MINIMAX ESTIMATION OF SMOOTH OPTIMAL TRANSPORT MAPS

• 成果类型: Article
• 署名作者: Huetter, Jan-Christian; Rigollet, Philippe
• 署名单位: Harvard University; Massachusetts Institute of Technology (MIT); Broad Institute; Massachusetts Institute of Technology (MIT)
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/20-AOS1997
• 发表日期: 2021
• 页码: 1166-1194
• 关键词: regularized optimal transport; wasserstein deconvolution; boundary-regularity; ergodic diffusions; geodesic pca; rates; CONVERGENCE; time

摘要

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 R-d 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.