We study the matrix denoising problem of estimating the singular vectors of a rank-$1$ signal corrupted by noise with both column and row correlations. Existing works are either unable to pinpoint the exact asymptotic estimation error or, when they do so, the resulting approaches (e.g., based on whitening or singular value shrinkage) remain vastly suboptimal. On top of this, most of the literature has focused on the special case of estimating the left singular vector of the signal when the noise only possesses row correlation (one-sided heteroscedasticity). In contrast, our work establishes the information-theoretic and algorithmic limits of matrix denoising with doubly heteroscedastic noise. We characterize the exact asymptotic minimum mean square error, and design a novel spectral estimator with rigorous optimality guarantees: under a technical condition, it attains positive correlation with the signals whenever information-theoretically possible and, for one-sided heteroscedasticity, it also achieves the Bayes-optimal error. Numerical experiments demonstrate the significant advantage of our theoretically principled method with the state of the art. The proofs draw connections with statistical physics and approximate message passing, departing drastically from standard random matrix theory techniques.

翻译：我们研究了在同时存在列相关和行相关噪声的情况下，估计秩-1信号奇异向量的矩阵去噪问题。现有研究要么无法精确确定渐近估计误差，要么在能够确定时，所提出的方法（例如基于白化或奇异值收缩的方法）仍然远非最优。此外，大多数文献都集中于噪声仅具有行相关性（单侧异方差）时估计信号左奇异向量的特殊情况。相比之下，我们的工作确立了具有双重异方差噪声的矩阵去噪问题的信息论与算法极限。我们刻画了精确的渐近最小均方误差，并设计了一种具有严格最优性保证的新型谱估计器：在满足一个技术条件时，只要信息论上可能，该估计器就能与信号达到正相关；对于单侧异方差情况，它还能达到贝叶斯最优误差。数值实验证明了我们这种具有理论原则的方法相较于现有技术的显著优势。证明过程与统计物理学和近似消息传递建立了联系，与标准的随机矩阵理论技术截然不同。