Unbalanced optimal transport: Dynamic and Kantorovich formulations
Unbalanced optimal transport: Dynamic and Kantorovich formulations
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DOI：
10.1016/j.jfa.2018.03.008
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发表时间：
2018-06-01
影响因子：
1.7
通讯作者：
Vialard, Francois-Xavier
中科院分区：
文献类型：
--
作者：
Chizat, Lenaic;Peyre, Gabriel;Vialard, Francois-Xavier
本文提出了一类受最优输运启发的任意非负氡测度之间的新距离。这些距离由两个等效的替代公式定义：(i)一个动态公式将距离定义为测量空间上的测地线距离（ii）一个静态的“Kantorovich”公式，其中距离是描述两个测量之间的传递（传输，创造和破坏）的耦合对上的优化问题的最小值。这两种公式都是凸优化问题，根据目标应用程序从一种公式切换到另一种公式的能力是我们模型的关键属性。特别有趣的是最近由[7,15]独立引入的Wasserstein Fisher Rao度量。最初通过动态公式定义，它属于这类度量，因此自动受益于静态Kantorovich公式。(C) 2018爱思唯尔公司版权所有。
This article presents a new class of distances between arbitrary nonnegative Radon measures inspired by optimal transport. These distances are defined by two equivalent alternative formulations: (i) a dynamic formulation defining the distance as a geodesic distance over the space of measures (ii) a static "Kantorovich" formulation where the distance is the minimum of an optimization problem over pairs of couplings describing the transfer (transport, creation and destruction) of mass between two measures. Both formulations are convex optimization problems, and the ability to switch from one to the other depending on the targeted application is a crucial property of our models. Of particular interest is the Wasserstein Fisher Rao metric recently introduced independently by [7,15]. Defined initially through a dynamic formulation, it belongs to this class of metrics and hence automatically benefits from a static Kantorovich formulation. (C) 2018 Elsevier Inc. All rights reserved.