论文标题
在多元正常和Q-正常分布上熵登记的最佳转运
Entropy-regularized optimal transport on multivariate normal and q-normal distributions
论文作者
论文摘要
概率度量的距离和差异在统计，机器学习和许多其他相关领域中起着核心作用。韦斯堡（Wasserstein）的距离近年来因其与其他距离或差异的区别而受到了很多关注。尽管〜计算Wasserstein的距离是昂贵的，但提出了熵调节的最佳运输，以在计算上有效地近似Wasserstein距离。这项研究的目的是了解熵登记的最佳运输的理论方面。在本文中，我们将重点放在多元正常分布和$ Q $ - 纳式分布上的熵登记的最佳传输上。我们获得了多元正常和$ q $ - 正常分布的熵调查最佳运输成本的明确形式；这提供了了解熵正则化的效果的观点，此前仅在实验中知道。此外，我们获得了满足某些条件的概率度量的熵调查的kantorovich估计器。我们还展示了在某些实验中熵正则化的熵效率，最佳耦合，几何结构和统计效率如何受到熵的影响。尤其是，我们关于多变量$ Q $ - 正常分布和熵调节化的Kantorovich估计值的TSALLIS熵调查最佳运输最佳耦合的明确形式的结果是新颖的，并且将成为对更一般环境理解的第一步。
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 $q$-normal distributions. We~obtain the explicit form of the entropy-regularized optimal transport cost on multivariate normal and $q$-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 $q$-normal distributions and the entropy-regularized Kantorovich estimator are novel and will become the first step towards the understanding of a more general setting.
