TITLE: Comparison of Information and Wasserstein Topologies
QUESTION [5 upvotes]: There are many possible metrics one can place on the space of Gaussian probability measures on $\mathbb{R}^n$, with strictly positive definite co-variance matrices.  Let's denote this space by $X$.
I'm particularly interested in the information geometric one (using the Fisher-Rao-Riemann metric) and the one induced by restricting the Wasserstein $2$ metric from $\mathcal{P}_2(\mathbb{R}^n)$ to the subspace $X$.  But how doe these compare?  Most specifically, is the Wasserstein $2$-distance dominated by the Fisher-Rao metric's induced distance function?

REPLY [2 votes]: Wasserstein distance has not good properties of invariances compared to Fisher Metric, that could be extended on convex cones by Jean-Louis Koszul tools (https://link.springer.com/chapter/10.1007/978-3-030-02520-5_12), and on homogeneous symplectic manifolds for Lie groups by Jean-Marie Souriau tools (https://www.mdpi.com/1099-4300/22/5/498).
These topics are developd in GSI conferences: www.gsi2021.org