Diffusion Maps, Spectral Clustering and Reaction

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 di#usion maps, defined as the embedding of complex (high dimensional) data onto a low dimensional Euclidian 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 di#usion map converge to the eigenfunctions of a corresponding di#erential operator defined on the support of the probability distribution. Di#erent normalizations of the Markov chain on the graph lead to di#erent limiting di#erential operators