Abstract
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 . 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.
Funding Statement
TM was supported in part by the Natural Sciences and Engineering Research Council of Canada, through a PGS D scholarship.
TM, SB and LW were supported in part by National Science Foundation grants DMS-1713003, DMS-2113684 and DMS-2310632.
SB was additionally supported by a Google Research Scholar Award and an Amazon Research Award.
JNW gratefully acknowledges the support of National Science Foundation grant DMS-2015291.
Acknowledgments
The authors would like to thank Alden Green for bringing their attention to the paper Hendriks (1990), and Ziv Goldfeld and Kengo Kato for a discussion related to the results of Section 5.2. The authors are grateful for the constructive comments of the Editor, Associate Editor and four anonymous reviewers, which significantly improved the quality of this manuscript. TM also wishes to thank Aram-Alexandre Pooladian for conversations related to this work, and for his comments on an earlier version of this manuscript.
Citation
Tudor Manole. Sivaraman Balakrishnan. Jonathan Niles-Weed. Larry Wasserman. "Plugin estimation of smooth optimal transport maps." Ann. Statist. 52 (3) 966 - 998, June 2024. https://doi.org/10.1214/24-AOS2379
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