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It is known that under the Log-Sobolev Inequality for $\pi$, i.e., if for all $\rho$, $$H_\pi(\rho):=\int \rho(x)\log\frac{\rho(x)}{\pi(x)}dx \leq \frac{1}{2\beta}\int \rho(x)\left\|\nabla \log\frac{\rho(x)}{\pi(x)}\right\|^2 dx$$ we have $$\frac{\beta}{2}W_{2}(\rho,\pi)^2 \leq H_\pi(\rho) \;\text{for}\; W_{2}(\rho,\pi)^2 := \inf_{x\sim\rho,x'\sim \pi} \mathbb{E}[\|x-x'\|_2^2]$$ the Wasserstein-2 distance and $H_\pi(\rho)$ is the KL divergence. I believe it's also sometimes referred to as Talagrand's inequality. My question is - is there an analogue of the inequality for a weighted version of the Log-Sobolev Inequality? More specifically, if for all $\rho$, we have $$\int \rho(x)\log\frac{\rho(x)}{\pi(x)}dx \leq \frac{1}{2\beta}\int \rho(x)\left\|\nabla \log\frac{\rho(x)}{\pi(x)}\right\|_{G(x)}^2 dx$$ for some $G(x) \succ 0$, does that imply $$\frac{\beta}{2}W_{2,G}(\rho,\pi)^2 \leq H_\pi(\rho)\; \text{where}\; W_{2,G}(\rho,\pi)^2 := \inf_{x\sim\rho,x'\sim \pi} \mathbb{E}[\|x-x'\|_{G(x)^{-1}}^2]?$$ It is true for the simplest case where $G$ is a fixed matrix but does it hold more generally? Thank you in advance for any pointers/help!

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  • $\begingroup$ Well, certainly if your Riemannian metric tensor $G(x)$ is uniformly bounded from below and from above you will get a similar inequality by paying a factor $C\approx (\lambda/\Lambda)^2$ or $C\approx (\Lambda/\lambda)^2$ in front of $\frac\beta 2$, where $\lambda\succ G(x)\succ \Lambda$ are the lower and upper bounds. In other words you can use the fact that the inequality holds true in the Euclidean setting. But perhaps you are really interested in the sharp $\frac{\beta}{2}$ factor? (in which case I don't know the answer, but I suspect this is actually not trivial at all) $\endgroup$ May 19 at 19:55
  • $\begingroup$ Thanks yes I understand the comment - but don't really want to reduce to the Euclidean setting in the bound :( $\endgroup$
    – user_qj
    May 19 at 20:09
  • $\begingroup$ I see... then I suspect this is really tricky, getting sharp constants for such highly nonlinear functional inequalities can be delicate. One last comment, though: in the Euclidean setting the $\beta/2$ factor is related to displacement convexity (in the sense of McCann) of the relative entropy $\rho\mapsto H(\rho|\pi)$, which in turn is related to log-concavity of the reference measure $\pi$. There are results about displacement convexity over manifolds, in which case the Ricci curvature plays a significant role. Perhaps it's worth looking into it? (I recommend Villani's "big book") $\endgroup$ May 19 at 20:13
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The answer is yes: this is exactly Theorem 22.17 in Villani's "big" book [1], page 585.

[1] Villani, C. (2008). Optimal transport: old and new (Vol. 338). Springer Science & Business Media.

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