Optimal entropy-transport problems and a new Hellinger--Kantorovich distance between positive measures
- Liero, Matthias
- Mielke, Alexander
- Savaré, Giuseppe
2010 Mathematics Subject Classification
- 28A33 54E35 49Q20 49J35 49J40 49K35 46G99
- Entropy-transport problem, Hellinger-Kantorovich distance, relative entropy, optimality conditions, cone over metric space
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.
- Inv. Math., 211 (2018), pp. 969--1117 (published online on 14.12.2017), DOI 10.1007/s00222-017-0759-8 .