1Bit Compressive Sensing: Reformulation and RRSPBased Sign Recovery Theory
Abstract
Recently, the 1bit compressive sensing (1bit CS) has been studied in the field of sparse signal recovery. Since the amplitude information of sparse signals in 1bit CS is not available, it is often the support or the sign of a signal that can be exactly recovered with a decoding method. In this paper, we first show that a necessary assumption (that has been overlooked in the literature) should be made for some existing theories and discussions for 1bit CS. Without such an assumption, the found solution by some existing decoding algorithms might be inconsistent with 1bit measurements. This motivates us to pursue a new direction to develop uniform and nonuniform recovery theories for 1bit CS with a new decoding method which always generates a solution consistent with 1bit measurements. We focus on an extreme case of 1bit CS, in which the measurements capture only the sign of the product of a sensing matrix and a signal. We show that the 1bit CS model can be reformulated equivalently as an $\ell_0$minimization problem with linear constraints. This reformulation naturally leads to a new linearprogrambased decoding method, referred to as the 1bit basis pursuit, which is remarkably different from existing formulations. It turns out that the uniqueness condition for the solution of the 1bit basis pursuit yields the socalled restricted range space property (RRSP) of the transposed sensing matrix. This concept provides a basis to develop sign recovery conditions for sparse signals through 1bit measurements. We prove that if the sign of a sparse signal can be exactly recovered from 1bit measurements with 1bit basis pursuit, then the sensing matrix must admit a certain RRSP, and that if the sensing matrix admits a slightly enhanced RRSP, then the sign of a $k$sparse signal can be exactly recovered with 1bit basis pursuit.
 Publication:

arXiv eprints
 Pub Date:
 December 2014
 DOI:
 10.48550/arXiv.1412.5514
 arXiv:
 arXiv:1412.5514
 Bibcode:
 2014arXiv1412.5514Z
 Keywords:

 Computer Science  Information Theory;
 Mathematics  Optimization and Control