The Gromov-Wasserstein (GW) distance is an effective measure of alignment between distributions supported on distinct ambient spaces. Calculating essentially the mutual departure from isometry, it has found vast usage in domain translation and network analysis. It has long been shown to be vulnerable to contamination in the underlying measures. All efforts to introduce robustness in GW have been inspired by similar techniques in optimal transport (OT), which predominantly advocate partial mass transport or unbalancing. In contrast, the cross-domain alignment problem being fundamentally different from OT, demands specific solutions to tackle diverse applications and contamination regimes. Deriving from robust statistics, we discuss three contextually novel techniques to robustify GW and its variants. For each method, we explore metric properties and robustness guarantees along with their co-dependencies and individual relations with the GW distance. For a comprehensive view, we empirically validate their superior resilience to contamination under real machine learning tasks against state-of-the-art methods.

翻译：Gromov-Wasserstein (GW) 距离是衡量定义在不同支撑空间上的分布之间对齐关系的有效度量。它本质上计算了与等距映射的相互偏离程度，在领域转换和网络分析中得到了广泛应用。长期以来，GW 距离已被证明易受基础测度污染的影响。现有为 GW 引入鲁棒性的所有尝试都受到最优传输 (OT) 中类似技术的启发，这些技术主要倡导部分质量传输或不平衡传输。相比之下，跨域对齐问题在根本上不同于 OT，需要针对不同的应用场景和污染机制提出专门的解决方案。基于鲁棒统计学的思想，我们讨论了三种在语境上新颖的技术，用于增强 GW 距离及其变体的鲁棒性。对于每种方法，我们探讨了其度量性质、鲁棒性保证，以及它们之间的相互依赖关系及其与 GW 距离的个体关联。为了获得全面的视角，我们在真实的机器学习任务中通过实验验证了这些方法相对于现有先进技术，在抗污染方面具有更优越的鲁棒性。