PLUGIN ESTIMATION OF SMOOTH OPTIMAL TRANSPORT MAPS

• 成果类型: Article
• 署名作者: Manole, Tudor; Balakrishnan, Sivaraman; Niles-Weed, Jonathan; Wasserman, Larry
• 署名单位: Carnegie Mellon University; New York University; New York University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/24-AOS2379
• 发表日期: 2024
• 页码: 966-998
• 关键词: CENTRAL LIMIT-THEOREMS; monge-ampere equation; Wasserstein Distance; probability density; boundary-regularity; empirical measures; Minimax Estimation; CONVERGENCE; rates; time

摘要

We analyze a number of natural estimators for the optimal transport map between two distributions and show that they are minimax optimal. We adopt the plugin approach: our estimators are simply optimal couplings between measures derived from our observations, appropriately extended so that they define functions on Rd. d . When the underlying map is assumed to be Lipschitz, we show that computing the optimal coupling between the empirical measures, and extending it using linear smoothers, already gives a minimax optimal estimator. When the underlying map enjoys higher regularity, we show that the optimal coupling between appropriate nonparametric density estimates yields faster rates. Our work also provides new bounds on the risk of corresponding plugin estimators for the quadratic Wasserstein distance, and we show how this problem relates to that of estimating optimal transport maps using stability arguments for smooth and strongly convex Brenier potentials. As an application of our results, we derive central limit theorems for plugin estimators of the squared Wasserstein distance, which are centered at their population counterpart when the underlying distributions have sufficiently smooth densities. In contrast to known central limit theorems for empirical estimators, this result easily lends itself to statistical inference for the quadratic Wasserstein distance.