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.

翻译：概率测度之间的切片Wasserstein（SW）距离定义为由关联的一维投影所导出的Wasserstein距离的平均值。因此，SW距离可表示为球面上均匀测度的积分，并可采用蒙特卡洛框架进行计算。球谐函数是球面上的多项式，构成球面上平方可积函数集合的标准正交基。结合这两个事实，本文提出一种新的蒙特卡洛方法，即球谐控制变量（SHCV）法，通过将球谐函数作为控制变量来近似SW距离。该方法具有良好的理论性质，例如当变量间存在特定线性依赖关系时，对高斯测度具有无误差特性。此外，与标准蒙特卡洛方法相比，该方法对一般测度建立了更优的收敛速率。该收敛性分析依赖于SW被积函数关联的Lipschitz性质。多项数值实验表明，SHCV在SW距离计算中的性能优于现有最先进方法。