SlicedWasserstein Estimation with Spherical Harmonics as Control Variates
Abstract
The SlicedWasserstein (SW) distance between probability measures is defined as the average of the Wasserstein distances resulting for the associated onedimensional 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 squareintegrable 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 noerror 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 stateoftheart methods for SW distance computation.
 Publication:

arXiv eprints
 Pub Date:
 February 2024
 DOI:
 10.48550/arXiv.2402.01493
 arXiv:
 arXiv:2402.01493
 Bibcode:
 2024arXiv240201493L
 Keywords:

 Statistics  Machine Learning;
 Computer Science  Machine Learning;
 65C05 (Primary) 65D30;
 68Txx;
 68Wxx (Secondary)
 EPrint:
 Accepted to ICML 2024