Sparse Signal Recovery from Random Measurements

TL;DR

This work addresses recovering a sparse signal $z\in\mathbb{R}^n$ from random linear measurements without optimization. It introduces a median-based recovery method that uses $O(\log n)$ Gaussian sensing matrices with $k = \Theta(s\log n)$ measurements, achieving accurate reconstruction in noise with time complexity $O(kn\log n)$ and high-probability guarantees. The authors extend the approach to support set estimation and benchmark against standard optimization-based methods on binary signals, showing competitive accuracy and substantial speedups, especially for larger sparsities. The method offers a scalable, non-optimization-driven alternative for compressed sensing with practical implications for fast sparse recovery in large-scale problems.

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 $Θ(\log n)$ random sensing matrices in $\mathbb{R}^{k \times n}$ and runs in $O(kn\log n)$ time, where $k = Θ(s\log n)$ 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.

Figures (6)

• Figure 1: Performance for $n=2000$.
• Figure 2: Performance for $n=4000$.
• Figure 3: Performance for $n=8000$.
• Figure 5: Performance for $n=2000$.
• Figure 6: Performance for $n=4000$.

Theorems & Definitions (12)

• Lemma 1
• Theorem 2
• proof
• Lemma 3
• Lemma 4
• Theorem 5
• proof
• Theorem 6
• proof
• proof