Spherical Sliced-Wasserstein

TL;DR

This paper introduces a novel Sliced-Wasserstein discrepancy tailored for spherical data, extending Wasserstein-based methods from Euclidean spaces to manifolds like the sphere, with applications in machine learning tasks involving spherical representations.

Contribution

It proposes the spherical Sliced-Wasserstein, a new discrepancy measure for data on the sphere, based on closed-form Wasserstein solutions and a spherical Radon transform, enabling manifold-aware Wasserstein computations.

Findings

Efficient algorithms for spherical Sliced-Wasserstein

Applications in sampling, density estimation, and auto-encoders on the sphere

Demonstrated advantages over Euclidean-based Wasserstein methods

Abstract

Many variants of the Wasserstein distance have been introduced to reduce its original computational burden. In particular the Sliced-Wasserstein distance (SW), which leverages one-dimensional projections for which a closed-form solution of the Wasserstein distance is available, has received a lot of interest. Yet, it is restricted to data living in Euclidean spaces, while the Wasserstein distance has been studied and used recently on manifolds. We focus more specifically on the sphere, for which we define a novel SW discrepancy, which we call spherical Sliced-Wasserstein, making a first step towards defining SW discrepancies on manifolds. Our construction is notably based on closed-form solutions of the Wasserstein distance on the circle, together with a new spherical Radon transform. Along with efficient algorithms and the corresponding implementations, we illustrate its properties in several machine learning use cases where spherical representations of data are at stake: sampling on the sphere, density estimation on real earth data or hyperspherical auto-encoders.