We consider the problem of signal estimation in a generalized linear model (GLM). GLMs include many canonical problems in statistical estimation, such as linear regression, phase retrieval, and 1-bit compressed sensing. Recent work has precisely characterized the asymptotic minimum mean-squared error (MMSE) for GLMs with i.i.d. Gaussian sensing matrices. However, in many models there is a significant gap between the MMSE and the performance of the best known feasible estimators. In this work, we address this issue by considering GLMs defined via spatially coupled sensing matrices. We propose an efficient approximate message passing (AMP) algorithm for estimation and prove that with a simple choice of spatially coupled design, the MSE of a carefully tuned AMP estimator approaches the asymptotic MMSE in the high-dimensional limit. To prove the result, we first rigorously characterize the asymptotic performance of AMP for a GLM with a generic spatially coupled design. This characterization is in terms of a deterministic recursion (`state evolution') that depends on the parameters defining the spatial coupling. Then, using a simple spatially coupled design and a judicious choice of functions for the AMP algorithm, we analyze the fixed points of the resulting state evolution and show that it achieves the asymptotic MMSE. Numerical results for phase retrieval and rectified linear regression show that spatially coupled designs can yield substantially lower MSE than i.i.d. Gaussian designs at finite dimensions when used with AMP algorithms.

翻译：本文研究广义线性模型中的信号估计问题。广义线性模型包含统计估计中的诸多经典问题，如线性回归、相位恢复和1位压缩感知。近期研究已精确刻画了具有独立同分布高斯感知矩阵的广义线性模型的渐近最小均方误差。然而，在许多模型中，最小均方误差与当前最优可行估计器的性能之间存在显著差距。为此，本文通过空间耦合感知矩阵定义的广义线性模型来解决该问题。我们提出一种高效的近似消息传递算法进行估计，并证明：通过简单的空间耦合设计，经精心调谐的近似消息传递估计器的均方误差在高维极限下趋近于渐近最小均方误差。为证明该结论，我们首先严格刻画了具有通用空间耦合设计的广义线性模型中近似消息传递算法的渐近性能，该刻画通过取决于空间耦合参数的确定性递推方程实现。随后，采用简单的空间耦合设计和近似消息传递算法的合理函数选择，我们分析了所得状态演化方程的固定点，并证明其能达到渐近最小均方误差。相位恢复和整流线性回归的数值结果表明：在有限维度下，结合近似消息传递算法时，空间耦合设计相比独立同分布高斯设计能显著降低均方误差。