Comparison of Information and Wasserstein Topologies 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?
 A: It is not the case that the Fisher-Rao distance dominates the Wasserstein distance. For instance, it fails for univariate normal distributions $\mathcal{N}(\mu,\sigma)$. In particular, the Wasserstein distance is the Euclidean distance on the half-plane $\mathbb{H}= \{(\mu,\sigma)~|~ \sigma >0\}$. On the other hand, the Fisher metric is hyperbolic, with metric $ds^2= \frac{1}{\sigma^2}(d \mu^2 + 2d \sigma^2).$ As a result, when the variance is large enough, the distance in the Fisher-Rao metric will be smaller than in the Wasserstein metric.
However, it is possible to bound the Wasserstein distance using the entropy, (or more precisely the KL-divergence). In particular, Talagrand proved that if $\gamma$ is the standard multivariate Gaussian with density $$d \gamma(x)=\frac{e^{-|x|^{2} / 2}}{(2 \pi)^{n / 2}} d x,$$ and $\mu$ is any measure with Radon-Nikodym derivative $\frac{d\mu}{d \gamma}= h$, then the following inequality holds
$$W(\mu, \gamma) \leq \sqrt{2 \int_{\mathbb{R}^{n}} h \log h \, d \gamma}.$$
For a more complete reference, Otto and Villani wrote an excellent paper generalizing this inequality and connecting it to the log-Sobolev inequality.
A: 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
