Regularity theory and geometry of unbalanced optimal transport

IF 1.6 2区 数学 Q1 MATHEMATICS
Thomas Gallouët , Roberta Ghezzi , François-Xavier Vialard
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引用次数: 0

Abstract

Using the dual formulation only, we show that the regularity of unbalanced optimal transport, also called entropy-transport, inherits from the regularity of standard optimal transport. We provide detailed examples of Riemannian manifolds and costs for which unbalanced optimal transport is regular. Among all entropy-transport formulations, the Wasserstein-Fisher-Rao (WFR) metric, also called Hellinger-Kantorovich, stands out since it admits a dynamical formulation, which extends the Benamou-Brenier formulation of optimal transport. After demonstrating the equivalence between dynamical and static formulations on a closed Riemannian manifold, we prove a polar factorization theorem, similar to the one due to Brenier and Mc-Cann. As a byproduct, we formulate the Monge-Ampère equation associated with the WFR metric, which also holds for more general costs. Last, we study the link between c-convex functions for the cost induced by the WFR metric and the cost on the cone. The main result is that the weak Ma-Trudinger-Wang condition on the cone implies the same condition on the manifold for the cost induced by the WFR metric.
不平衡最优运输的正则性理论与几何
仅使用对偶公式,我们就证明了不平衡最优输运的规律性,也称为熵输运,继承自标准最优输运的规律性。我们提供了详细的黎曼流形和成本的例子,其中不平衡最优运输是规则的。在所有熵输运公式中,Wasserstein-Fisher-Rao (WFR)度量,也称为Hellinger-Kantorovich,因为它承认一个动态公式,它扩展了最优输运的Benamou-Brenier公式,而脱颖而出。在证明了封闭黎曼流形上动态和静态公式之间的等价性之后,我们证明了一个极分解定理,类似于Brenier和Mc-Cann的定理。作为副产品,我们制定了与WFR指标相关的monge - ampantere方程,该方程也适用于更一般的成本。最后,我们研究了由WFR度量引起的代价与锥上代价的c-凸函数之间的联系。主要结果是,锥上的弱Ma-Trudinger-Wang条件意味着由WFR度量引起的代价在流形上的相同条件。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
3.20
自引率
5.90%
发文量
271
审稿时长
7.5 months
期刊介绍: The Journal of Functional Analysis presents original research papers in all scientific disciplines in which modern functional analysis plays a basic role. Articles by scientists in a variety of interdisciplinary areas are published. Research Areas Include: • Significant applications of functional analysis, including those to other areas of mathematics • New developments in functional analysis • Contributions to important problems in and challenges to functional analysis
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