A signal recovery algorithm for sparse matrix based compressed sensing

TL;DR

This paper introduces a low-cost approximate signal recovery algorithm for compressed sensing using sparse measurement matrices, capable of achieving near-optimal recovery thresholds with limited matrix density.

Contribution

The paper presents a novel approximate recovery algorithm that performs well with sparse matrices and low computational cost, extending recovery capabilities beyond dense matrices.

Findings

Algorithm saturates Donoho-Tanner threshold for dense matrices

Effective with sparse matrices with limited non-zero entries

Achieves perfect recovery with O(N) computational cost

Abstract

We have developed an approximate signal recovery algorithm with low computational cost for compressed sensing on the basis of randomly constructed sparse measurement matrices. The law of large numbers and the central limit theorem suggest that the developed algorithm saturates the Donoho-Tanner weak threshold for the perfect recovery when the matrix becomes as dense as the signal size $N$ and the number of measurements $M$ tends to infinity keep $\alpha=M/N \sim O(1)$, which is supported by extensive numerical experiments. Even when the numbers of non-zero entries per column/row in the measurement matrices are limited to $O(1)$, numerical experiments indicate that the algorithm can still typically recover the original signal perfectly with an $O(N)$ computational cost per update as well if the density $\rho$ of non-zero entries of the signal is lower than a certain critical value $\rho_{\rm th}(\alpha)$ as $N,M \to \infty$.