It is now known [Otto et Villani 2000; Cordero et al 2006; etc.] that on an $n$-dimensional smooth Riemannian manifold $X$ and a probability measure $\mu$ on $X$ with density $d\mu \propto e^{-V}dvol$ satisfying the curvature condition

$$ \operatorname{Hess}_x(V) + \operatorname{Ric}_x \succeq (1/c) I_n,\forall x \in X, $$

the transportation-cost inequality $$W_2(\nu,\mu) \le \sqrt{2cH(\nu\|\mu)} $$ holds for all other distributions $\nu$ on $X$ absolutely continuous w.r.t $\mu$. Here $W_2$ is the Wasserstein-2 distance induced by the geodesic metric on $X$.

Question

  • (A) What are the most general conditions under which the above transportation-cost inequality holds.

  • (B) Can the curvature condition be relaxed to "piecewise" version. That is, what if we instead assume a convex mixture $d\mu = \sum_{i=1}^k \pi_i d\mu_i$ where each $d\mu_i$ is log-concave and satisfies the curvature condition on some piece $X_i$ of $X$ ?

  • (B') Are there concentration inequalities for mixtures of Gaussians ?


Partial answer

  • Log-concave distributions satisfying the Bakry-Eméry $\operatorname{CD}(n,\infty)$ curvature condition on manifolds
  • Distributions with densities on compact homogeneous Riemannian manifolds. See this paper of Rothaus.
  • Distributions which can be realized as pushforwards of distributions with some $\text{T}_2(c)$, under Lipschitz maps. If $\mu$ has $\text{T}_2(c)$ property and $\varphi: X \rightarrow Y$ is $L$-Lipschitz, then $\varphi_\#\mu$ has $\text{T}_2(L^2c)$.
  • Finite tensor product $\mu_1 \otimes \mu_2 \otimes \ldots \otimes \mu_k$ of distributions having $\text{T}_2(c)$ also has $\text{T}_2(c)$.
  • Could you provide a reference to your second partial answer on distributions on compact manifolds? Thanks! – O. Richard Oct 18 at 16:44
  • @O.Richard This follows from Pinker's inequality and the fact that $W \le \text{diam}(X)TV$ (see Theorem 4 of this review math.hmc.edu/~su/papers.dir/metrics.pdf). – dohmatob Oct 18 at 18:02
  • Oops, good catch. Ya, what I just said holds for $W_1$, not $W_2$. On a finite space though, $W_2 \le \text{diam}^2(X)W_1^{1/2}$. I don't think the $1/2$ power can be replaced with $1$. My false claim about $TV$ is likely unrepairable :/ – dohmatob Oct 18 at 18:21
  • Yeah... Perhaps the best bound is in Bolley and Villani (2012) Particular Case 5: $W_p(\mu,\nu)\leq 2^{\frac{1}{2p}} \mathrm{diam}(X)H(\mu||\nu)^{\frac{1}{2p}}$. – O. Richard Oct 18 at 18:31
  • @O.Richard It turns out that on a compact homogeneous Riemannian mainfold $X$, every measure which has density w.r.t Lebesgue satisfies Log-Sobolev inequality at all levels $\rho < $ hyper-contractivity constant of $X$, and therefore the Talagrand transportation-cost inequality $\text{T}_2(\rho)$ ams.org/journals/proc/1998-126-10/S0002-9939-98-04406-2/… – dohmatob Oct 26 at 16:33

(A) Nathael Gozlan proved that a distribution $\mu$ satisfies a $T_2$ inequality if and only if all finite tensor products of $\mu$ satisfy a subgaussian concentration property.

(B) and (B') For finite mixtures you could certainly get something by viewing them as pushforwards of product measures. However, you would necessarily get something that depends badly on $k$, since any distribution on $\mathbb{R}^n$ --- including those with no good concentration properties --- can be approximated by a mixture of Gaussians.

  • Interesting (upvoted). Thanks! – dohmatob Oct 18 at 17:19

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