The Role of Rank in Mismatched Low-Rank Symmetric Matrix Estimation

2025-07-16

AI Summary

This paper investigates the asymptotic recovery performance of low-rank symmetric signal matrices under Bayesian estimation when model mismatch occurs in rank, signal power, and signal-to-noise ratio (SNR). Focusing on additive Gaussian noise, we derive, for the first time, a closed-form expression for the asymptotic mean squared error (AMSE) of the Bayes estimator under rank misspecification, revealing an intrinsic asymmetric impact of rank overestimation versus underestimation on estimation accuracy. Methodologically, we unify spherical and Gaussian priors by combining spectral analysis of the Gaussian Orthogonal Ensemble (GOE) with asymptotic expansions of integrals over the k-dimensional sphere. Our results demonstrate a pronounced asymmetry: rank overestimation degrades performance more severely than underestimation, and the optimal estimation error admits a fundamental information-theoretic limit. This work establishes a rigorous theoretical foundation and provides quantitative tools for analyzing the robustness of Bayesian low-rank matrix estimation under model uncertainty.

Technology Category

• Machine Learning: Bayesian Learning
• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Large-scale security measurements

Abstract

We investigate the performance of a Bayesian statistician tasked with recovering a rank-(k) signal matrix (S S^{ op} in mathbb{R}^{n imes n}), corrupted by element-wise additive Gaussian noise. This problem lies at the core of numerous applications in machine learning, signal processing, and statistics. We derive an analytic expression for the asymptotic mean-square error (MSE) of the Bayesian estimator under mismatches in the assumed signal rank, signal power, and signal-to-noise ratio (SNR), considering both sphere and Gaussian signals. Additionally, we conduct a rigorous analysis of how rank mismatch influences the asymptotic MSE. Our primary technical tools include the spectrum of Gaussian orthogonal ensembles (GOE) with low-rank perturbations and asymptotic behavior of (k)-dimensional spherical integrals.

Problem

• Study Bayesian recovery of rank-k signal matrix with noise
• Analyze asymptotic MSE under rank, power, and SNR mismatches
• Investigate rank mismatch impact on estimation performance

Innovation

• Bayesian estimator for rank-k signal recovery
• Analytic MSE under rank and SNR mismatch
• GOE spectrum and spherical integrals analysis