Limit Laws for Gromov-Wasserstein Alignment with Applications to Testing Graph Isomorphisms

TL;DR

This paper develops the first limit laws for the empirical Gromov-Wasserstein distance, enabling statistical inference and hypothesis testing for graph isomorphism problems based on metric measure space comparisons.

Contribution

It derives the first limit laws for the GW distance in various settings, providing a foundation for statistical testing of graph isomorphisms.

Findings

Established limit laws for empirical GW distance in discrete, semi-discrete, and general settings.

Proposed an efficient estimation procedure for the limiting distribution in the discrete case.

Applied the theory to test graph isomorphism from unlabelled graph collections.

Abstract

The Gromov-Wasserstein (GW) distance enables comparing metric measure spaces based solely on their internal structure, making it invariant to isomorphic transformations. This property is particularly useful for comparing datasets that naturally admit isomorphic representations, such as unlabelled graphs or objects embedded in space. However, apart from the recently derived empirical convergence rates for the quadratic GW problem, a statistical theory for valid estimation and inference remains largely obscure. Pushing the frontier of statistical GW further, this work derives the first limit laws for the empirical GW distance across several settings of interest: (i)~discrete, (ii)~semi-discrete, and (iii)~general distributions under moment constraints under the entropically regularized GW distance. The derivations rely on a novel stability analysis of the GW functional in the marginal distributions. The limit laws then follow by an adaptation of the functional delta method. As asymptotic normality fails to hold in most cases, we establish the consistency of an efficient estimation procedure for the limiting law in the discrete case, bypassing the need for computationally intensive resampling methods. We apply these findings to testing whether collections of unlabelled graphs are generated from distributions that are isomorphic to each other.