Recovery of Binary Sparse Signals from Structured Biased Measurements

TL;DR

This paper demonstrates that binary sparse signals can be accurately reconstructed from partial random circulant measurements using least-squares, matching basis pursuit performance, with stability under noise.

Contribution

It shows least-squares reconstruction is as effective as basis pursuit for binary sparse signals from structured measurements, and establishes measurement bounds and noise stability.

Findings

Least-squares performs as well as basis pursuit for binary sparse signals

Number of measurements needed is comparable to dense signal recovery

Reconstruction is stable under noisy measurements

Abstract

In this paper we study the reconstruction of binary sparse signals from partial random circulant measurements. We show that the reconstruction via the least-squares algorithm is as good as the reconstruction via the usually used program basis pursuit. We further show that we need as many measurements to recover an $s$-sparse signal $x_0\in\mathbb{R}^N$ as we need to recover a dense signal, more-precisely an $N-s$-sparse signal $x_0\in\mathbb{R}^N$. We further establish stability with respect to noisy measurements.