On Wasserstein geometry of the space of Gaussian measures

TL;DR

This paper explores the geometric structure of the space of Gaussian measures under the $L^2$-Wasserstein metric, providing a Riemannian geometric perspective and explicit curvature formulas.

Contribution

It introduces a Riemannian metric inducing the Wasserstein distance on Gaussian measures and derives explicit formulas for sectional curvatures based on covariance eigenvalues.

Findings

Constructed a Riemannian metric for Gaussian measures in Wasserstein space

Derived explicit formulas for sectional curvatures in terms of covariance eigenvalues

Provided detailed geometric descriptions of the Wasserstein space of Gaussian measures

Abstract

The space of Gaussian measures on a Euclidean space is geodesically convex in the $L^2$-Wasserstein space. This space is a finite dimensional manifold since Gaussian measures are parameterized by means and covariance matrices. By restricting to the space of Gaussian measures inside the $L^2$-Wasserstein space, we manage to provide detailed descriptions of the $L^2$-Wasserstein geometry from a Riemannian geometric viewpoint. We first construct a Riemannian metric which induces the $L^2$-Wasserstein distance. Then we obtain a formula for the sectional curvatures of the space of Gaussian measures, which is written out in terms of the eigenvalues of the covariance matrix.