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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].

ball of fix ed radius, or by using more sophisticated rules based on local metrics.) Reconstruction errors are then measured by the cost function: (1) which adds up the squared distances between all the data points and their recon-structions. The weights summarize the contribution of the th data point to the th reconstruction. To compute the ...

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