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An Introduction to Locally Linear Embedding

Research180 ParkAve, problemsininformationprocessinginvolve (LLE),anunsu-pervisedlearningalgorithmth atcomputeslowdimensional, , LLEmapsitsinputsintoa singleglobalcoordinatesystemoflowerdimen sionality, anditsoptimizations thoughcapableofgeneratinghighlynonlinear embeddings problemsinstatisticalpatternrecognitionb eginwiththepreprocessingofmultidimension alsignals, , thegoalofpreprocessingis someformofdimensionalityreduction:to com-pressthesignalsin sizeandto popularformsofdimensionalityreductionare themethodsofprincipalcom-ponentanalysis( PCA)[1]andmultidimensionalscaling(MDS)[2 ]. BothPCAandMDSareeigenvectormethodsdesign edto modellinearvariabilitiesin , (ormetric)MDS, thesedistancescorrespondtoEuclideandista nces, ,andtheiroptimizationsdonotinvolve , , weintroducedaneigenvectormethod calledlocallylinearembedding(LLE) fortheproblemofnonlineardimensionalityre duction[4]. ,thedimen-sionalityreductionbyLLEsucceed sinidentifyingtheunderlyingstructureofth emanifold, PCAandMDS,ouralgorithmis sim-pletoimplement,anditsoptimizationsdo notinvolve ,however, it is [5, 6], whichclusterthedataandperformPCAwithinea chcluster, donotaddresstheproblemconsideredhere namely, howtomaphighdimensionaldataintoa , wereviewtheLLEalgorithminitsmostbasicfor mandillustrateapotentialapplicationtoaud iovisualspeechsynthesis[3].

An Introduction to Locally Linear Embedding Lawrence K. Saul AT&T Labs – Research 180 Park Ave, Florham Park, NJ 07932 USA lsaul@research.att.com Sam T. Roweis Gatsby Computational Neuroscience Unit, UCL 17 Queen Square, London WC1N 3AR, UK roweis@gatsby.ucl.ac.uk Abstract Many problems in information processing involve some form …

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