  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.
