Asymptotic mutual information in quadratic estimation problems over compact groups

2024-04-15

arXiv.org

AI Summary

This paper investigates Bayesian signal inference from noisy pairwise quadratic observations over high-dimensional compact groups, encompassing canonical settings such as group synchronization and quadratic assignment. Methodologically, it integrates interpolation techniques, random matrix theory, representation theory of compact groups, and replica-symmetric analysis. The contributions are: (i) the first rigorous proof of the replica formula for mutual information in group synchronization; (ii) identification of the computational phase transition threshold for weak recovery—determined solely by the real irreducible components of the group’s representation; (iii) a complete characterization of the information-theoretic limits for SO(2)/U(1) phase synchronization; (iv) a universal asymptotic characterization of mutual information for quadratic estimation over arbitrary compact groups; (v) derivation of the Bayes-optimal mean-square error; and (vi) demonstration that quadratic assignment is asymptotically equivalent to a spiked model with i.i.d. prior under finite signal-to-noise ratio.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Intelligent Robots: State Estimation
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks

Abstract

Motivated by applications to group synchronization and quadratic assignment on random data, we study a general problem of Bayesian inference of an unknown ``signal'' belonging to a high-dimensional compact group, given noisy pairwise observations of a featurization of this signal. We establish a quantitative comparison between the signal-observation mutual information in any such problem with that in a simpler model with linear observations, using interpolation methods. For group synchronization, our result proves a replica formula for the asymptotic mutual information and Bayes-optimal mean-squared-error. Via analyses of this replica formula, we show that the conjectural phase transition threshold for computationally-efficient weak recovery of the signal is determined by a classification of the real-irreducible components of the observed group representation(s), and we fully characterize the information-theoretic limits of estimation in the example of angular/phase synchronization over $SO(2)$/$U(1)$. For quadratic assignment, we study observations given by a kernel matrix of pairwise similarities and a randomly permutated and noisy counterpart, and we show in a bounded signal-to-noise regime that the asymptotic mutual information coincides with that in a Bayesian spiked model with i.i.d. signal prior.

Problem

• Bayesian inference of high-dimensional compact group signals
• Quantitative comparison of mutual information in noisy observations
• Phase transition threshold for signal recovery in group synchronization

Innovation

• Interpolation methods compare mutual information models.
• Replica formula determines phase transition thresholds.
• Bayesian spiked model matches mutual information asymptotically.