发表机构
Imperial College London; Cornell University（帝国理工学院; 康奈尔大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究提出适用于有限支撑度量测度空间的离散GW对偶结果，推导经验GW距离的样本复杂度与极限分布，给出带收敛保证的正则化GW算法，构建图分布同构测试的高效框架。
AI 中文摘要
格罗莫夫-瓦瑟斯坦（GW）距离提供了一种仅基于内禀结构对齐度量测度（mm）空间的原则性框架，其识别跨空间分布同构表示的能力，使其适用于比较图这类自然出现同构等价的数据，或更广泛地说，比较图上的分布。最近，针对欧氏分布间带平方欧氏距离或内积代价的GW距离，已推导出一类对偶形式，推动了该场景下新统计与算法结果的发展。本研究为带熵正则化和不带熵正则化的GW距离提供了适用于所有有限支撑mm空间的新型对偶结果。利用该结果，我们推导了有限mm空间间经验GW距离的样本复杂度，以及在适当中心化和缩放下的极限分布。此外，我们提出了求解正则化GW问题的新算法，这些算法具有形式化收敛保证。这些统计与算法进展形成了一个原则性且高效的框架，用于基于样本测试节点数固定的图集合上的两个分布是否同构。
英文摘要
The Gromov-Wasserstein (GW) distance provides a principled framework for aligning metric measure (mm) spaces based solely on their intrinsic structure. Its ability to identify isomorphic representations of distributions across spaces renders it valuable for comparing data where equality up to isomorphism occurs naturally such as in graphs or, more generally, distributions on graphs. Recently, a type of dual form for the GW distance between Euclidean distributions with the squared Euclidean or inner product costs was derived, spurring the development of new statistical and algorithmic results for this setting. This work furnishes a novel duality result for GW distances with and without entropic regularization that is applicable to all finitely supported mm spaces. Leveraging this result, we derive the sample complexity of empirical GW distances between finite mm spaces, as well as limit distributions under proper centering and scaling. Furthermore, we propose new algorithms for solving the regularized GW problem which are subject to formal convergence guarantees. These statistical and algorithmic advancements give rise to a principled and efficient framework for testing whether two distributions on the set of graphs with a fixed number of nodes are isomorphic based on samples.
Comments72 pages, 6 figures, 1 table
