  We consider a collection of $n$ points in $\mathbb{R}^d$ measured at $m$
times, which are encoded in an $n \times d \times m$ 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 construct a
corresponding sequence of diffusion operators and study their product. Via this
product, we introduce the notion of time coupled diffusion distance and 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 over a manifold with time dependent metric.
