Analysis of Hard-Thresholding for Distributed Compressed Sensing with One-Bit Measurements

TL;DR

This paper demonstrates that a simple hard-thresholding method can effectively recover multiple signals sharing a common support from one-bit measurements, requiring fewer measurements per non-zero entry than previous single-signal methods.

Contribution

It introduces a novel analysis showing that hard-thresholding can recover multiple signals with shared support from fewer measurements, improving upon existing bounds for single-signal recovery.

Findings

Successful recovery of multiple signals sharing support from one-bit measurements.

Reduction in measurements needed per non-zero entry compared to single-signal bounds.

Numerical experiments validate theoretical results.

Abstract

A simple hard-thresholding operation is shown to be able to recover $L$ signals $\mathbf{x}_1,...,\mathbf{x}_L \in \mathbb{R}^n$ that share a common support of size $s$ from $m = \mathcal{O}(s)$ one-bit measurements per signal if $L \ge \log(en/s)$. This result improves the single signal recovery bounds with $m = \mathcal{O}(s\log(en/s))$ measurements in the sense that asymptotically fewer measurements per non-zero entry are needed. Numerical evidence supports the theoretical considerations.