Robust random graph matching in Gaussian models via vector approximate message passing

TL;DR

This paper introduces a robust polynomial-time algorithm for matching correlated Gaussian Wigner matrices that withstands significant adversarial perturbations, advancing the reliability of graph matching in noisy and adversarial environments.

Contribution

The paper presents the first efficient algorithm for robust random graph matching under large adversarial perturbations, combining vector AMP with spectral cleaning techniques.

Findings

Algorithm succeeds with constant correlation and small perturbation size

Robustness against adversarial perturbations of size up to nearly linear in n

Extends previous graph matching methods to adversarial settings

Abstract

In this paper, we focus on the matching recovery problem between a pair of correlated Gaussian Wigner matrices with a latent vertex correspondence. We are particularly interested in a robust version of this problem such that our observation is a perturbed input $(A+E,B+F)$ where $(A,B)$ is a pair of correlated Gaussian Wigner matrices and $E,F$ are adversarially chosen matrices supported on an unknown $\epsilon n * \epsilon n$ principle minor of $A,B$, respectively. We propose a vector approximate message passing (vector AMP) algorithm that succeeds in polynomial time as long as the correlation $\rho$ between $(A,B)$ is a non-vanishing constant and $\epsilon = o\big( \tfrac{1}{(\log n)^{20}} \big)$. The main methodological inputs for our result are the iterative random graph matching algorithm proposed in \cite{DL22+, DL23+} and the spectral cleaning procedure proposed in \cite{IS24+}. To the best of our knowledge, our algorithm is the first efficient random graph matching type algorithm that is robust under any adversarial perturbations of $n^{1-o(1)}$ size.