Phase transition in binary compressed sensing based on $L_{1}$-norm minimization

TL;DR

This paper analyzes the performance of binary compressed sensing with $L_{1}$-norm minimization using the replica method, revealing that biased Gaussian measurement matrices improve reconstruction success conditions.

Contribution

It provides a statistical mechanical analysis of binary compressed sensing with biased Gaussian matrices, extending previous results and offering new insights into measurement matrix design.

Findings

Biased Gaussian measurement matrices enhance reconstruction success.

The replica method effectively evaluates binary compressed sensing performance.

Results align with prior studies, confirming the analysis.

Abstract

Compressed sensing is a signal processing scheme that reconstructs high-dimensional sparse signals from a limited number of observations. In recent years, various problems involving signals with a finite number of discrete values have been attracting attention in the field of compressed sensing. In particular, binary compressed sensing, which restricts signal elements to binary values $\{0, 1\}$, is the most fundamental and straightforward analysis subject in such problem settings. We evaluate the typical performance of noiseless binary compressed sensing based on $L_{1}$-norm minimization using the replica method, a statistical mechanical approach. We analyze a general setting where the elements of the observation matrix follow a Gaussian distribution, including a non-zero mean. We demonstrate that the biased observation matrix indicates more reconstruction success conditions in binary compressed sensing. Our results are consistent with the outcomes of several prior studies.