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For a problem I've been working on, I'm thinking about couplings between true and empirical distrubutions.

I have two datasets $S$ and $T$ with underlying measures $\mu_S,\,\mu_T$. And then I have some coupling on those, say $\pi\in \Pi(\mu_S,\mu_T)$ with $\mu_S$ and $\mu_T$ as marginals.

Now if I come up with empirical distributions $\hat{\mu}_S$ and $\hat{\mu}_T$ based off of sample draws from $S$, $T$, is there anything I can say about the couplings $\hat{\pi} \in \Pi(\hat{\mu}_S,\hat{\mu}_T)$?

Or at least any literature out there that discusses this?

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  • $\begingroup$ I don't think there is an easy answer to this, such matters are fairly delicate $\endgroup$
    – alesia
    Aug 17, 2022 at 2:13

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If the support of your distributions are compact, or if the distributions decay fast enough, then empirical distributions will converge to the distributions. Rate estimates are available in the literature.

Then comes the question of convergence of optimal couplings. This is more delicate, but some results exist. You can look at Delalande's and Merigot "Quantitative stability of optimal transport maps under variations of the target measure" and references therein.

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If $\pi$ is known and you want $\hat\pi$ to be close to $\pi$, then you can define $\hat\pi$ as a minimizer of a distance $d(\hat\pi,\pi)$ from $\hat\pi$ to $\pi$ over all $\hat\pi\in\Pi(\hat{\mu_S},\hat{\mu_T})$. For $d$ you can use e.g. a Wasserstein distance.

On the other hand, if $\pi$ is known (and hence $\mu_S$ and $\mu_T$ are known), why would you need empirical measures at all? Empirical measures are used to estimate their true, but unknown counterparts.

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  • $\begingroup$ My bad, I meant to say $\pi$ is the Wasserstein coupling—i.e. $W(S,T)=\int c(x_s,x_t)\, d\pi(x_s,x_t)$ where $c$ is some “cost” function. $\endgroup$
    – Kashif
    Mar 5, 2019 at 17:09
  • $\begingroup$ "$\pi$ is the Wasserstein coupling"... between what and what? If between the measures $\mu_S$ and $\mu_T$, and you know these measures, why would you then need $\hat\mu_S$ and $\hat\mu_T$? $\endgroup$ Mar 5, 2019 at 17:17
  • $\begingroup$ It’s for a research problem I have where I’m trying to understand empirical losses. $\pi$ is the Wasserstein coupling between $\mu_S$ and $\mu_T$. We don’t necessarily know the those measures but we can use concentration bounds between empirical and true losses. $\endgroup$
    – Kashif
    Mar 5, 2019 at 18:05

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