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-瓦瑟斯坦（SW）距离定义为由相关一维投影产生的瓦瑟斯坦距离的平均值。因此，SW距离可以表示为关于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，即球谐函数控制变量法（SHCV），通过将球谐函数作为控制变量来近似SW距离。该方法具有优良的理论性质，例如在变量间存在某种线性依赖形式时，对高斯测度具有无误差特性。此外，对于一般测度，本文证明了该方法相比蒙特卡洛方法具有更优的收敛速度。收敛性分析依赖于SW被积函数的Lipschitz性质。多项数值实验表明，SHCV在SW距离计算中相较于现有最优方法具有更优越的性能。