On Wasserstein geometry of the space of Gaussian measures

Details On Wasserstein geometry of the space of Gaussian measures On Wasserstein geometry of the space of Gaussian measures

2008-01-15

arxiv.org/abs/0801.2250v3

Mathematics

Differential Geometry

15pages, 1figures

Scientific paper

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.

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