April 2021 Minimax estimation of smooth optimal transport maps
Jan-Christian Hütter, Philippe Rigollet
Author Affiliations +
Ann. Statist. 49(2): 1166-1194 (April 2021). DOI: 10.1214/20-AOS1997


Brenier’s theorem is a cornerstone of optimal transport that guarantees the existence of an optimal transport map T between two probability distributions P and Q over Rd under certain regularity conditions. The main goal of this work is to establish the minimax estimation rates for such a transport map from data sampled from P and Q under additional smoothness assumptions on T. To achieve this goal, we develop an estimator based on the minimization of an empirical version of the semidual optimal transport problem, restricted to truncated wavelet expansions. This estimator is shown to achieve near minimax optimality using new stability arguments for the semidual and a complementary minimax lower bound. Furthermore, we provide numerical experiments on synthetic data supporting our theoretical findings and highlighting the practical benefits of smoothness regularization. These are the first minimax estimation rates for transport maps in general dimension.


Download Citation

Jan-Christian Hütter. Philippe Rigollet. "Minimax estimation of smooth optimal transport maps." Ann. Statist. 49 (2) 1166 - 1194, April 2021. https://doi.org/10.1214/20-AOS1997


Received: 1 May 2019; Revised: 1 June 2020; Published: April 2021
First available in Project Euclid: 2 April 2021

Digital Object Identifier: 10.1214/20-AOS1997

Primary: 62G05

Keywords: Minimax rates , nonparametric estimation , Optimal transport , wavelet estimator

Rights: Copyright © 2021 Institute of Mathematical Statistics


This article is only available to subscribers.
It is not available for individual sale.

Vol.49 • No. 2 • April 2021
Back to Top