Bi-stochastic kernels via asymmetric affinity functions

In this short letter we present the construction of a bi-stochastic kernel p for an arbitrary data set X that is derived from an asymmetric affinity function {\alpha}. The affinity function {\alpha} measures the similarity between points in X and some reference set Y. Unlike other methods that construct bi-stochastic kernels via some convergent iteration process or through solving an optimization problem, the construction presented here is quite simple. Furthermore, it can be viewed through the lens of out of sample extensions, making it useful for massive data sets.Comment: 5 pages. v2: Expanded upon the first paragraph of subsection 2.1. v3: Minor changes and edits. v4: Edited comments and added DO

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

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.Comment: submitted to NIPS 200