[Paper Review] Bayesian matching of unlabelled point sets using Procrustes and configuration models

Kim Kenobi, Ian L. Dryden|arXiv (Cornell University)|Sep 16, 2010

Morphological variations and asymmetry3 references3 citations

TL;DR

This paper proposes an improved Bayesian MCMC approach for matching unlabelled point sets using Procrustes and configuration models, enhancing convergence via strategic large jumps in the burn-in phase. It compares both models on simulated and real protein binding site data, finding that performance depends on variance: the configuration model excels with low variance, while Procrustes performs better with higher variance, with both showing similar results under Laplace approximation.

ABSTRACT

The problem of matching unlabelled point sets using Bayesian inference is considered. Two recently proposed models for the likelihood are compared, based on the Procrustes size-and-shape and the full configuration. Bayesian inference is carried out for matching point sets using Markov chain Monte Carlo simulation. An improvement to the existing Procrustes algorithm is proposed which improves convergence rates, using occasional large jumps in the burn-in period. The Procrustes and configuration methods are compared in a simulation study and using real data, where it is of interest to estimate the strengths of matches between protein binding sites. The performance of both methods is generally quite similar, and a connection between the two models is made using a Laplace approximation.

Motivation & Objective

• To improve convergence of Bayesian MCMC algorithms for matching unlabelled point sets by introducing large jumps during the burn-in phase.
• To compare the performance of Procrustes-based and configuration-based Bayesian models in matching protein binding sites.
• To evaluate the robustness and accuracy of both models under varying levels of noise (standard deviation) in the data.
• To investigate the relationship between the two models using Laplace approximation, establishing a theoretical connection.
• To provide a practical framework for identifying structural similarities in molecular configurations, particularly in protein active sites.

Proposed method

• Uses Markov chain Monte Carlo (MCMC) simulation to sample from posterior distributions over match matrices and nuisance parameters (rotation, translation).
• Applies partial Procrustes registration to align matched point pairs and define a size-and-shape distance between configurations.
• Introduces occasional large, irreversible jumps in the MCMC chain during the burn-in phase to escape local modes and improve convergence.
• Employs a match matrix Λ to represent one-to-many or one-to-one correspondences between points in two configurations X and μ, with a dummy column for unmatched points.
• Uses a likelihood model based on Procrustes size-and-shape for one approach and full configuration likelihood for the other, both within a Bayesian hierarchical framework.
• Applies Laplace approximation to link the Procrustes and configuration models, showing their asymptotic equivalence under certain conditions.

Experimental results

Research questions

• RQ1 How does the introduction of large jumps during burn-in improve convergence in MCMC-based point set matching?
• RQ2 Which model—Procrustes or configuration—yields more accurate match probability estimates under low-variance conditions?
• RQ3 How does model performance vary with increasing noise (standard deviation) in the data?
• RQ4 What is the relationship between the Procrustes and configuration models, and can they be linked via approximation methods?
• RQ5 Can the MCMC methods reliably identify true matches in real protein binding site data, and how do they compare to existing benchmarks?

Key findings

• The improved MCMC algorithm with large jumps in burn-in significantly enhances convergence rates and reduces the risk of getting trapped in local modes.
• For low variance (standard deviation ≤ d_min/5), the configuration model estimates match probabilities more reliably, especially for unmatched points, outperforming the Procrustes model.
• For higher variance (standard deviation = d_min/2), the Procrustes model provides more accurate and higher match probability estimates for correctly matched points than the configuration model.
• There exists a critical variance threshold between d_min/5 and d_min/2 where the two models swap performance dominance, depending on the match type (matched vs. unmatched).
• The Laplace approximation establishes a strong theoretical connection between the Procrustes and configuration models, explaining their similar practical performance in many cases.
• Both models are effective for pairwise and extendable to multiple molecule alignment, though computational cost remains high for large datasets, suggesting use in combination with fast screening methods.