Diffusion maps, spectral clustering and eigenfunctions of fokker-planck operators

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Diffusion maps, spectral clustering and eigenfunctions of fokker-planck operators (2005)

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by Boaz Nadler , Stéphane Lafon , Ronald R. Coifman , Ioannis G. Kevrekidis

Venue:	in Advances in Neural Information Processing Systems 18
Citations:	110 - 14 self

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@INPROCEEDINGS{Nadler05diffusionmaps,,
    author = {Boaz Nadler and Stéphane Lafon and Ronald R. Coifman and Ioannis G. Kevrekidis},
    title = {Diffusion maps, spectral clustering and eigenfunctions of fokker-planck operators},
    booktitle = {in Advances in Neural Information Processing Systems 18},
    year = {2005},
    pages = {955--962},
    publisher = {MIT Press}
}

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Abstract

This paper presents a diffusion based probabilistic interpretation of spectral clustering and dimensionality reduction algorithms that use the eigenvectors of the normalized graph Laplacian. Given the pairwise adjacency matrix of all points, we define a diffusion distance between any two data points and show that the low dimensional representation of the data by the first few eigenvectors of the corresponding Markov matrix is optimal under a certain mean squared error criterion. Furthermore, assuming that data points are random samples from a density p(x) = e −U(x) we identify these eigenvectors as discrete approximations of eigenfunctions of a Fokker-Planck operator in a potential 2U(x) with reflecting boundary conditions. Finally, applying known results regarding the eigenvalues and eigenfunctions of the continuous Fokker-Planck operator, we provide a mathematical justification for the success of spectral clustering and dimensional reduction algorithms based on these first few eigenvectors. This analysis elucidates, in terms of the characteristics of diffusion processes, many empirical findings regarding spectral clustering algorithms.

Keyphrases

spectral clustering fokker-planck operator diffusion map data point first eigenvectors mathematical justification continuous fokker-planck operator spectral clustering algorithm error criterion certain mean dimensionality reduction pairwise adjacency matrix probabilistic interpretation random sample diffusion process dimensional reduction algorithm boundary condition discrete approximation low dimensional representation many empirical finding diffusion distance corresponding markov matrix normalized graph laplacian

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