Let $X=(X,d_X)$ and $Y=(X,d_Y)$ be metric spaces and $\varphi: X\rightarrow Y$ be an $L$-Lipschitz map, with $0 \le L < \infty$. Suppose $\mu$ is a probability measure on $X$ which satisfies Talagrand transportation-cost inequality, namely

There exists a constant $c_\mu > 0$ such that $$ W(\nu,\mu) \le \sqrt{2c_\mu H(\nu\|\mu)}, $$ for every other probability measure $\nu$ on $X$.

Here, $H$ denotes relative entropy (KL divergence) and $W$ denotes the Wasserstein $2$-distance on the space of probability measures on $X$ (with finite 2nd moment).


Does the pushforward $\varphi_\#\mu$ of $\mu$ under $\varphi$ satisfy such an inequality ? My guess is that it does, with constant $c_{\varphi_\#\mu} \le L c_\mu$.


The result you want (but of course with $c_{\varphi_\#\mu} \le L^2 c_\mu$ rather than $c_{\varphi_\#\mu} \le L c_\mu$) is Lemma 2.1 in the paper at https://arxiv.org/pdf/math/0410172

  • $\begingroup$ That's awesome. Thanks. Also thanks for catching the $L \implies L^2$ thinko :) $\endgroup$
    – dohmatob
    Sep 3 '18 at 18:13

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.