Diffusion maps, spectral clustering and reaction coordinates of dynamical systems

Abstract

A central problem in data analysis is the low dimensional representation of high dimensional data and the concise description of its underlying geometry and density. In the analysis of large scale simulations of complex dynamical systems, where the notion of time evolution comes into play, important problems are the identification of slow variables and dynamically meaningful reaction coordinates that capture the long time evolution of the system. In this paper we provide a unifying view of these apparently different tasks, by considering a family of diffusion maps, defined as the embedding of complex (high dimensional) data onto a low dimensional Euclidean space, via the eigenvectors of suitably defined random walks defined on the given datasets. Assuming that the data is randomly sampled from an underlying general probability distribution p ( x ) = e − U ( x ) , we show that as the number of samples goes to infinity, the eigenvectors of each diffusion map converge to the eigenfunctions of a corresponding differential operator defined on the support of the probability distribution. Different normalizations of the Markov chain on the graph lead to different limiting differential operators. Specifically, the normalized graph Laplacian leads to a backward Fokker–Planck operator with an underlying potential of 2 U ( x ) , best suited for spectral clustering. A different anisotropic normalization of the random walk leads to the backward Fokker–Planck operator with the potential U ( x ) , best suited for the analysis of the long time asymptotics of high dimensional stochastic systems governed by a stochastic differential equation with the same potential U ( x ) . Finally, yet another normalization leads to the eigenfunctions of the Laplace–Beltrami (heat) operator on the manifold in which the data resides, best suited for the analysis of the geometry of the dataset regardless of its possibly non-uniform density.