论文标题
关于Kantorovich潜力的独特性
On the Uniqueness of Kantorovich Potentials
论文作者
论文摘要
Kantorovich电位表示著名最佳运输问题的双重解决方案。这些解决方案的唯一性来自理论和算法的观点，并且最近在统计和熵最佳运输的背景下成为渐近结果的必要条件。在这项工作中，我们挑战了一个共同的看法，即连续设置中的独特性依赖于至少一项涉及措施的支持的连接性，即使这两个措施都脱离了支持，我们也为独特性提供了温和的条件。由于我们的主要发现是基于坎托维奇在相互关联的组件上具有独特性的基础，因此我们重新审视相应的论点，并提供了众所周知的结果的概括。为此，我们介绍了诱发规律性的概念，并利用它扩展了Gangbo和McCann（1996）提出的Kantorovich潜力的规律性理论，以在$ \ Mathbb {R}^d $中更一般的成本函数和地球空间。
Kantorovich potentials denote the dual solutions of the renowned optimal transportation problem. Uniqueness of these solutions is relevant from both a theoretical and an algorithmic point of view, and has recently emerged as a necessary condition for asymptotic results in the context of statistical and entropic optimal transport. In this work, we challenge the common perception that uniqueness in continuous settings is reliant on the connectedness of the support of at least one of the involved measures, and we provide mild sufficient conditions for uniqueness even when both measures have disconnected support. Since our main finding builds upon the uniqueness of Kantorovich potentials on connected components, we revisit the corresponding arguments and provide generalizations of well-known results. To this end, we introduce the notion of induced regularity and employ it to extend the regularity theory of Kantorovich potentials advanced by Gangbo and McCann (1996) to more general cost functions in $\mathbb{R}^d$ and to geodesic spaces.
