June 2022 Minimax estimation of smooth densities in Wasserstein distance
Jonathan Niles-Weed, Quentin Berthet
Author Affiliations +
Ann. Statist. 50(3): 1519-1540 (June 2022). DOI: 10.1214/21-AOS2161


We study nonparametric density estimation problems where error is measured in the Wasserstein distance, a metric on probability distributions popular in many areas of statistics and machine learning. We give the first minimax-optimal rates for this problem for general Wasserstein distances for two classes of densities: smooth probability densities on [0,1]d bounded away from 0, and sub-Gaussian densities lying in the Hölder class Cs, s(0,1). Unlike classical nonparametric density estimation, these rates depend on whether the densities in question are bounded below, even in the compactly supported case. Motivated by variational problems involving the Wasserstein distance, we also show how to construct discretely supported measures, suitable for computational purposes, which achieve the minimax rates. Our main technical tool is an inequality giving a nearly tight dual characterization of the Wasserstein distances in terms of Besov norms.

Funding Statement

Part of this research was conducted while at the Institute for Advanced Study. JNW gratefully acknowledges its support.
Most of this work was conducted while QB was at University of Cambridge and was supported in part by The Alan Turing Institute under the EPSRC Grant EP/N510129/1.


Download Citation

Jonathan Niles-Weed. Quentin Berthet. "Minimax estimation of smooth densities in Wasserstein distance." Ann. Statist. 50 (3) 1519 - 1540, June 2022. https://doi.org/10.1214/21-AOS2161


Received: 1 May 2020; Revised: 1 December 2021; Published: June 2022
First available in Project Euclid: 16 June 2022

MathSciNet: MR4441130
zbMATH: 07547940
Digital Object Identifier: 10.1214/21-AOS2161

Primary: 62F99
Secondary: 62H99

Keywords: Density estimation , High-dimensional statistics , Optimal transport , Wasserstein distance

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.50 • No. 3 • June 2022
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