Optimal Entropy-Transport problems and a new Hellinger-Kantorovich distance between positive measures
Optimal Entropy-Transport problems and a new Hellinger-Kantorovich distance between positive measures
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DOI：
10.1007/s00222-017-0759-8
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发表时间：
2018-03-01
影响因子：
3.1
通讯作者：
Savare, Giuseppe
中科院分区：
文献类型：
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作者：
Liero, Matthias;Mielke, Alexander;Savare, Giuseppe
我们为在一般拓扑空间中的非负ra和有限ra之间的新一类最佳熵传输问题开发了一个完整的理论。这些问题很自然地通过放松最佳运输问题的边际约束：给定两对有限的措施（可能具有不同的总质量），一个人寻找最小化的线性传输功能和两个凸熵功能的总和，这些功能量化了在某种程度上，运输计划的边缘偏离了指定的措施。作为该理论的强大应用，我们研究了对数熵传输问题的特殊情况，并引入了公制空间中的措施之间的新的Hellinger-Kantorovich距离。这两个看似远的主题之间的惊人联系使得对新的大地距离的几何特性进行了深入的分析，这在某种程度上是著名的Hellinger-Kakutani和Kantorovich-Wasserstein距离之间的。
We develop a full theory for the new class of Optimal Entropy-Transport problems between nonnegative and finite Radon measures in general topological spaces. These problems arise quite naturally by relaxing the marginal constraints typical of Optimal Transport problems: given a pair 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, which 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.