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?



Comparison of Information and Wasserstein Topologies
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](http://cedricvillani.org/sites/dev/files/old_images//2012/08/014.OV-Talagrand.pdf) generalizing this inequality and connecting it to the log-Sobolev inequality.


