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

Gabriel Rioux, Ziv Goldfeld, Kengo Kato

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.

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

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.

Paper Structure

This paper contains 52 sections, 46 theorems, 231 equations, 11 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Literature Review
  4. Background and Preliminary Results
  5. Notation
  6. Optimal Transport
  7. Entropic Optimal Transport
  8. Gromov-Wasserstein Distances
  9. Discrete Gromov-Wasserstein Distance
  10. Stability analysis
  11. Distributional limits
  12. Direct estimation of the limit
  13. Semi-Discrete Gromov-Wasserstein Distance
  14. Existence of Gromov-Monge maps
  15. Stability and limit distributions
  16. ...and 37 more sections

Key Result

Theorem 1

Let $(\mu_0,\mu_1)\in\mathcal{P}(\mathbb R^{d_0})\times \mathcal{P}(\mathbb R^{d_1})$ be compactly supported. The following statements hold:

Figures (11)

  • Figure 1: Distributions from \ref{['sec:testingBinary']}. $\nu_0$ is generated by sampling $(p_{0,ij})_{\substack{i,j\in[N]\\i<j}}$ from the uniform distribution on $[0,1]$. $\nu_1$ is obtained from $\nu_0$ by a randomly chosen permutation. The modified probability for $\nu_2$ is highlighted in orange. Edge probabilities are presented with two significant figures for ease of reading.
  • Figure 2: These plots compile the type 1 and type 2 error of the proposed test for varying numbers of samples and desired significance levels by following the methodology described in \ref{['sec:testingBinary']}.
  • Figure 3: Distributions from \ref{['sec:testingWeighted']}. First, edge weight probabilities for $\nu_0$ were sampled from the Dirichlet distribution, then a random permutation was chosen to generate $\nu_1$ from $\nu_0$ as depicted. Edge weight probabilities are given as vectors whose $k$-th entry represents the probability (truncated to two significant figures) that the edge takes the weight $k-1$.
  • Figure 4: These plots compile the type 1 and type 2 error of the proposed test for varying numbers of samples and desired significance levels by following the methodology described in \ref{['sec:testingWeighted']}.
  • Figure 5: Distributions from \ref{['sec:compactDists']}. Here $\mathrm B_{a,b}$ denotes the beta distribution with parameters $(a,b)$ whereas $\mathcal{U}$ is the uniform distribution on $[0,1]$.
  • ...and 6 more figures

Theorems & Definitions (92)

  • Theorem 1: On minimizers of \ref{['eq:Objective']}
  • Remark 1: Uniqueness of cross-correlation matrix
  • Theorem 2: Stability for discrete GW
  • Remark 2: Comparison with OT stability
  • Theorem 3: Limit distributions for discrete GW
  • Remark 3: Extension of limits
  • Remark 4: Removing the square
  • Remark 5: Estimating the limit
  • Corollary 1: Limit distribution under the null
  • Theorem 4
  • ...and 82 more