低秩的讲座课件.pdf

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1、SomeRecentAdvancesintheTheoryofLow-rankModelingEmmanuelCandesMachineLearningSummerSchool(MLSS2001)CarcansMaubuisson,September2011ObjectiveExplosionofresearchontheoryoflow-rankmodelingOurgoalistodiscusssomerecentworksSomeofitisoursSomeofitisnotMatrixCompletionThe

2、Net ixproblemNet ixdatabaseAbouthalfamillionusersAbout18,000moviesPeopleratemoviesSparselysampledentriesTheNet ixproblemNet ixdatabaseAbouthalfamillionusers2Movies3About18,000movies67Peopleratemovies6767Users67Sparselysampledentries676745TheNet ixprobl

3、emNet ixdatabaseAbouthalfamillionusers2Movies3About18,000movies67Peopleratemovies6767Users67Sparselysampledentries676745ChallengeCompletetheNet ixmatrix"Manysuchproblems!collaborativeltering,partiallylledoutsurveys...Globalpositioningfromlocaldistan

4、cesPointsfxg2Rdj1jnPartialinformationaboutdistancesL=kxxk2ijijj23Example(Singer,Biswasetal.)Low-poweredwirelesslynetworkedsensors667767Eachsensorcanconstructadistanceestimatei6767fromnearestneighbor6745GlobalpositioningfromlocaldistancesPointsfxg2Rd

5、j1jnPartialinformationaboutdistancesL=kxxk2ijijj23Example(Singer,Biswasetal.)Low-poweredwirelesslynetworkedsensors667767Eachsensorcanconstructadistanceestimatei6767fromnearestneighbor6745ProblemLocatethesensorsOtherproblemsofthiskindLinearsystemiden

6、tication(Vandenbergheetal.)Quantum-statetomography(Grossetal.)Partiallyobservedcovariancematrix(Vaidyanathanetal.)Low-rankmatrixcompletioninmachinelearning(Srebroetal.Vertetal.)Structure-from-motionproblemincomputervision(Tomasietal.)...Matrixcompletion23????6

7、????7MatrixL2Rn1n26766????77Observesubsetofentries66????77Canweguessthemissingentries?4?????5????Matrixcompletion23????6????7MatrixL2Rn1n26766????77Observesubsetofentries66????77Canweguessthemissingentries?4?????5????Everybodywouldagreethisl

8、ooksimpossibleMassivehigh-dimensionaldataEngineering/scienticapplications:unknownmatrixhasoften(approx.)lowrankImagesVideosHigh-dimensionalitybutoftenlow-dime