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Article

Entropy-Regularized Optimal Transport on Multivariate Normal and q-normal Distributions

Department of Mathematics, Faculty of Science and Technology, Keio University, Yokohama 223-8522, Japan
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Academic Editor: Steeve Zozor
Entropy 2021, 23(3), 302; https://doi.org/10.3390/e23030302
Received: 19 January 2021 / Revised: 19 February 2021 / Accepted: 26 February 2021 / Published: 3 March 2021
(This article belongs to the Special Issue Entropies, Divergences, Information, Identities and Inequalities)
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. View Full-Text
Keywords: optimal transportation; entropy regularization; Wasserstein distance; Tsallis entropy; q-normal distribution optimal transportation; entropy regularization; Wasserstein distance; Tsallis entropy; q-normal distribution
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MDPI and ACS Style

Tong, Q.; Kobayashi, K. Entropy-Regularized Optimal Transport on Multivariate Normal and q-normal Distributions. Entropy 2021, 23, 302. https://doi.org/10.3390/e23030302

AMA Style

Tong Q, Kobayashi K. Entropy-Regularized Optimal Transport on Multivariate Normal and q-normal Distributions. Entropy. 2021; 23(3):302. https://doi.org/10.3390/e23030302

Chicago/Turabian Style

Tong, Qijun, and Kei Kobayashi. 2021. "Entropy-Regularized Optimal Transport on Multivariate Normal and q-normal Distributions" Entropy 23, no. 3: 302. https://doi.org/10.3390/e23030302

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