论文标题
部分可观测时空混沌系统的无模型预测
Central limit theorem for the Sliced 1-Wasserstein distance and the max-Sliced 1-Wasserstein distance
论文作者
论文摘要
在许多领域，Wasserstein距离一直是一个有吸引力的工具。但是，由于其高度计算复杂性和维度诅咒的现象在经验估计中，已经提出了瓦斯坦距离的各种延伸，以克服诸如切成薄片的瓦斯坦斯坦距离之类的缺点。它具有低计算成本和无尺寸样本复杂性的效果，但分布限制的结果很少。在本文中，我们专注于切成薄片的1-wasserstein距离及其变体最大1-wasserstein距离。我们利用Banach空间中的中心极限定理来得出切成薄片的1-wasserstein距离的极限分布。通过将经验最大的1-wasserstein距离视为某些功能类别索引的经验过程的至上，我们证明该功能类是在轻度矩假设下的p-donsker。此外，对于基于蒙特卡洛方法的计算切片的p-wasserstein距离，我们探讨了多少个随机投影可以确保误差很小。我们还提供了在不同条件下的真实和经验概率度量之间的预期最大1-wasserstein的上限，并且还提出了最大1-wasserstein距离的浓度不平等。作为理论的应用，我们将其用于两样本测试问题。
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.
