In this paper, we consider the problem of computing the integral of a function on the unit sphere, in any dimension, using Monte Carlo methods. Although the methods we present are general, our guiding thread is the sliced Wasserstein distance between two measures on $\mathbb{R}^d$, which is precisely an integral on the $d$-dimensional sphere. The sliced Wasserstein distance (SW) has gained momentum in machine learning either as a proxy to the less computationally tractable Wasserstein distance, or as a distance in its own right, due in particular to its built-in alleviation of the curse of dimensionality. There has been recent numerical benchmarks of quadratures for the sliced Wasserstein, and our viewpoint differs in that we concentrate on quadratures where the nodes are repulsive, i.e. negatively dependent. Indeed, negative dependence can bring variance reduction when the quadrature is adapted to the integration task. Our first contribution is to extract and motivate quadratures from the recent literature on determinantal point processes (DPPs) and repelled point processes, as well as repulsive quadratures from the literature specific to the sliced Wasserstein distance. We then numerically benchmark these quadratures. Moreover, we analyze the variance of the UnifOrtho estimator, an orthogonal Monte Carlo estimator. Our analysis sheds light on UnifOrtho's success for the estimation of the sliced Wasserstein in large dimensions, as well as counterexamples from the literature. Our final recommendation for the computation of the sliced Wasserstein distance is to use randomized quasi-Monte Carlo in low dimensions and UnifOrtho in large dimensions. DPP-based quadratures only shine when quasi-Monte Carlo also does, while repelled quadratures show moderate variance reduction in general, but more theoretical effort is needed to make them robust.

翻译：本文研究利用蒙特卡洛方法计算任意维度单位球面上函数积分的问题。尽管所提方法具有普适性，但我们的核心线索是$\mathbb{R}^d$上两个度量间的切片Wasserstein距离——该距离本质上正是$d$维球面上的积分。切片Wasserstein距离（SW）在机器学习领域日益受到重视：它既可作为计算难度更高的Wasserstein距离的替代度量，亦可作为独立距离度量，特别是因其天然具备缓解维度灾难的特性。近期已有针对切片Wasserstein数值积分法的基准测试，而本文的视角不同——我们专注于节点具有排斥性（即负相关）的积分方法。事实上，当积分方案与具体积分任务适配时，负相关性能够带来方差缩减效应。我们的首要贡献是从近期关于行列式点过程（DPPs）与排斥点过程的文献中提取并论证积分方案，同时从切片Wasserstein距离的专门文献中梳理排斥型积分方法。随后我们对这些积分方案进行数值基准测试。此外，我们分析了UnifOrtho估计器（一种正交蒙特卡洛估计器）的方差特性。该分析不仅揭示了UnifOrtho在高维切片Wasserstein估计中取得成功的原因，也解释了文献中某些反例的成因。对于切片Wasserstein距离的计算，我们的最终建议是：低维情形采用随机拟蒙特卡洛方法，高维情形采用UnifOrtho方法。基于DPP的积分方案仅在拟蒙特卡洛同样有效的场景中表现突出，而排斥型积分方案总体上仅能实现有限的方差缩减，其稳健性仍需更多理论研究的支撑。