Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: CRII: CIF: New Paradigms in Generalization and Information-Theoretic Analysis of Deep Neural Networks, Extreme Value Analysis for Non-Stationary Processes, EPSRC Centre for Doctoral Training in Statistics and Machine Learning, CAREER: Smooth statistical distances for a scalable learning theory, New Challenges in Statistical Inference with Regularized Optimal Transport
OpenAlex ID: W7208758929
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