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 judicious choice of functions defining the AMP, 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：我们研究了广义线性模型（GLM）中的信号估计问题。GLM涵盖了统计估计中的许多经典问题，例如线性回归、相位恢复和1比特压缩感知。近期研究精确刻画了独立同分布高斯感知矩阵下GLM的渐近最小均方误差（MMSE）。然而，在许多模型中，MMSE与已知最佳可行估计器的性能之间存在显著差距。本文通过考虑由空间耦合感知矩阵定义的GLM来解决此问题。我们提出了一种高效的近似消息传递（AMP）估计算法，并证明，通过简单选择空间耦合设计，精心调谐的AMP估计器在高维极限下的均方误差可逼近渐近MMSE。为证明该结果，我们首先严格刻画了具有一般空间耦合设计的GLM中AMP的渐近性能。该刻画基于一个确定性递归（“状态演化”），该递归依赖于定义空间耦合的参数。随后，利用简单的空间耦合设计和AMP定义函数的审慎选择，我们分析了所得状态演化的不动点，并证明其达到了渐近MMSE。相位恢复和修正线性回归的数值结果表明，当与AMP算法结合使用时，空间耦合设计在有限维度下能够获得比独立同分布高斯设计显著更低的MSE。