Compressed sensing with sparse, structured matrices

TL;DR

This paper introduces a new class of sparse, structured matrices for compressed sensing that enable linear-time acquisition and recovery of sparse signals, approaching theoretical limits with efficient algorithms.

Contribution

It proposes a novel ensemble of sparse matrices for compressed sensing that achieves near-optimal recovery with linear computational complexity.

Findings

Achieves perfect recovery of sparse signals at near-theoretical limits.

Runs in linear time for large N, making it highly efficient.

Transforms dense matrix ensembles into sparse ones with similar thresholds.

Abstract

In the context of the compressed sensing problem, we propose a new ensemble of sparse random matrices which allow one (i) to acquire and compress a {\rho}0-sparse signal of length N in a time linear in N and (ii) to perfectly recover the original signal, compressed at a rate {\alpha}, by using a message passing algorithm (Expectation Maximization Belief Propagation) that runs in a time linear in N. In the large N limit, the scheme proposed here closely approaches the theoretical bound {\rho}0 = {\alpha}, and so it is both optimal and efficient (linear time complexity). More generally, we show that several ensembles of dense random matrices can be converted into ensembles of sparse random matrices, having the same thresholds, but much lower computational complexity.