Fast Approximation of the Sliced-Wasserstein Distance Using Concentration of Random Projections

TL;DR

This paper introduces a deterministic approximation method for the Sliced-Wasserstein distance that leverages measure concentration, reducing computational complexity and providing guarantees as data dimension grows.

Contribution

The authors propose a novel deterministic approximation for SW based on measure concentration, eliminating the need for sampling random projections.

Findings

The approximation error diminishes as data dimension increases.

The method is both accurate and computationally efficient.

Theoretical guarantees are provided for the approximation's convergence.

Abstract

The Sliced-Wasserstein distance (SW) is being increasingly used in machine learning applications as an alternative to the Wasserstein distance and offers significant computational and statistical benefits. Since it is defined as an expectation over random projections, SW is commonly approximated by Monte Carlo. We adopt a new perspective to approximate SW by making use of the concentration of measure phenomenon: under mild assumptions, one-dimensional projections of a high-dimensional random vector are approximately Gaussian. Based on this observation, we develop a simple deterministic approximation for SW. Our method does not require sampling a number of random projections, and is therefore both accurate and easy to use compared to the usual Monte Carlo approximation. We derive nonasymptotical guarantees for our approach, and show that the approximation error goes to zero as the dimension increases, under a weak dependence condition on the data distribution. We validate our theoretical findings on synthetic datasets, and illustrate the proposed approximation on a generative modeling problem.