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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].

as linear methods. Recently, we introduced an eigenvector method—called locally linear embedding (LLE)—for the problem of nonlinear dimensionality reduction[4]. This problem is illustrated by the nonlinear manifold in Figure 1. In this example, the dimen-sionality reduction by LLE succeeds in identifying the underlying structure of the

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