Time coupled diffusion maps

We consider a collection of n points in Rd measured at m times, which are encoded in an n×d×m data tensor. Our objective is to define a single embedding of the n points into Euclidean space which summarizes the geometry as described by the data tensor. In the case of a fixed data set, diffusion maps and related graph Laplacian methods define such an embedding via the eigenfunctions of a diffusion operator constructed on the data. Given a sequence of m measurements of n points, we introduce the notion of time coupled diffusion maps which have natural geometric and probabilistic interpretations. To frame our method in the context of manifold learning, we model evolving data as samples from an underlying manifold with a time-dependent metric, and we describe a connection of our method to the heat equation on such a manifold.