Diffusion Maps, Spectral Clustering and Eigenfunctions of Fokker-Planck operators

TL;DR

This paper offers a probabilistic interpretation of spectral clustering using diffusion processes, connecting eigenvectors of graph Laplacians to eigenfunctions of Fokker-Planck operators, and justifies their effectiveness mathematically.

Contribution

It introduces a diffusion-based framework linking spectral clustering eigenvectors to Fokker-Planck eigenfunctions, providing a theoretical foundation for their success.

Findings

Eigenvectors approximate Fokker-Planck eigenfunctions

Diffusion distance optimally represents data in low dimensions

Mathematical justification for spectral clustering effectiveness

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.