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 $$W(\nu,\mu) \le \sqrt{2cH(\nu\|\mu)} $$ holds for all other distributions $\nu$ on $X$. Here $W$ is the Wasserstein 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 $\operatorname{CD}(n,\infty)$ curvature condition on manifolds
  • Distributions on compact manifolds: Take $c=\operatorname{diam}(X)^2/2$
  • Distributions which can be realized as pushforwards of distributions with som $\text{T}_2(c)$, 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 $\text{T}_2(c)$.

(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.

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