On the Minimax Optimality of Estimating the Wasserstein Metric

We study the minimax optimal rate for estimating the Wasserstein-$1$ metric between two unknown probability measures based on $n$ i.i.d. empirical samples from them. We show that estimating the Wasserstein metric itself between probability measures, is not significantly easier than estimating the probability measures under the Wasserstein metric. We prove that the minimax optimal rates for these two problems are multiplicatively equivalent, up to a $\log \log (n)\mathrm{/}\log (n)$ factor.