The Gromov-Wasserstein (GW) distances define a family of metrics, based on ideas from optimal transport, which enable comparisons between probability measures defined on distinct metric spaces. They are particularly useful in areas such as network analysis and geometry processing, as computation of a GW distance involves solving for registration between the objects which minimizes geometric distortion. Although GW distances have proven useful for various applications in the recent machine learning literature, it has been observed that they are inherently sensitive to outlier noise and cannot accommodate partial matching. This has been addressed by various constructions building on the GW framework; in this article, we focus specifically on a natural relaxation of the GW optimization problem, introduced by Chapel et al., which is aimed at addressing exactly these shortcomings. Our goal is to understand the theoretical properties of this relaxed optimization problem, from the viewpoint of metric geometry. While the relaxed problem fails to induce a metric, we derive precise characterizations of how it fails the axioms of non-degeneracy and triangle inequality. These observations lead us to define a novel family of distances, whose construction is inspired by the Prokhorov and Ky Fan distances, as well as by the recent work of Raghvendra et al.\ on robust versions of classical Wasserstein distance. We show that our new distances define true metrics, that they induce the same topology as the GW distances, and that they enjoy additional robustness to perturbations. These results provide a mathematically rigorous basis for using our robust partial GW distances in applications where outliers and partial matching are concerns.

翻译：Gromov-Wasserstein (GW) 距离是基于最优传输思想定义的一族度量，可用于比较定义在不同度量空间上的概率测度。它们在网络分析和几何处理等领域尤为有用，因为计算GW距离需要求解使几何畸变最小化的对象间配准问题。尽管GW距离在近期机器学习文献中已被证明适用于多种应用，但人们发现其本质上对异常值噪声敏感且无法处理部分匹配问题。基于GW框架的各种构造已针对此问题提出改进；本文特别关注由Chapel等人引入的GW优化问题的自然松弛形式，该形式旨在直接解决这些缺陷。我们的目标是从度量几何的角度理解这一松弛优化问题的理论性质。虽然该松弛问题未能构成度量，但我们精确刻画了其违反非退化性与三角不等式公理的方式。这些观察引导我们定义了一族新的距离，其构造灵感来源于Prokhorov距离、Ky Fan距离，以及Raghvendra等人近期关于经典Wasserstein距离鲁棒版本的研究。我们证明新距离构成真正的度量，其诱导的拓扑与GW距离相同，且对扰动具有更强的鲁棒性。这些结果为在存在异常值和部分匹配问题的应用中使用我们提出的鲁棒部分GW距离提供了严格的数学基础。