Compressed Sensing of Approximately-Sparse Signals: Phase Transitions and Optimal Reconstruction

TL;DR

This paper analyzes the phase transitions and optimal reconstruction methods for approximately sparse signals in compressed sensing, using replica calculations and spatially-coupled matrices to improve performance.

Contribution

It models approximately sparse signals with Gaussian small components and demonstrates how spatially-coupled matrices can achieve optimal reconstruction.

Findings

Replica calculations determine mean-squared error for Bayes-optimal reconstruction.

G-AMP algorithm's optimality region is characterized.

Spatially-coupled measurement matrices restore optimality where homogeneous matrices fail.

Abstract

Compressed sensing is designed to measure sparse signals directly in a compressed form. However, most signals of interest are only "approximately sparse", i.e. even though the signal contains only a small fraction of relevant (large) components the other components are not strictly equal to zero, but are only close to zero. In this paper we model the approximately sparse signal with a Gaussian distribution of small components, and we study its compressed sensing with dense random matrices. We use replica calculations to determine the mean-squared error of the Bayes-optimal reconstruction for such signals, as a function of the variance of the small components, the density of large components and the measurement rate. We then use the G-AMP algorithm and we quantify the region of parameters for which this algorithm achieves optimality (for large systems). Finally, we show that in the region where the GAMP for the homogeneous measurement matrices is not optimal, a special "seeding" design of a spatially-coupled measurement matrix allows to restore optimality.