Motivated by the growing popularity of variants of the Wasserstein distance in statistics and machine learning, we study statistical inference for the Sliced Wasserstein distance—an easily computable variant of the Wasserstein distance. Specifically, we construct confidence intervals for the Sliced Wasserstein distance which have finite-sample validity under no assumptions or under mild moment assumptions. These intervals are adaptive in length to the regularity of the underlying distributions. We also bound the minimax risk of estimating the Sliced Wasserstein distance, and as a consequence establish that the lengths of our proposed confidence intervals are minimax optimal over appropriate distribution classes. To motivate the choice of these classes, we also study minimax rates of estimating a distribution under the Sliced Wasserstein distance. These theoretical findings are complemented with a simulation study demonstrating the deficiencies of the classical bootstrap, and the advantages of our proposed methods. We also show strong correspondences between our theoretical predictions and the adaptivity of our confidence interval lengths in simulations. We conclude by demonstrating the use of our confidence intervals in the setting of simulator-based likelihood-free inference. In this setting, contrasting popular approximate Bayesian computation methods, we develop uncertainty quantification methods with rigorous frequentist coverage guarantees.
This work was supported in part by the National Science Foundation grants DMS-1713003, DMS-2113684 and CIF-1763734. TM acknowledges the support of the Natural Sciences and Engineering Research Council of Canada (NSERC), through a PGS D scholarship. SB was supported by a Google Research Scholar Award and an Amazon Research Award.
The authors would like to thank an anonymous referee for thoughtful comments which significantly helped improve the content and presentation of this paper. Tudor Manole would like to thank Niccolò Dalmasso for conversations which inspired the likelihood-free inference application in Section 7, and Arun Kumar Kuchibhotla for suggesting references on concentration inequalities for self-normalized empirical processes.
"Minimax confidence intervals for the Sliced Wasserstein distance." Electron. J. Statist. 16 (1) 2252 - 2345, 2022. https://doi.org/10.1214/22-EJS2001