TY - JOUR

T1 - Entropy-regularized optimal transport on multivariate normal and q-normal distributions

AU - Tong, Qijun

AU - Kobayashi, Kei

N1 - Funding Information:
This work was supported by RIKEN AIP and JSPS KAKENHI (JP19K03642, JP19K00912).
Publisher Copyright:
© 2021 by the authors. Licensee MDPI, Basel, Switzerland.

PY - 2021/3

Y1 - 2021/3

N2 - 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.

AB - 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.

KW - Entropy regularization

KW - Optimal transportation

KW - Q-normal distribution

KW - Tsallis entropy

KW - Wasserstein distance

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U2 - 10.3390/e23030302

DO - 10.3390/e23030302

M3 - Article

AN - SCOPUS:85102687239

SN - 1099-4300

VL - 23

SP - 1

EP - 20

JO - Entropy

JF - Entropy

IS - 3

M1 - 302

ER -