Minimax Confidence Intervals for the Sliced Wasserstein Distance

TL;DR

This paper develops finite-sample valid confidence intervals for the Sliced Wasserstein distance, establishing their minimax optimality, and demonstrates their practical utility in likelihood-free inference with rigorous coverage guarantees.

Contribution

It introduces adaptive, minimax optimal confidence intervals for the Sliced Wasserstein distance with theoretical guarantees and practical applications in likelihood-free inference.

Findings

Confidence intervals have finite-sample validity without strong assumptions.

Intervals are minimax optimal over certain distribution classes.

Simulation studies show advantages over classical bootstrap methods.

Abstract

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.