论文标题
Gromov-Wasserstein和Gromov-Monge距离的比较结果
Comparison Results for Gromov-Wasserstein and Gromov-Monge Distances
论文作者
论文摘要
Gromov-Wasserstein（GW）的概率度量之间的最佳运输距离的启发，Gromov-Wasserstein（GW）距离构成了一个指标的度量家族，这是指标度量空间的同构型范围。在以前的工作中，作者引入了这种结构的一种变体，该变体灵感来自原始的Monge最佳运输公式。所产生的家族的元素被称为Gromov-Monge（GM）距离。从理论和面向应用的角度来看，这些转基因距离及相关思想已成为感兴趣的主题。在本说明中，我们建立了GM距离的几种理论特性，重点是GM和GW距离之间的比较。特别是，我们表明，对于非原子度量测量空间，GM和GW距离相等。我们还考虑了GM距离的变体，例如Sturm的$ L_P $ -Transportion距离的Monge版本，并与GW距离进行了精确的比较。最后，我们在GM距离和等轴测不变的Monge最佳传输距离之间建立了Bi-Hölder等效性，在形状和图像分析应用中已使用的欧几里得度量度量空间之间。
Inspired by the Kantorovich formulation of optimal transport distance between probability measures on a metric space, Gromov-Wasserstein (GW) distances comprise a family of metrics on the space of isomorphism classes of metric measure spaces. In previous work, the authors introduced a variant of this construction which was inspired by the original Monge formulation of optimal transport; elements of the resulting family are referred to Gromov-Monge (GM) distances. These GM distances, and related ideas, have since become a subject of interest from both theoretical and applications-oriented perspectives. In this note, we establish several theoretical properties of GM distances, focusing on comparisons between GM and GW distances. In particular, we show that GM and GW distances are equal for non-atomic metric measure spaces. We also consider variants of GM distance, such as a Monge version of Sturm's $L_p$-transportion distance, and give precise comparisons to GW distance. Finally, we establish bi-Hölder equivalence between GM distance and an isometry-invariant Monge optimal transport distance between Euclidean metric measure spaces that has been utilized in shape and image analysis applications.
