Let $(X,d)$ be a complete separable metric space, and let $(\mathcal{P}_2 (X), W_2)$ be the space of probability measures on $X$ with finite second moments, equipped with the 2-Wasserstein distance. It is known that discrete measures are dense inside $(\mathcal{P}_2 (X), W_2)$ - namely, given any $\mu \in \mathcal{P}_2 (X)$, and $\delta>0$, one can find a discrete measure $\mu_\delta$ with $W_2 (\mu, \mu_\delta)<\delta$.

Now, let $\mu_0, \mu_1 \in \mathcal{P}_2 (X)$, and let $\mu_t$ be a $W_2$ geodesic connecting $\mu_0$ and $\mu_1$ (a.k.a. $\mu_t$ is a [not necessarily unique] displacement interpolation between $\mu_0$ and $\mu_1$). Is the displacement interpolation stable under discrete approximation? That is, can one pick discrete $\mu_{0,n}, \mu_{1,n}$ such that $\mu_{t,n}$ is close to $\mu_t$ for all $t\in[0,1]$?


1 Answer 1


The displacement interpolation $\mu_t$ should not be fixed a priori, due to nonuniqueness of Wasserstein Geodesics. Thus, the correct question should be: fix the approximating sequences $(\mu_{0,n}),(\mu_{1,n})$ and $W_2$ geodesics $\mu_{t,n}$, and ask if there exists one $\mu_t$ close to $\mu_{t,n}$ for $t \in [0,1]$.

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    $\begingroup$ This is a comment rather than an answer, but I could not post it as a comment. Anyway, something useful in this direction can be found in Lemma 4.4 arxiv.org/pdf/1609.00782.pdf which, combined with Proposition 4.8 of arxiv.org/pdf/1311.4907.pdf gives you $W_2$ close $\mu_{t,n}$'s. $\endgroup$ Commented Aug 19, 2020 at 8:10
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    $\begingroup$ Certainly there are non-uniqueness issues. I actually meant something like: given a Wasserstein geodesic $\mu_t$, can we produce sequences $(\mu_{0,n})$ and $(\mu_{1,n})$ such that $(\mu_{t,n})$ converges to $\mu_t$ in some suitable sense. $\endgroup$ Commented Aug 19, 2020 at 8:35
  • $\begingroup$ These links do look quite helpful, thank you! $\endgroup$ Commented Aug 19, 2020 at 8:36
  • $\begingroup$ I think is very hard to fix $\mu_t$ and produce, afterwards, approximating marginals $(\mu_{0,n}),(\mu_{1,n})$. If you stick to the other way, as in my answer, you can try to argue essentially by tightness to get $\mu_t$ from $\mu_{t,n}$. Finally, you just need to show that the limit $\mu_t$ is a $W_2$ geodesic. The first link I gave you follows this path. With this approach, is very hard to control whether you are converging to your fixed a priori Wasserstein geodesic, or to another. $\endgroup$ Commented Aug 19, 2020 at 8:43
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    $\begingroup$ There is not only the problem of uniqueness, but also of existence. Under the present assumptions, $\mu_{0,n}$ and $\mu_{1,n}$ may not be connected by a $W_2$-geodesic. $\endgroup$
    – MaoWao
    Commented Aug 19, 2020 at 9:21

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