Time-Coupled Diffusion Maps

We investigate the problem of creating concise geometric descriptions of data which smoothly changes with respect to a single parameter. Specifically, our objective is to construct a diffusion maps framework for such data sets. We assume smoothly changing data represents a sequence of embeddings from an underlying manifold with a smooth 1-parameter family of Riemannian metrics. We construct a time inhomogeneous Markov chain which approximates the heat kernel on the underlying evolving manifold. This construction is achieved by associating “diffusion time” with the data’s evolution parameter, and thus we call the resulting class of embeddings time-coupled diffusion maps (Joint work with Matthew J. Hirn).