Entropy-Regularized $2$-Wasserstein Distance between Gaussian Measures

TL;DR

This paper explores the geometry of Gaussian measures under entropy-regularized 2-Wasserstein distance, providing closed-form solutions, fixed-point characterizations, and analyzing limiting behaviors with numerical illustrations.

Contribution

It introduces closed-form solutions for the entropy-regularized Wasserstein distance between Gaussians and a fixed-point method for Gaussian barycenters, advancing the understanding of Gaussian geometry.

Findings

Closed-form expressions for the regularized Wasserstein distance and interpolations.

A fixed-point algorithm for Gaussian barycenters.

Analysis of the limits of the Sinkhorn divergence as regularization varies.

Abstract

Gaussian distributions are plentiful in applications dealing in uncertainty quantification and diffusivity. They furthermore stand as important special cases for frameworks providing geometries for probability measures, as the resulting geometry on Gaussians is often expressible in closed-form under the frameworks. In this work, we study the Gaussian geometry under the entropy-regularized 2-Wasserstein distance, by providing closed-form solutions for the distance and interpolations between elements. Furthermore, we provide a fixed-point characterization of a population barycenter when restricted to the manifold of Gaussians, which allows computations through the fixed-point iteration algorithm. As a consequence, the results yield closed-form expressions for the 2-Sinkhorn divergence. As the geometries change by varying the regularization magnitude, we study the limiting cases of vanishing and infinite magnitudes, reconfirming well-known results on the limits of the Sinkhorn divergence. Finally, we illustrate the resulting geometries with a numerical study.