We study the problem of estimating a rank one signal matrix from an observed matrix generated by corrupting the signal with additive rotationally invariant noise. We develop a new class of approximate message-passing algorithms for this problem and provide a simple and concise characterization of their dynamics in the high-dimensional limit. At each iteration, these algorithms exploit prior knowledge about the noise structure by applying a non-linear matrix denoiser to the eigenvalues of the observed matrix and prior information regarding the signal structure by applying a non-linear iterate denoiser to the previous iterates generated by the algorithm. We exploit our result on the dynamics of these algorithms to derive the optimal choices for the matrix and iterate denoisers. We show that the resulting algorithm achieves the smallest possible asymptotic estimation error among a broad class of iterative algorithms under a fixed iteration budget.

翻译：本文研究从受加性旋转不变噪声干扰的观测矩阵中估计秩为一信号矩阵的问题。我们针对该问题提出了一类新的近似消息传递算法，并在高维极限下给出了其动力学行为的简洁刻画。在每次迭代中，这些算法通过以下两种方式利用先验信息：对观测矩阵特征值施加非线性矩阵去噪器以利用噪声结构先验，对算法前序迭代结果施加非线性迭代去噪器以利用信号结构先验。基于对算法动力学行为的分析结果，我们推导出矩阵去噪器与迭代去噪器的最优选择方案。研究表明，在固定迭代次数约束下，该算法能在广泛迭代算法类中实现最小的渐近估计误差。