# On the Minimax Optimality of Estimating the Wasserstein Metric

arXiv: https://arxiv.org/abs/1908.10324 · HTML: https://mlsanity.com/paper/1908.10324/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Aug 2019.

## Built on by (most cited first, 2 of 2)

- [Plugin Estimation of Smooth Optimal Transport Maps](https://mlsanity.com/paper/2107.12364/) (2021)
- [Estimation and inference for the Wasserstein distance between mixing measures in topic models](https://mlsanity.com/paper/2206.12768/) (2022)

## Abstract (arXiv)

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.