Monte Carlo (MC) approximation has been used as the standard computation approach for the Sliced Wasserstein (SW) distance, which has an intractable expectation in its analytical form. However, the MC method is not optimal in terms of minimizing the 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 verify various ways of constructing QMC points sets on the 3D unit-hypersphere, including Gaussian-based mapping, equal area mapping, generalized spiral points, and optimizing discrepancy energies. Furthermore, to obtain an unbiased estimation for stochastic optimization, we extend QSW into Randomized Quasi-Sliced Wasserstein (RQSW) by introducing randomness to the discussed low-discrepancy sequences. For theoretical properties, 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）逼近已成为切片沃瑟斯坦（SW）距离的标准计算方法，该距离在其解析形式中具有难以求解的期望值。然而，MC方法在最小化绝对逼近误差方面并非最优。为了提供更优的经验SW类别，我们提出依赖拟蒙特卡洛（QMC）方法的拟切片沃瑟斯坦（QSW）逼近。为系统研究QMC在SW中的应用，我们聚焦于三维场景，具体计算三维空间中概率测度之间的SW距离。详细而言，我们通过实验验证了在三维单位超球面上构建QMC点集的各种方法，包括高斯映射、等面积映射、广义螺旋点以及优化不一致性能量。此外，为在随机优化中获得无偏估计，我们通过向所讨论的低差异序列引入随机性，将QSW扩展为随机化拟切片沃瑟斯坦（RQSW）。在理论性质方面，我们证明了QSW的渐近收敛性以及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QSW与RQSW变体的优越性能。