  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.
