On the Minimax Optimality of Estimating the Wasserstein Metric
This work addresses a fundamental statistical problem for researchers in machine learning and statistics, providing theoretical insights into the hardness of Wasserstein metric estimation, but it is incremental as it builds on existing minimax theory.
The paper tackles the problem of estimating the Wasserstein-1 metric between two unknown probability measures from empirical samples, showing that the minimax optimal rate for this estimation is multiplicatively equivalent to that of estimating the measures themselves under the Wasserstein metric, up to a log log(n)/log(n) factor.
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 $\log \log (n)/\log (n)$ factor.