Estimation of Wasserstein Distances
This is a note on the estimation of Wasserstein distances.
The Wasserstein Law of Large Numbers
Suppose that $X_1,\dots,X_n\overset{i.i.d}{\sim}\mu$, where $\mu$ is a probability measure on a compact subset of $\RR^d$, which we can assume to be the unit cube $[0,1]^d$ for convenience. The law of large numbers implies that the empirical measure \(\mu_n=\frac{1}{n}\sum_{i=1}^n\delta_{X_i},\) converges
If the support of $\mu$ lies in $[0,1]^d$, then \(\EE W_1(\mu_n,\mu)\lessim \sqrt{d}\)