Generalized Orthogonal Approximate Message-Passing for Sublinear Sparsity

TL;DR

This paper introduces GOAMP, a novel algorithm for reconstructing sublinearly sparse signals from generalized linear measurements, addressing convergence issues of AMP with non-Gaussian matrices.

Contribution

The paper proposes GOAMP with Onsager correction for sublinear sparsity, extending AMP's applicability and improving reconstruction performance with ill-conditioned matrices.

Findings

GOAMP achieves error-free reconstruction when measurement dimension exceeds a threshold.

GOAMP outperforms existing algorithms like generalized AMP in ill-conditioned scenarios.

Numerical simulations validate GOAMP's effectiveness for linear and 1-bit compressed sensing.

Abstract

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.