Monte Carlo (MC) integration has been employed as the standard approximation method for the Sliced Wasserstein (SW) distance, whose analytical expression involves an intractable expectation. However, MC integration is not optimal in terms of absolute approximation error. To provide a better class of empirical SW, we propose quasi-sliced Wasserstein (QSW) approximations that rely on Quasi-Monte Carlo (QMC) methods. For a comprehensive investigation of QMC for SW, we focus on the 3D setting, specifically computing the SW between probability measures in three dimensions. In greater detail, we empirically evaluate various methods to construct QMC point sets on the 3D unit-hypersphere, including the Gaussian-based and equal area mappings, generalized spiral points, and optimizing discrepancy energies. Furthermore, to obtain an unbiased estimator for stochastic optimization, we extend QSW to Randomized Quasi-Sliced Wasserstein (RQSW) by introducing randomness in the discussed point sets. Theoretically, we prove the asymptotic convergence of QSW and the unbiasedness of RQSW. Finally, we conduct experiments on various 3D tasks, such as point-cloud comparison, point-cloud interpolation, image style transfer, and training deep point-cloud autoencoders, to demonstrate the favorable performance of the proposed QSW and RQSW variants.

翻译：蒙特卡洛（MC）积分是切片Wasserstein（SW）距离的标准近似方法，其解析表达式涉及难以计算的期望。然而，MC积分在绝对逼近误差方面并非最优。为提供更优的经验SW类别，我们提出基于准蒙特卡洛（QMC）方法的准切片Wasserstein（QSW）近似。为全面研究QMC在SW中的应用，我们聚焦三维场景，具体计算三维空间中概率测度之间的SW距离。详言之，我们通过实验评估了在三维单位超球面上构造QMC点集的多种方法，包括高斯映射、等面积映射、广义螺旋点以及基于差异能量的优化方法。此外，为获得适用于随机优化的无偏估计量，我们通过引入讨论点集的随机性，将QSW扩展为随机化准切片Wasserstein（RQSW）。理论上，我们证明了QSW的渐近收敛性与RQSW的无偏性。最后，我们在点云比较、点云插值、图像风格迁移及深度点云自编码器训练等多种三维任务上进行实验，验证了所提出的QSW及RQSW变体的优越性能。