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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].

The embeddings discovered by LLE are easiest to visualize for intrinsically two dimensional manifolds. In Fig. 1, for example, theinput to LLE consisted 546 7 data points sampled off the S-shapedmanifold. The resulting embedding shows how the algorithm, using 8 :9 neighbors per data point, successfully unraveled the underlying two dimensional ...

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