Tong Qijun, Kobayashi Kei
Department of Mathematics, Faculty of Science and Technology, Keio University, Yokohama 223-8522, Japan.
Entropy (Basel). 2021 Mar 3;23(3):302. doi: 10.3390/e23030302.
The distance and divergence of the probability measures play a central role in statistics, machine learning, and many other related fields. The Wasserstein distance has received much attention in recent years because of its distinctions from other distances or divergences. Although computing the Wasserstein distance is costly, entropy-regularized optimal transport was proposed to computationally efficiently approximate the Wasserstein distance. The purpose of this study is to understand the theoretical aspect of entropy-regularized optimal transport. In this paper, we focus on entropy-regularized optimal transport on multivariate normal distributions and -normal distributions. We obtain the explicit form of the entropy-regularized optimal transport cost on multivariate normal and -normal distributions; this provides a perspective to understand the effect of entropy regularization, which was previously known only experimentally. Furthermore, we obtain the entropy-regularized Kantorovich estimator for the probability measure that satisfies certain conditions. We also demonstrate how the Wasserstein distance, optimal coupling, geometric structure, and statistical efficiency are affected by entropy regularization in some experiments. In particular, our results about the explicit form of the optimal coupling of the Tsallis entropy-regularized optimal transport on multivariate -normal distributions and the entropy-regularized Kantorovich estimator are novel and will become the first step towards the understanding of a more general setting.
概率测度的距离和散度在统计学、机器学习以及许多其他相关领域中起着核心作用。近年来，瓦瑟斯坦距离因其与其他距离或散度的差异而备受关注。尽管计算瓦瑟斯坦距离成本高昂，但人们提出了熵正则化最优传输来在计算上有效地近似瓦瑟斯坦距离。本研究的目的是理解熵正则化最优传输的理论方面。在本文中，我们专注于多元正态分布和 - 正态分布上的熵正则化最优传输。我们得到了多元正态分布和 - 正态分布上熵正则化最优传输成本的显式形式；这提供了一个视角来理解熵正则化的效果，而此前仅通过实验得知。此外，我们得到了满足某些条件的概率测度的熵正则化康托罗维奇估计量。我们还在一些实验中展示了熵正则化如何影响瓦瑟斯坦距离、最优耦合、几何结构和统计效率。特别是，我们关于多元 - 正态分布上 Tsallis 熵正则化最优传输的最优耦合的显式形式以及熵正则化康托罗维奇估计量的结果是新颖的，将成为迈向理解更一般情形的第一步。
