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arXiv · 0801.2250

On Wasserstein geometry of the space of Gaussian measures

Asuka Takatsu
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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.

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Asuka Takatsu. 2009-02-11. On Wasserstein geometry of the space of Gaussian measures. https://arxiv.org/abs/0801.2250

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