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I asked this same question on MSE, but with no luck, so I am trying to ask here.

Consider two measures $\mu , \nu$ on $\mathbb{R}^n$. Now consider a map (a priori only measurable, but feel free to add some regularity if you need) $f$ and the push forward measures $f_\# \mu, f_\# \nu$. Is there any estimate between the Wasserstein distance (say, 1-Wasserstein) between $\mu, \nu$ and the Wasserstein distance between $f_\# \mu, f_\# \nu$ other than $W(f_\# \mu , f_\# \nu) \leq Lip(f) W(\mu, \nu)$ if $f$ is Lipschitz? Also, is the optimal transport plan $\pi$ between $\mu$ and $\nu$ related to the optimal transport plan $\pi'$ between $f_\# \mu$ and $f_\# \nu$? I would be more interested in the case where $\mu$ and $\nu$ are diffuse, but anything is useful.

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    $\begingroup$ For this, you need conditions on $f$, say a Lipschitz condition. $\endgroup$
    – Iosif Pinelis
    Commented Sep 8 at 0:22
  • $\begingroup$ @IosifPinelis do you perhaps know some counterexample that shows that, if $f$ is not Lipschitz, we can say nothing on the distance between $f_\# \mu, f_\# \nu$? Or do you at least believe this to be true? $\endgroup$
    – tommy1996q
    Commented Sep 8 at 10:37
  • $\begingroup$ Your comment is unclear to me. There is no such thing as "we can say nothing on". About any object A, we can always say "A is A". $\endgroup$
    – Iosif Pinelis
    Commented Sep 8 at 13:02
  • $\begingroup$ @IosifPinelis I mean if using the Lipschitz constant on $f$ is the only known (at leat, well known) way to estimate the distance between the two push-forward measures, not requiring anything specific on $\mu$ and $\nu$ $\endgroup$
    – tommy1996q
    Commented Sep 8 at 14:58
  • $\begingroup$ I think your post should be edited to make the question specific. "[A]ny relation" is very non-specific. The best would be to state your question formally. Also, there should be only one question in one post. $\endgroup$
    – Iosif Pinelis
    Commented Sep 8 at 15:43

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Is there any estimate between the Wasserstein distance (say, 1-Wasserstein) between $\mu, \nu$ and the Wasserstein distance between $f_\# \mu, f_\# \nu$ other than $W(f_\# \mu , f_\# \nu) \leq Lip(f) W(\mu, \nu)$ if $f$ is Lipschitz?

Yes. More generally, suppose that $f$ satisfies the Hölder condition $$|f(x)-f(y)|\le C|x-y|^a$$ for some real $C\ge0$, some $a\in(0,1]$, and all $x$ and $y$ in $\R^n$, where $|\cdot|$ is the Euclidean norm.

Then for any random vectors $X$ and $Y$ in $\R^n$ with respective distributions $\mu$ and $\nu$, $$E|f(X)-f(Y)|\le CE|X-Y|^a\le C(E|X-Y|)^a,$$ whence $$W_1(f_{\#}\mu,f_{\#}\nu)\le CW_1(\mu,\nu)^a.$$

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