Distributional Convergence of the Sliced Wasserstein Process

TL;DR

This paper introduces the Sliced Wasserstein Process, a stochastic process capturing the empirical Wasserstein distances across all one-dimensional projections, and proves a uniform distributional limit theorem for it, unifying various projection-based Wasserstein distances.

Contribution

It defines the Sliced Wasserstein Process and establishes a unified distributional limit theorem, enabling analysis of all projection-based Wasserstein distances.

Findings

Proves a uniform distributional limit theorem for the Sliced Wasserstein Process.

Provides a unified framework for distributional limits of projection-based Wasserstein distances.

Illustrates results on examples with previously unknown distributional limits.

Abstract

Motivated by the statistical and computational challenges of computing Wasserstein distances in high-dimensional contexts, machine learning researchers have defined modified Wasserstein distances based on computing distances between one-dimensional projections of the measures. Different choices of how to aggregate these projected distances (averaging, random sampling, maximizing) give rise to different distances, requiring different statistical analyses. We define the \emph{Sliced Wasserstein Process}, a stochastic process defined by the empirical Wasserstein distance between projections of empirical probability measures to all one-dimensional subspaces, and prove a uniform distributional limit theorem for this process. As a result, we obtain a unified framework in which to prove distributional limit results for all Wasserstein distances based on one-dimensional projections. We illustrate these results on a number of examples where no distributional limits were previously known.