingcenterswithveryspecicprobabilities.Specically,wechooseapointpasacenterwithprobabilityproportionaltop'scontributiontotheoverallpotential.Lettingdenotethepotentialafterchoosingcentersinthisway,weshowthefollowing.Theorem1.1.Foranysetofdatapoints,E[]8(lnk+2)OPT.Thissamplingisbothfastandsimple,anditalreadyachievesapproximationguaranteesthatk-meanscan-not.Weproposeusingittoseedtheinitialcentersfork-means,leadingtoacombinedalgorithmwecallk-means++.ThiscomplementsaveryrecentresultofOstrovskyetal.[24],whoindependentlyproposedmuchthesamealgorithm.Whereastheyshowedthisrandomizedseed-ingisO(1)-competitiveondatasetsfollowingacertainseparationcondition,weshowitisO(logk)-competitiveonalldatasets.WealsoshowthattheanalysisforTheorem1.1istightuptoaconstantfactor,andthatitcanbeeas-ilyextendedtovariouspotentialfunctionsinarbitrarymetricspaces.Inparticular,wecanalsogetasim-pleO(logk)approximationalgorithmforthek-medianobjective.Furthermore,weprovidepreliminaryexperi-mentaldatashowingthatinpractice,k-means++reallydoesoutperformk-meansintermsofbothaccuracyandspeed,oftenbyasubstantialmargin.1.1RelatedworkAsafundamentalprobleminmachinelearning,k-meanshasarichhistory.Becauseofitssimplicityanditsobservedspeed,Lloyd'smethod[20]remainsthemostpopularapproachinpractice, despiteitslimitedaccuracy.TheconvergencetimeofLloyd'smethodhasbeenthesubjectofarecentseriesofpapers[2,4,8,14];inthisworkwefocusonimprovingitsaccuracy.Inthetheorycommunity,Inabaetal.[16]werethersttogiveanexactalgorithmforthek-meansproblem,withtherunningtimeofO(nkd).Sincethen,anumberofpolynomialtimeapproximationschemeshavebeendeveloped(see[9,13,19,21]andthereferencestherein).Whiletheauthorsdevelopinterestinginsightsintothestructureoftheclusteringproblem,theiralgorithmsarehighlyexponential(orworse)ink,andareunfortunatelyimpracticalevenforrelativelysmalln,kandd.Kanungoetal.[17]proposedanO(n3d)algorithmthatis(9+)-competitive.However,n3comparesunfavorablywiththealmostlinearrunningtimeofLloyd'smethod,andtheexponentialdependenceondcanalsobeproblematic.Forthesereasons,Kanungoetal.alsosuggestedawayofcombiningtheirtechniqueswithLloyd'salgorithm,butinordertoavoidtheexponentialdependenceond,theirapproachsacricesallapproximationguarantees.MettuandPlaxton[22]alsoachievedaconstant-probabilityO(1)approximationusingatechniquecalledsuccessivesampling.TheymatchourrunningtimeofO(nkd),butonlyifkissucientlylargeandthespreadissucientlysmall.Inpractice,ourapproachissimpler,andourexperimentalresultsseemtobebetterintermsofbothspeedandaccuracy.Veryrecently,Ostrovskyetal.[24]independentlyproposedanalgorithmthatisessentiallyidenticaltoours,althoughtheiranalysisisquitedierent.LettingOPT;kdenotetheoptimalpotentialforak-clusteringonagivendataset,theyprovek-means++isO(1)-competitiveinthecasewhereOPT;k OPT;k12.Theintuitionhereisthatifthisconditiondoesnothold,thenthedataisnotwellsuitedforclusteringwiththegivenvaluefork.Combiningthisresultwithoursgivesastrongcharacterizationofthealgorithm'sperformance.Inparticular,k-means++isneverworsethanO(logk)-competitive,andonverywellformeddatasets,itimprovestobeingO(1)-competitive.Overall,theseedingtechniqueweproposeissimilarinspirittothatusedbyMeyerson[23]foronlinefacilitylocation,andMishraetal.[12]andCharikaretal.[6]inthecontextofk-medianclustering.However,ouranalysisisquitedierentfromthoseworks.2PreliminariesInthissection,weformallydenethek-meansproblem,aswellasthek-meansandk-means++algorithms.Forthek-meansproblem,wearegivenanintegerkandasetofndatapointsXRd.WewishtochoosekcentersCsoastominimizethepotentialfunction,=Xx2Xminc2Ckxck2:Choosingthesecentersimplicitlydenesaclustering{foreachcenter,wesetoneclustertobethesetofdatapointsthatareclosertothatcenterthantoanyother.Asnotedabove,ndinganexactsolutiontothek-meansproblemisNP-hard.Throughoutthepaper,wewillletCOPTdenotetheoptimalclusteringforagiveninstanceofthek-meansproblem,andwewillletOPTdenotethecorrespondingpotential.GivenaclusteringCwithpotential,wealsolet(A)denotethecontributionofAXtothepotential(i.e.,(A)=Px2Aminc2Ckxck2).2.1Thek-meansalgorithmThek-meansmethodisasimpleandfastalgorithmthatattemptstolocallyimproveanarbitraryk-meansclustering.Itworksasfollows.1.ArbitrarilychoosekinitialcentersC=fc1;:::;ckg.2.Foreachi2f1;:::;kg,settheclusterCitobethesetofpointsinXthatareclosertocithantheyaretocjforallj=i.3.Foreachi2f1;:::;kg,setcitobethecenterofmassofallpointsinCi:ci=1 jCijPx2Cix.4.RepeatSteps2and3untilCnolongerchanges.ItisstandardpracticetochoosetheinitialcentersuniformlyatrandomfromX.ForStep2,tiesmaybebrokenarbitrarily,aslongasthemethodisconsistent.Steps2and3arebothguaranteedtodecrease,sothealgorithmmakeslocalimprovementstoanarbitraryclusteringuntilitisnolongerpossibletodoso.ToseethatStep3doesinfactdecreases,itishelpfultorecallastandardresultfromlinearalgebra(see[14]).Lemma2.1.LetSbeasetofpointswithcenterofmassc(S),andletzbeanarbitrarypoint.Then,Px2Skxzk2Px2Skxc(S)k2=jSj
kc(S)zk2.MonotonicityforStep3followsfromtakingStobeasingleclusterandztobeitsinitialcenter.Asdiscussedabove,thek-meansalgorithmisat-tractiveinpracticebecauseitissimpleanditisgener-allyfast.Unfortunately,itisguaranteedonlytondalocaloptimum,whichcanoftenbequitepoor.2.2Thek-means++algorithmThek-meansalgo-rithmbeginswithanarbitrarysetofclustercenters.Weproposeaspecicwayofchoosingthesecenters.At anygiventime,letD(x)denotetheshortestdistancefromadatapointxtotheclosestcenterwehaveal-readychosen.Then,wedenethefollowingalgorithm,whichwecallk-means++.1a.Chooseaninitialcenterc1uniformlyatrandomfromX.1b.Choosethenextcenterci,selectingci=x02XwithprobabilityD(x0)2 x2XD(x)2.1c.RepeatStep1buntilwehavechosenatotalofkcenters.2-4.Proceedaswiththestandardk-meansalgorithm.WecalltheweightingusedinStep1bsimply\D2weighting".3k-means++isO(logk)-competitiveInthissection,weproveourmainresult.Theorem3.1.IfCisconstructedwithk-means++,thenthecorrespondingpotentialfunctionsatisesE[]8(lnk+2)OPT.Infact,weprovethisholdsafteronlyStep1ofthealgorithmabove.Steps2through4canthenonlydecrease.Notsurprisingly,ourexperimentsshowthislocaloptimizationisimportantinpractice,althoughitisdiculttoquantifythistheoretically.Ouranalysisconsistsoftwoparts.First,weshowthatk-means++iscompetitiveinthoseclustersofCOPTfromwhichitchoosesacenter.Thisiseasiestinthecaseofourrstcenter,whichischosenuniformlyatrandom.Lemma3.1.LetAbeanarbitraryclusterinCOPT,andletCbetheclusteringwithjustonecenter,whichischosenuniformlyatrandomfromA.Then,E[(A)]=2OPT(A).Proof.Letc(A)denotethecenterofmassofthedatapointsinA.ByLemma2.1,weknowthatsinceCOPTisoptimal,itmustbeusingc(A)asthecentercorrespondingtotheclusterA.Usingthesamelemmaagain,weseeE[(A)]isgivenby,Xa02A1 jAj
Xa2Akaa0k2!=1 jAjXa02A Xa2Akac(A)k2+jAj
ka0c(A)k2!=2Xa2Akac(A)k2;andtheresultfollows.OurnextstepistoproveananalogofLemma3.1fortheremainingcenters,whicharechosenwithD2weighting.Lemma3.2.LetAbeanarbitraryclusterinCOPT,andletCbeanarbitraryclustering.IfweaddarandomcentertoCfromA,chosenwithD2weighting,thenE[(A)]8OPT(A).Proof.Theprobabilitythatwechoosesomexeda0asourcenter,giventhatwearechoosingourcenterfromA,ispreciselyD(a0)2 a2AD(a)2.Furthermore,afterchoos-ingthecentera0,apointawillcontributepreciselymin(D(a);kaa0k)2tothepotential.Therefore,E[(A)]=Xa02AD(a0)2 Pa2AD(a)2Xa2Amin(D(a);kaa0k)2:NotebythetriangleinequalitythatD(a0)D(a)+kaa0kforalla;a0.Fromthis,thepower-meaninequality1impliesthatD(a0)22D(a)2+2kaa0k2.Summingoveralla,wethenhavethatD(a0)22 jAjPa2AD(a)2+2 jAjPa2Akaa0k2,andhence,E[(A)]isatmost,2 jAj
Xa02APa2AD(a)2 Pa2AD(a)2
Xa2Amin(D(a);kaa0k)2+2 jAj
Xa02APa2Akaa0k2 Pa2AD(a)2
Xa2Amin(D(a);kaa0k)2:Intherstexpression,wesubstitutemin(D(a);kaa0k)2kaa0k2,andinthesecondexpression,wesubstitutemin(D(a);kaa0k)2D(a)2.Simplifying,wethenhave,E[(A)]4 jAj
Xa02AXa2Akaa0k2=8OPT(A):ThelaststepherefollowsfromLemma3.1.WehavenowshownthatseedingbyD2weightingiscompetitiveaslongasitchoosescentersfromeachclusterofCOPT,whichcompletesthersthalfofourargument.WenowuseinductiontoshowthetotalerroringeneralisatmostO(logk). 1Thepower-meaninequalitystatesforanyrealnumbersa1;


;amthata2i1 m(ai)2.ItfollowsfromtheCauchy-Schwarzinequality.Weareonlyusingthecasem=2here,butwewillneedthegeneralcaseforLemma3.3. Lemma3.3.LetCbeanarbitraryclustering.Chooseu�0\uncovered"clustersfromCOPT,andletXudenotethesetofpointsintheseclusters.AlsoletXc=XXu.NowsupposeweaddturandomcenterstoC,chosenwithD2weighting.LetC0denotethetheresultingclustering,andlet0denotethecorrespondingpotential.Then,E[0]isatmost,(Xc)+8OPT(Xu)
(1+Ht)+ut u
(Xu):Here,Htdenotestheharmonicsum,1+1 2+


+1 t.Proof.Weprovethisbyinduction,showingthatiftheresultholdsfor(t1;u)and(t1;u1),thenitalsoholdsfor(t;u).Therefore,itsucestocheckt=0;u�0andt=u=1asourbasecases.Ift=0andu�0,theresultfollowsfromthefactthat1+Ht=ut u=1.Next,supposet=u=1.Wechooseouronenewcenterfromtheoneuncoveredclusterwithprobabilityexactly(Xu) .Inthiscase,Lemma3.2guaranteesthatE[0](Xc)+8OPT(Xu).Since0evenifwechooseacenterfromacoveredcluster,wehave,E[0](Xu) 
(Xc)+8OPT(Xu)+(Xc) 
2(Xc)+8OPT(Xu):Since1+Ht=2here,wehaveshowntheresultholdsforbothbasecases.Wenowproceedtoprovetheinductivestep.Itisconvenientheretoconsidertwocases.Firstsupposewechooseourrstcenterfromacoveredcluster.Asabove,thishappenswithprobabilityexactly(Xc) .Notethatthisnewcentercanonlydecrease.Bearingthisinmind,applytheinductivehypothesiswiththesamechoiceofcoveredclusters,butwithtdecreasedbyone.ItfollowsthatourcontributiontoE[0]inthiscaseisatmost,(Xc) 
(Xc)+8OPT(Xu)
(1+Ht1)+ut+1 u
(Xu):Ontheotherhand,supposewechooseourrstcenterfromsomeuncoveredclusterA.Thishappenswithprobability(A) .Letpadenotetheprobabilitythatwechoosea2Aasourcenter,giventhecenterissomewhereinA,andletadenote(A)afterwechooseaasourcenter.Onceagain,weapplyourinductivehypothesis,thistimeaddingAtothesetofcoveredclusters,aswellasdecreasingbothtanduby1.ItfollowsthatourcontributiontoE[OPT]inthiscaseisatmost,(A) 
Xa2Apa(Xc)+a+8OPT(Xu)8OPT(A)
(1+Ht1)+ut u1
(Xu)(A)(A) 
(Xc)+8OPT(Xu)
(1+Ht1)+ut u1
(Xu)(A):ThelaststepherefollowsfromthefactthatPa2Apaa8OPT(A),whichisimpliedbyLemma3.2.Now,thepower-meaninequalityimpliesthatPAXu(A)21 u
(Xu)2.Therefore,ifwesumoveralluncoveredclustersA,weobtainapotentialcontri-butionofatmost,(Xu) 
(Xc)+8OPT(Xu)
(1+Ht1)+1 
ut u1
(Xu)21 u
(Xu)2=(Xu) 
(Xc)+8OPT(Xu)
(1+Ht1)+ut u
(Xu):CombiningthepotentialcontributiontoE[0]frombothcases,wenowobtainthedesiredbound:E[0](Xc)+8OPT(Xu)
(1+Ht1)+ut u
(Xu)+(Xc) 
(Xu) u(Xc)+8OPT(Xu)
1+Ht1+1 u+ut u
(Xu):Theinductivestepnowfollowsfromthefactthat1 u1 t.WespecializeLemma3.3toobtainourmainresult.Theorem3.1IfCisconstructedwithk-means++,thenthecorrespondingpotentialfunctionsatisesE[]8(lnk+2)OPT.Proof.ConsidertheclusteringCafterwehavecom-pletedStep1.LetAdenotetheCOPTclusterinwhichwechosetherstcenter.ApplyingLemma3.3with t=u=k1andwithAbeingtheonlycoveredclus-ter,wehave,E[OPT](A)+8OPT8OPT(A)
(1+Hk1):TheresultnowfollowsfromLemma3.1,andfromthefactthatHk11+lnk.4AmatchinglowerboundInthissection,weshowthattheD2seedingusedbyk-means++isnobetterthan\n(logk)-competitiveinexpectation,therebyprovingTheorem3.1istightwithinaconstantfactor.Fixk,andthenchoosen,
,withnkand
.WeconstructXwithnpoints.Firstchoosekcentersc1;c2;:::;cksuchthatkcicjk2=
2nk n

2foralli=j.Now,foreachci,adddatapointsxi;1;xi;2;


;xi;n karrangedinaregularsimplexwithcenterci,sidelength,andradiusq nk 2n
.Ifwedothisinorthogonaldimensionsforeachi,wethenhave,kxi;i0xj;j0k=ifi=j,or
otherwise.Weproveourseedingtechniqueisinexpectation\n(logk)worsethantheoptimalclusteringinthiscase.Clearly,theoptimalclusteringhascentersfcig,whichleadstoanoptimalpotentialofOPT=nk 2
2.Conversely,usinganinductionsimilartothatofLemma3.3,weshowD2seedingcannotmatchthisbound.Asbefore,weboundtheexpectedpotentialintermsofthenumberofcenterslefttochooseandthenumberofuncoveredclusters(thoseclustersofC0fromwhichwehavenotchosenacenter).Lemma4.1.LetCbeanarbitraryclusteringonXwithkt1centers,butwithuclustersfromCOPTuncovered.NowsupposeweaddtrandomcenterstoC,chosenwithD2weighting.LetC0denotethetheresultingclustering,andlet0denotethecorrespondingpotential.Furthermore,let=nk2 n,=
22k2
2andH0u=Pui=1ki ki.Then,E[0]isatleast,t+1
n2
(1+H0u)
+n k
22n2
(ut):Proof.Weprovethisbyinductionont.Ift=0,notethat,0==nu
n kk
2+u
n k

2:Sincenu
n kn k,wehavenu
n kk nu
n kn kk n k=.Also,;1.Therefore,0
nu
n k
2
+u
n k

2:Finally,sincen2uu
n k
2
andn2un2H0u,wehave,0
n2
(1+H0u)
+n k
22n2
u:Thiscompletesthebasecase.Wenowproceedtoprovetheinductivestep.AswithLemma3.3,weconsidertwocases.Theprobabilitythatourrstcenterischosenfromanuncoveredclusteris,u
n k

2 u
n k

2+(ku)
n k
2(kt)2u
2 u
2+(ku)2
u
2 u
2+(ku)2:Applyingourinductivehypothesiswithtandubothdecreasedby1,weobtainapotentialcontributionfromthiscaseofatleast,u
2 u
2+(ku)2
t+1
n2
(1+H0u1)
+n k
22n2
(ut):Theprobabilitythatourrstcenterischosenfromacoveredclusteris(ku)
n k
2(kt)2 u
n k

2+(ku)
n k
2(kt)2(ku)
n k
2(kt)2 (ku)
n k
2
(ku)2 u
2+(ku)2
(ku)2 u
2+(ku)2:Applyingourinductivehypothesiswithtdecreasedby1butwithuconstant,weobtainapotentialcontributionfromthiscaseofatleast,(ku)2 u
2+(ku)2
t+1
n2
(1+H0u)
+n k
22n2
(ut+1):Therefore,E[0]isatleast,t+1
n2
(1+H0u)
+n k
22n2
(ut)+t+1 u
2+(ku)2
(ku)2
n k
22n2u
2
H0(u)H0(u1)
n2
: However,H0uH0u1=ku kuand=
22k2
2,sou
2
H0(u)H0(u1)
n2
=(ku)2
n k
22n2;andtheresultfollows.Asintheprevioussection,weobtainthedesiredresultbyspecializingtheinduction.Theorem4.1.D2seedingisnobetterthan2(lnk)-competitive.Proof.Supposeaclusteringwithpotentialiscon-structedusingk-means++onXdescribedabove.Ap-plyLemma4.1withu=t=k1aftertherstcenterhasbeenchosen.Notingthat1+H0k1=1+Pk1i=11 i1 k
=Hk�lnk,wethenhave,E[]k
n2
lnk:Now,xkandbutletnand
approachinnity.Thenandbothapproach1,andtheresultfollowsfromthefactthatOPT=nk 2
2.5GeneralizationsAlthoughthek-meansalgorithmitselfappliesonlyinvectorspaceswiththepotentialfunction=Px2Xminc2Ckxck2,wenotethatourseedingtech-niquedoesnothavethesamelimitations.Inthissec-tion,wediscussextendingourresultstoarbitrarymet-ricspaceswiththemoregeneralpotentialfunction,[`]=Px2Xminc2Ckxck`for`1.Inparticular,notethatthecaseof`=1isthek-medianspotentialfunction.Thesegeneralizationsrequireonlyonechangetothealgorithmitself.InsteadofusingD2seeding,weswitchtoD`seeding{i.e.,wechoosex0asacenterwithprobabilityD(x0)` x2XD(x)`.Fortheanalysis,themostimportantchangeap-pearsinLemma3.1.Ouroriginalproofusesaninnerproductstructurethatisnotavailableinthegeneralcase.However,aslightlyweakerresultcanbeprovenusingonlythetriangleinequality.Lemma5.1.LetAbeanarbitraryclusterinCOPT,andletCbetheclusteringwithjustonecenter,whichischo-senuniformlyatrandomfromA.Then,E[[`](A)]2`[`]OPT(A).Proof.LetcdenotethecenterofAinCOPT.Then,E[[`](A)]=1 jAjXa02AXa2Akaa0k`2`1 jAjXa02AXa2Akack`+ka0ck`
=2`[`]OPT(A):Thesecondstepherefollowsfromthetriangleinequalityandthepower-meaninequality.Therestofourupperboundanalysiscarriesthroughwithoutchange,exceptthatintheproofofLemma3.2,weloseafactorof2`1fromthepower-meaninequality,insteadofjust2.Puttingeverythingtogether,weobtainthegeneraltheorem.Theorem5.1.IfCisconstructedwithD`seeding,thenthecorrespondingpotentialfunction[`]satises,E[[`]]22`(lnk+2)[`]OPT.6EmpiricalresultsInordertoevaluatek-means++inpractice,wehaveimplementedandtesteditinC++[3].Inthissection,wediscusstheresultsofthesepreliminaryexperiments.WefoundthatD2seedingsubstantiallyimprovesboththerunningtimeandtheaccuracyofk-means.6.1DatasetsWeevaluatedtheperformanceofk-meansandk-means++onfourdatasets.Therstdataset,Norm25,issynthetic.Togenerateit,wechose25\true"centersuniformlyatrandomfroma15-dimensionalhypercubeofsidelength500.WethenaddedpointsfromGaussiandistributionsofvariance1aroundeachtruecenter.Thus,weobtainedanumberofwellseparatedGaussianswiththethetruecentersprovidingagoodapproximationtotheoptimalclustering.Wechosetheremainingdatasetsfromreal-worldexamplesotheUC-IrvineMachineLearningReposi-tory.TheClouddataset[7]consistsof1024pointsin10dimensions,anditisPhilippeCollard'srstcloudcoverdatabase.TheIntrusiondataset[18]consistsof494019pointsin35dimensions,anditrepresentsfeaturesavail-abletoanintrusiondetectionsystem.Finally,theSpamdataset[25]consistsof4601pointsin58dimensions,anditrepresentsfeaturesavailabletoane-mailspamdetectionsystem.Foreachdataset,wetestedk=10;25;and50.6.2MetricsSinceweweretestingrandomizedseed-ingprocesses,weran20trialsforeachcase.Wereport Average Minimum AverageT k k-meansk-means++ k-meansk-means++ k-meansk-means++ 10 1:365
105 8.47% 1:174
105 0.93% 0.12 46.72% 25 4:233
104 99.96% 1:914
104 99.92% 0.90 87.79% 50 7:750
103 99.81% 1:474
101 0.53% 2.04 1.62% Table1:ExperimentalresultsontheNorm25dataset(n=10000,d=15).Fork-means,welisttheactualpotentialandtimeinseconds.Fork-means++,welistthepercentageimprovementoverk-means:100%
1k-means++value k-meansvalue
. Average Minimum AverageT k k-meansk-means++ k-meansk-means++ k-meansk-means++ 10 7:921
103 22.33% 6:284
103 10.37% 0.08 51.09% 25 3:637
103 42.76% 2:550
103 22.60% 0.11 43.21% 50 1:867
103 39.01% 1:407
103 23.07% 0.16 41.99% Table2:ExperimentalresultsontheClouddataset(n=1024,d=10).Fork-means,welisttheactualpotentialandtimeinseconds.Fork-means++,welistthepercentageimprovementoverk-means. Average Minimum AverageT k k-meansk-means++ k-meansk-means++ k-meansk-means++ 10 3:387
108 93.37% 3:206
108 94.40% 63.94 44.49% 25 3:149
108 99.20% 3:100
108 99.32% 257.34 49.19% 50 3:079
108 99.84% 3:076
108 99.87% 917.00 66.70% Table3:ExperimentalresultsontheIntrusiondataset(n=494019,d=35).Fork-means,welisttheactualpotentialandtimeinseconds.Fork-means++,welistthepercentageimprovementoverk-means. Average Minimum AverageT k k-meansk-means++ k-meansk-means++ k-meansk-means++ 10 3:698
104 49.43% 3:684
104 54.59% 2.36 69.00% 25 3:288
104 88.76% 3:280
104 89.58% 7.36 79.84% 50 3:183
104 95.35% 2:384
104 94.30% 12.20 75.76% Table4:ExperimentalresultsontheSpamdataset(n=4601,d=58).Fork-means,welisttheactualpotentialandtimeinseconds.Fork-means++,welistthepercentageimprovementoverk-means. theminimumandtheaveragepotential(actuallydi-videdbythenumberofpoints),aswellasthemeanrunningtime.Ourimplementationsarestandardwithnospecialoptimizations.6.3ResultsTheresultsfork-meansandk-means++aredisplayedinTables1through4.Welisttheabsoluteresultsfork-means,andthepercentageimprovementachievedbyk-means++(e.g.,a90%improvementintherunningtimeisequivalenttoafactor10speedup).Weobservethatk-means++consistentlyoutperformedk-means,bothbyachievingalowerpotentialvalue,insomecasesbyseveralordersofmagnitude,andalsobyhavingafasterrunningtime.TheD2seedingisslightlyslowerthanuniformseeding,butitstillleadstoafasteralgorithmsinceithelpsthelocalsearchconvergeafterfeweriterations.Thesyntheticexampleisacasewherestandardk-meansdoesverybadly.Eventhoughthereisan\obvious"clustering,theuniformseedingwillinevitablymergesomeoftheseclusters,andthelocalsearchwillneverbeabletosplitthemapart(see[12]forfurtherdiscussionofthisphenomenon).Thecarefulseedingmethodofk-means++avoidedthisproblemaltogether,anditalmostalwaysattainedtheoptimalclusteringonthesyntheticdataset.Thedierencebetweenk-meansandk-means++onthereal-worlddatasetswasalsosubstantial.Ineverycase,k-means++achievedatleasta10%accuracyimprovementoverk-means,anditoftenperformedmuchbetter.Indeed,ontheSpamandIntrusiondatasets,k-means++achievedpotentials20to1000timessmallerthanthoseachievedbystandardk-means.Eachtrialalsocompletedtwotothreetimesfaster,andeachindividualtrialwasmuchmorelikelytoachieveagoodclustering.7ConclusionandfutureworkWehavepresentedanewwaytoseedthek-meansalgorithmthatisO(logk)-competitivewiththeoptimalclustering.Furthermore,ourseedingtechniqueisasfastandassimpleasthek-meansalgorithmitself,whichmakesitattractiveinpractice.Towardsthatend,weranpreliminaryexperimentsonseveralreal-worlddatasets,andweobservedthatk-means++substantiallyoutperformedstandardk-meansintermsofbothspeedandaccuracy.AlthoughouranalysisoftheexpectedpotentialE[]achievedbyk-means++istighttowithinacon-stantfactor,afewopenquestionsstillremain.Mostimportantly,itisstandardpracticetorunthek-meansalgorithmmultipletimes,andthenkeeponlythebestclusteringfound.Thisraisesthequestionofwhetherk-means++achievesasymptoticallybetterresultsifitisallowedseveraltrials.Forexample,ifk-means++isrun2ktimes,ourargumentscanbemodiedtoshowitislikelytoachieveaconstantapproximationatleastonce.Weaskwhetherasimilarboundcanbeachievedforasmallernumberoftrials.Also,experimentsshowedthatk-means++generallyperformedbetterifitselectedseveralnewcentersduringeachiteration,andthengreedilychosetheonethatdecreasedasmuchaspossible.Unfortunately,ourproofsdonotcarryovertothisscenario.Itwouldbeinterestingtoseeacomparable(orbetter)asymptoticresultprovenhere.Finally,wearecurrentlyworkingonamorethor-oughexperimentalanalysis.Inparticular,wearemea-suringtheperformanceofnotonlyk-means++andstan-dardk-means,butalsoothervariantsthathavebeensuggestedinthetheorycommunity.AcknowledgementsWewouldliketothankRajeevMotwaniforhishelpfulcomments.References[1]PankajK.AgarwalandNabilH.Mustafa.k-meansprojectiveclustering.InPODS'04:Proceedingsofthetwenty-thirdACMSIGMOD-SIGACT-SIGARTsym-posiumonPrinciplesofdatabasesystems,pages155{165,NewYork,NY,USA,2004.ACMPress.[2]D.ArthurandS.Vassilvitskii.Worst-caseandsmoothedanalysisoftheICPalgorithm,withanap-plicationtothek-meansmethod.InSymposiumonFoundationsofComputerScience,2006.[3]DavidArthurandSergeiVassilvitskii.k-means++testcode.http://www.stanford.edu/~darthur/kMeansppTest.zip.[4]DavidArthurandSergeiVassilvitskii.Howslowisthek-meansmethod?InSCG'06:Proceedingsofthetwenty-secondannualsymposiumoncomputationalgeometry.ACMPress,2006.[5]PavelBerkhin.Surveyofclusteringdataminingtechniques.Technicalreport,AccrueSoftware,SanJose,CA,2002.[6]MosesCharikar,LiadanO'Callaghan,andRinaPani-grahy.Betterstreamingalgorithmsforclusteringprob-lems.InSTOC'03:Proceedingsofthethirty-fthan-nualACMsymposiumonTheoryofcomputing,pages30{39,NewYork,NY,USA,2003.ACMPress.[7]PhilippeCollard'scloudcoverdatabase.ftp://ftp.ics.uci.edu/pub/machine-learning-databases/undocumented/taylor/cloud.data.[8]SanjoyDasgupta.Howfastis-means?InBernhardScholkopfandManfredK.Warmuth,editors,COLT,volume2777ofLectureNotesinComputerScience,page735.Springer,2003. 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