I know there are concentration bounds on $W(\mu,\hat{\mu})$ where $\mu$ and $\hat{\mu}$ are true and empirical distributions respectively, but is there anything out there on $W(\mu,\nu)$ versus $W(\hat{\mu},\hat{\nu})$ where $\hat{\mu}$ and $\hat{\nu}$ are the empirical distributions corresponding to $\mu$ and $\nu$ respectively?
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$\begingroup$ Wouldn’t you expect the empirical bound to be dominated by the limit? What happens around that is just noise. $\endgroup$– Anthony QuasCommented Feb 21, 2021 at 14:53
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$\begingroup$ Right—what I’m thinking of is a concentration bound on the Wasserstein between two empiricals. $\endgroup$– KashifCommented Feb 21, 2021 at 15:15
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1$\begingroup$ You can use the triangle inequality to relate $W ( \mu, \hat{\mu}), W(\nu, \hat{\nu}), W(\mu, \nu)$, and $W( \hat{\mu}, \hat{\nu})$. This should give a bound, if maybe not an optimal one. $\endgroup$– Will SawinCommented Feb 21, 2021 at 15:17
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$\begingroup$ Yeah that’s what I had so far. Just wondered if there were any tighter ones. $\endgroup$– KashifCommented Feb 21, 2021 at 20:23
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1 Answer
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Yes, it turns out you can do better than the triangle inequality in this case. See section 3.1 of "Faster Wasserstein Distance Estimation with the Sinkhorn Divergence".
