The Role of Rank in Mismatched Low-Rank Symmetric Matrix Estimation
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arXiv
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| Main Authors: | , , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915393081180160 |
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| author | Niu, Panpan Liu, Yuhao Fu, Teng Fan, Jie Deng, Chaowen Huang, Zhongyi |
| author_facet | Niu, Panpan Liu, Yuhao Fu, Teng Fan, Jie Deng, Chaowen Huang, Zhongyi |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_12019 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Role of Rank in Mismatched Low-Rank Symmetric Matrix Estimation Niu, Panpan Liu, Yuhao Fu, Teng Fan, Jie Deng, Chaowen Huang, Zhongyi Information Theory Signal Processing 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. |
| title | The Role of Rank in Mismatched Low-Rank Symmetric Matrix Estimation |
| topic | Information Theory Signal Processing |
| url | https://arxiv.org/abs/2507.12019 |