Sparse Signal Recovery from Random Measurements

TL;DR

This paper introduces a simple, efficient method for sparse signal recovery from random measurements that avoids optimization, using logarithmic measurements and running in near-linear time, with applications to support detection.

Contribution

The paper presents a novel non-optimization-based approach for sparse signal recovery using a logarithmic number of random measurements and efficient algorithms.

Findings

Method runs in O(kn log n) time.

Uses Θ(log n) random sensing matrices.

Effective in support detection for binary signals.

Abstract

Given the compressed sensing measurements of an unknown vector $z \in \mathbb{R}^n$ using random matrices, we present a simple method to determine $z$ without solving any optimization problem or linear system. Our method uses $\Theta(\log n)$ random sensing matrices in $\mathbb{R}^{k \times n}$ and runs in $O(kn\log n)$ time, where $k = \Theta(s\log n)$ and $s$ is the number of nonzero coordinates in $z$. We adapt our method to determine the support set of $z$ and experimentally compare with some optimization-based methods on binary signals.