Discrete Gromov-Wasserstein Duality: Algorithms and Isomorphism Testing

2026-09-02

AI Summary

本文解决了图分布同构性测试问题，通过提出新的Gromov-Wasserstein距离对偶形式及相应算法，并分析了样本复杂度和极限分布。

Abstract

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.

Problem

• Gromov-Wasserstein
• isomorphism testing
• metric measure spaces

Innovation

• Gromov-Wasserstein Distance
• Duality Result
• Isomorphism Testing
• Entropic Regularization
• Algorithmic Convergence