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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].

mixture models for local dimensionality reduction[5, 6], which cluster the data and perform PCA within each cluster, do not address the problem considered here— namely, how to map high dimensional data into a single global coordinate system of lower dimensionality. In this paper, we review the LLE algorithm in its most basic form and illustrate a

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