The Gromov-Wasserstein (GW) distance is a powerful tool for comparing metric measure spaces which has found broad applications in data science and machine learning. Driven by the need to analyze datasets whose objects have increasingly complex structure (such as node and edge-attributed graphs), several variants of GW distance have been introduced in the recent literature. With a view toward establishing a general framework for the theory of GW-like distances, this paper considers a vast generalization of the notion of a metric measure space: for an arbitrary metric space $Z$, we define a $Z$-network to be a measure space endowed with a kernel valued in $Z$. We introduce a method for comparing $Z$-networks by defining a generalization of GW distance, which we refer to as $Z$-Gromov-Wasserstein ($Z$-GW) distance. This construction subsumes many previously known metrics and offers a unified approach to understanding their shared properties. This paper demonstrates that the $Z$-GW distance defines a metric on the space of $Z$-networks which retains desirable properties of $Z$, such as separability, completeness, and geodesicity. Many of these properties were unknown for existing variants of GW distance that fall under our framework. Our focus is on foundational theory, but our results also include computable lower bounds and approximations of the distance which will be useful for practical applications.

翻译：Gromov-Wasserstein (GW) 距离是一种用于比较度量测度空间的强大工具，在数据科学和机器学习领域已获得广泛应用。受分析对象结构日益复杂（例如节点和边属性图）的数据集需求驱动，近期文献中已引入多种 GW 距离的变体。为了建立类 GW 距离理论的通用框架，本文考虑对度量测度空间概念进行广泛推广：对于任意度量空间 $Z$，我们将 $Z$-网络定义为一个配备有取值于 $Z$ 的核的测度空间。我们通过定义 GW 距离的推广形式来引入比较 $Z$-网络的方法，称之为 $Z$-Gromov-Wasserstein ($Z$-GW) 距离。该构造涵盖了许多先前已知的度量，并为理解它们的共同属性提供了统一方法。本文证明了 $Z$-GW 距离在 $Z$-网络空间上定义了一个度量，该度量保留了 $Z$ 的理想性质，例如可分性、完备性和测地性。在我们框架下涵盖的现有 GW 距离变体中，许多这些性质此前是未知的。我们的研究侧重于基础理论，但结果也包含了该距离的可计算下界和近似方法，这些将对实际应用有所助益。