Optimal Entropy-Transport problems and a new Hellinger-Kantorovich distance between positive measures

TL;DR

This paper introduces a comprehensive theory for Optimal Entropy-Transport problems, relaxing traditional constraints and leading to a new Hellinger-Kantorovich distance that bridges existing measure distances.

Contribution

It develops a full theoretical framework for entropy-transport problems and introduces the Hellinger-Kantorovich distance, connecting entropy methods with optimal transport geometry.

Findings

Established a general theory for entropy-transport problems

Defined the Hellinger-Kantorovich distance between measures

Analyzed geometric properties of the new distance

Abstract

We develop a full theory for the new class of Optimal Entropy-Transport problems between nonnegative and finite Radon measures in general topological spaces. They arise quite naturally by relaxing the marginal constraints typical of Optimal Transport problems: given a couple of finite measures (with possibly different total mass), one looks for minimizers of the sum of a linear transport functional and two convex entropy functionals, that quantify in some way the deviation of the marginals of the transport plan from the assigned measures. As a powerful application of this theory, we study the particular case of Logarithmic Entropy-Transport problems and introduce the new Hellinger-Kantorovich distance between measures in metric spaces. The striking connection between these two seemingly far topics allows for a deep analysis of the geometric properties of the new geodesic distance, which lies somehow between the well-known Hellinger-Kakutani and Kantorovich-Wasserstein distances.