On Wasserstein geometry of the 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.

15pages, 1figures

Cited by in corpus (1)

• Cohomological gauge theory, quiver matrix models and Donaldson-Thomas theory