Time Coupled Diffusion Maps

TL;DR

This paper introduces time coupled diffusion maps, a method to embed evolving data points collected over time into a single Euclidean space, capturing the underlying geometry and dynamics of the data.

Contribution

The paper develops a novel approach to analyze time-dependent data using product of diffusion operators, linking it to heat equations on manifolds with changing metrics.

Findings

Defines time coupled diffusion distance and maps with probabilistic interpretation

Models evolving data as samples from a manifold with a time-dependent metric

Connects the method to heat equations on manifolds with dynamic geometry

Abstract

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.