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arXiv · 2507.12019

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

Panpan Niu·Yuhao Liu·Teng Fu·Jie Fan·Chaowen Deng·Zhongyi Huang
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Abstract

We investigate the performance of a Bayesian statistician tasked with recovering a rank-\(k\) signal matrix \(\bS \bS^{\top} \in \mathbb{R}^{n \times n}\), 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.

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Panpan Niu, Yuhao Liu, Teng Fu, Jie Fan, Chaowen Deng, Zhongyi Huang. 2025-07-16. The Role of Rank in Mismatched Low-Rank Symmetric Matrix Estimation. https://arxiv.org/abs/2507.12019

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