剪切 1 - Wasserstein 距离和最大剪切 1 - Wasserstein 距离的中央限制定理 (Central limit theorem for the Sliced 1-Wasserstein distance and the max-Sliced 1-Wasserstein distance)

The Wasserstein distance has been an attractive tool in many fields. But due to its high computational complexity and the phenomenon of the curse of dimensionality in empirical estimation, various extensions of the Wasserstein distance have been proposed to overcome the shortcomings such as the Sliced Wasserstein distance. It enjoys a low computational cost and dimension-free sample complexity, but there are few distributional limit results of it. In this paper, we focus on Sliced 1-Wasserstein distance and its variant max-Sliced 1-Wasserstein distance. We utilize the central limit theorem in Banach space to derive the limit distribution for the Sliced 1-Wasserstein distance. Through viewing the empirical max-Sliced 1-Wasserstein distance as a supremum of an empirical process indexed by some function class, we prove that the function class is P-Donsker under mild moment assumption. Moreover, for computing Sliced p-Wasserstein distance based on Monte Carlo method, we explore that how many random projections that can make sure the error small in high probability. We also provide upper bound of the expected max-Sliced 1-Wasserstein between the true and the empirical probability measures under different conditions and the concentration inequalities for max-Sliced 1-Wasserstein distance are also presented. As applications of the theory, we utilize them for two-sample testing problem.

翻译：瓦瑟斯坦距离在许多领域都是一个有吸引力的工具。但是,由于它计算复杂程度高,且在实证估计中存在对维度的诅咒现象,因此提出了瓦瑟斯坦距离的各种扩展,以克服Slied Vasserstein距离等缺陷。它具有低计算成本和无维度的样本复杂性,但很少有分配限制结果。在本文中,我们侧重于Sliced 1-Wasserstein 距离及其变异的峰值1-Wasserstein距离。我们利用Banach空间的中央限值理论来得出Sliced 1-Wasserstein距离的限值分布。我们通过将经验性最高限值1-Wasserstein距离作为某些功能类指数化的经验性进程的一个精华,我们证明功能类在轻度假设下是P-Donsker。此外,我们探索了多少随机的预测能确保高概率小的错误。我们还提供了Aservicle 最高标准1和标准1号标准下的不同标准级标准值的上,我们还提出了标准级标准级标准1和标准级标准标准1下标准级标准中标准标准标准标准标准中的标准标准标准标准标准标准的高级标准。

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