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

2015/08/31 by Matthias Liero, Alexander Mielke, Giuseppe Savaré · 170 citations
Mathematics · #Class (philosophy) #Entropy (arrow of time) #Finite set #Geometric Analysis and Curvature Flows #Large deviations theory #Nonlinear Partial Differential Equations #Point processes and geometric inequalities #Probability measure #Radon measure #Regular polygon #math.OC

paper · pdf · open access · doi:10.1007/s00222-017-0759-8

published in Inventiones mathematicae 211(3), 969-1117 (Springer Science+Business Media) · Revision includes slight modifications in the presentation and corrections of typos

arxiv created 2016/01/08 · openalex created_date 2016/06/24 · openalex publication_date 2017/12/14 · arxiv updated 2018/10/16 · openalex updated_date 2026/08/05

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.

Citations

• On Information and Sufficiency
• Ricci curvature for metric-measure spaces via optimal transport
• On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces
• Divergence measures based on the Shannon entropy

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