Sliced-Wasserstein Estimation with Spherical Harmonics as Control Variates

TL;DR

This paper introduces a novel Monte Carlo method called Spherical Harmonics Control Variates (SHCV) for estimating the Sliced-Wasserstein distance, leveraging spherical harmonics to improve accuracy and convergence.

Contribution

The paper proposes a new Monte Carlo approach using spherical harmonics as control variates for SW distance estimation, with proven theoretical advantages and enhanced convergence rates.

Findings

SHCV achieves better accuracy than existing methods.

The approach has a no-error property for Gaussian measures.

Numerical experiments confirm superior performance.

Abstract

The Sliced-Wasserstein (SW) distance between probability measures is defined as the average of the Wasserstein distances resulting for the associated one-dimensional projections. As a consequence, the SW distance can be written as an integral with respect to the uniform measure on the sphere and the Monte Carlo framework can be employed for calculating the SW distance. Spherical harmonics are polynomials on the sphere that form an orthonormal basis of the set of square-integrable functions on the sphere. Putting these two facts together, a new Monte Carlo method, hereby referred to as Spherical Harmonics Control Variates (SHCV), is proposed for approximating the SW distance using spherical harmonics as control variates. The resulting approach is shown to have good theoretical properties, e.g., a no-error property for Gaussian measures under a certain form of linear dependency between the variables. Moreover, an improved rate of convergence, compared to Monte Carlo, is established for general measures. The convergence analysis relies on the Lipschitz property associated to the SW integrand. Several numerical experiments demonstrate the superior performance of SHCV against state-of-the-art methods for SW distance computation.