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1、ONTENSORS,SPARSITY,ANDNONNEGATIVEFACTORIZATIONSERICC.CHIyANDTAMARAG.KOLDAzAbstract.Tensorshavefoundapplicationinavarietyof elds,rangingfromchemometricstosignalprocessingandbeyond.Inthispaper,weconsidertheproblemofmultilinearmodelingofsparsecountdata.Ourgoalistodevelopadescriptivetensorf
2、actorizationmodelofsuchdata,alongwithappropriatealgorithmsandtheory.Todoso,weproposethattherandomvariationisbestdescribedviaaPoissondistribution,whichbetterdescribesthezerosobservedinthedataascomparedtothetypicalassumptionofaGaussiandistribution.UnderaPoissonassumption,we tamodeltoobserv
3、eddatausingthenegativelog-likelihoodscore.WepresentanewalgorithmforPoissontensorfactorizationcalledCANDECOMP{PARAFACAlternatingPoissonRegression(CP-APR)thatisbasedonamajorization-minimizationapproach.ItcanbeshownthatCP-APRisageneralizationoftheLee-Seungmultiplicativeupdates.Weshowhowtopr
4、eventthealgorithmfromconvergingtonon-KKTpointsandproveconvergenceofCP-APRundermildconditions.WealsoexplainhowtoimplementCP-APRforlarge-scalesparsetensorsandpresentresultsonseveraldatasets,bothrealandsimulated.Keywords.Nonnegativetensorfactorization,nonnegativeCANDECOMP-PARAFAC,Poissonten
5、sorfactorization,Lee-Seungmultiplicativeupdates,majorization-minimizationalgorithms1.Introduction.Tensorshavefoundapplicationinavarietyof elds,rangingfromchemometricstosignalprocessingandbeyond.Inthispaper,weconsidertheproblemofmultilinearmodelingofsparsecountdata.Forinstance,wemayconsid
6、erthenumberofpaperspublishedbyaspeci cauthorataspeci cconference[10],thenumberofpacketssentfromoneIPaddresstoanotherusingaspeci cport[42],orto/fromandtermcountsonemails[1].Ourgoalistodevelopadescriptivemodelofsuchdata,alongwithappropriatealgorithmsandtheory.LetXrepresentanN-waydatatensor
7、ofsizeI1I2IN.WeareinterestedinanR-componentnonnegativeCANDECOMP/PARAFAC[6,18]factormodelXRM=a(1)a(N);(1.1)rrrr=1(n)whererepresentsouterproductandarrepresentstherthcolumnofthenonneg-(n)ativefactormatrixAofsizeInR.Werefertoeachsummandasacomponent.Assumingeachf
