Exchangeable random permutations with an application to Bayesian graph matching

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.

💡 Research Summary

This paper introduces a comprehensive Bayesian framework for graph matching built upon a novel theory of exchangeable random permutations (ERPs). By exploiting the cycle representation of permutations, the authors show that exchangeability—defined as invariance of the distribution under relabeling of elements—implies that the cycle structure of a random permutation forms an exchangeable partition. Consequently, the law of the cycles completely determines the permutation distribution, and ERPs constitute the least informative priors respecting the required symmetry.
To provide a constructive prior, the authors develop the Position‑Aware Generalized Chinese Restaurant Process (PA‑gCRP), a sequential predictive scheme that extends the classic Chinese Restaurant Process. PA‑gCRP not only decides which existing cycle a new element joins but also its exact position within that cycle, thereby yielding a Kolmogorov‑consistent family of permutation laws for growing sample sizes. This sequential construction enables tractable posterior computation and natural extension beyond the observed data.
The Bayesian graph‑matching model couples PA‑gCRP with a correlated stochastic block model (CSBM). Two graphs sharing the same vertex set are linked by a latent permutation π and a common block partition of vertices. The ERP prior forces the cycle structure of π to align with the block partition, ensuring that nodes are matched only within the same stochastic block—a formal expression of the homogeneity assumption intrinsic to graph matching.
Posterior inference is performed via a node‑wise blocked Gibbs sampler. At each iteration, given current block assignments, the permutation is updated according to the PA‑gCRP predictive rule; then block parameters are sampled conditioned on the observed adjacency matrices and the current permutation. Because the ERP prior restricts the permutation space to a subset consistent with the block structure, the Gibbs sampler explores a dramatically reduced state space, leading to efficient mixing.
To summarize posterior uncertainty, the paper proposes perSALSO, an adaptation of the SALSO algorithm to the permutation domain. perSALSO searches for a single representative permutation that maximizes the posterior expected loss while respecting the full posterior distribution over cycles and blocks. This yields point estimates that are genuine permutations and provides interpretable measures of uncertainty (e.g., marginal matching probabilities for individual node pairs).
Empirical studies demonstrate that the proposed method outperforms traditional optimization‑based graph‑matching algorithms in both accuracy and uncertainty quantification. Moreover, the authors discuss extensions to record linkage, shuffled regression, and other problems where the labeling of objects is irrelevant, highlighting the broad applicability of ERPs as flexible non‑parametric priors over permutations.
In summary, the contributions are: (1) a full probabilistic characterization of exchangeable random permutations via cycles; (2) the PA‑gCRP sequential predictive construction; (3) a Bayesian hierarchical model that jointly learns the matching permutation and latent block structure; (4) an efficient blocked Gibbs sampler; and (5) perSALSO for principled posterior summarization. These advances provide a unified, theoretically grounded, and computationally feasible approach to inference and uncertainty quantification over permutations.