Exchangeable random permutations with an application to Bayesian graph matching

TL;DR

This paper develops a Bayesian framework for graph matching using exchangeable random permutations, introducing new probabilistic models, inference algorithms, and uncertainty quantification methods for permutation-based problems.

Contribution

It introduces a novel theory of exchangeable random permutations, a sequential construction called the position-aware generalized Chinese restaurant process, and applies these to Bayesian graph matching with efficient inference and uncertainty assessment.

Findings

Developed a new class of exchangeable permutation priors.

Created a Gibbs sampler for posterior inference in graph matching.

Introduced perSALSO for interpretable posterior summaries.

Abstract

We introduce a general Bayesian framework for graph matching grounded in a new theory of exchangeable random permutations. Leveraging the cycle representation of permutations and the literature on exchangeable random partitions, we define, characterize, and study the structural and predictive properties of these probabilistic objects. A novel sequential metaphor, the position-aware generalized Chinese restaurant process, provides a constructive foundation for this theory and supports practical algorithmic design. Exchangeable random permutations offer flexible priors for a wide range of inferential problems centered on permutations. As an application, we develop a Bayesian model for graph matching that integrates a correlated stochastic block model with our novel class of priors. The cycle structure of the matching is linked to latent node partitions that explain connectivity patterns, an assumption consistent with the homogeneity requirement underlying the graph matching task itself. Posterior inference is performed through a node-wise blocked Gibbs sampler directly enabled by the proposed sequential construction. To summarize posterior uncertainty, we introduce perSALSO, an adaptation of SALSO to the permutation domain that provides principled point estimation and interpretable posterior summaries. Together, these contributions establish a unified probabilistic framework for modeling, inference, and uncertainty quantification over permutations.