Phase transition in binary compressed sensing based on $L_1$-norm minimization

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.

14 pages, 2 Figures