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

Abstract

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.

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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

Information

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

Subjects:
Primary: 62G05

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.49 • No. 2 • April 2021
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