This paper addresses the reconstruction of sparse signals from generalized linear measurements. Signal sparsity is assumed to be sublinear in the signal dimension while it was proportional to the signal dimension in conventional research. Approximate message-passing (AMP) has poor convergence properties for sensing matrices beyond standard Gaussian matrices. To solve this convergence issue, generalized orthogonal AMP (GOAMP) is proposed for signals with sublinear sparsity. The main feature of GOAMP is the so-called Onsager correction to realize asymptotic Gaussianity of estimation errors. The Onsager correction in GOAMP is designed via state evolution for orthogonally invariant sensing matrices in the sublinear sparsity limit, where the signal sparsity and measurement dimension tend to infinity at sublinear speed in the signal dimension. When the support of non-zero signals does not contain a neighborhood of the origin, GOAMP using Bayesian denoisers is proved to achieve error-free signal reconstruction for linear measurements if and only if the measurement dimension is larger than a threshold, which is equal to that of AMP for standard Gaussian sensing matrices. Numerical simulations are also presented for linear measurements and 1-bit compressed sensing. When ill-conditioned sensing matrices are used, GOAMP for sublinear sparsity is shown to outperform existing reconstruction algorithms, including generalized AMP for sublinear sparsity.

翻译：本文研究广义线性测量中的稀疏信号重构问题。与传统研究中信号稀疏度与信号维度成正比不同，本文假设稀疏度相对于信号维度呈次线性。针对标准高斯矩阵之外的传感矩阵，近似消息传递（AMP）存在收敛性较差的问题。为解决这一收敛问题，本文提出适用于次线性稀疏信号的广义正交近似消息传递（GOAMP）方法。GOAMP的主要特征在于引入所谓的Onsager校正项，以实现估计误差的渐近高斯性。在次线性稀疏极限下（即信号稀疏度与测量维度以信号维度的次线性速度趋于无穷时），针对正交不变传感矩阵，通过状态演化设计了GOAMP中的Onsager校正项。当非零信号支撑集不包含原点邻域时，对于线性测量，采用贝叶斯去噪器的GOAMP能够实现无误信号重构，其充要条件为测量维度大于某个阈值——该阈值与标准高斯传感矩阵下AMP的阈值相等。本文还针对线性测量和1比特压缩感知进行了数值仿真实验。结果表明，当使用病态传感矩阵时，本文提出的次线性稀疏GOAMP优于现有重构算法，包括适用于次线性稀疏性的广义AMP。