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 {\em diffusion 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 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. One normalization gives the FokkerPlanck operators with the same potential U(x), best suited for the study of stochastic differential equations as well as for clustering. Another normalization gives the LaplaceBeltrami (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 nonuniform density.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 March 2005
 arXiv:
 arXiv:math/0503445
 Bibcode:
 2005math......3445N
 Keywords:

 Mathematics  Numerical Analysis;
 Mathematics  Probability;
 65c40;
 62h30
 EPrint:
 submitted to journal of applied and computational harmonic analysis