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arXiv · 1508.07941

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

Matthias Liero·Alexander Mielke·Giuseppe Savaré
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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.

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Matthias Liero, Alexander Mielke, Giuseppe Savaré. 2015-08-31. Optimal Entropy-Transport problems and a new Hellinger-Kantorovich distance between positive measures. https://doi.org/10.1007/s00222-017-0759-8

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