We consider a collection of @math points in @math measured at @math times, which are encoded in an @math data tensor. Our objective is to define a single embedding of the @math 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 @math measurements of @math 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.
Dynamical systems often exhibit the emergence of long-lived coherent sets, which are regions in state space that keep their geometric integrity to a high extent and thus play an important role in transport. In this article, we provide a method for extracting coherent sets from possibly sparse Lagrangian trajectory data. Our method can be seen as an extension of diffusion maps to trajectory space, and it allows us to construct "dynamical coordinates" which reveal the intrinsic low-dimensional organization of the data. The only a priori knowledge about the dynamics that we require is a locally valid notion of distance, which renders our method highly suitable for automated data analysis. We show convergence of our method to the analytic transfer operator framework of coherence in the infinite data limit, and illustrate its potential on several two- and three-dimensional examples as well as real world data.