- 1-Bit Compressive Sensing: Reformulation and RRSP-Based Sign Recovery Theory Yun-Bin Zhao(y.zhao.2bham.ac.uk) Chunlei Xu(xucfor.mat.bham.ac.uk) Abstract: Recently, the 1-bit compressive sensing (1-bit CS) has been studied in the field of sparse signal recovery. Since the amplitude information of sparse signals in 1-bit 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 1-bit CS. Without such an assumption, the found solution by some existing decoding algorithms might be inconsistent with 1-bit measurements. This motivates us to pursue a new direction to develop uniform and nonuniform recovery theories for 1-bit CS with a new decoding method which always generates a solution consistent with 1-bit measurements. We focus on an extreme case of 1-bit CS, in which the measurements capture only the sign of the product of a sensing matrix and a signal. We show that the 1-bit CS model can be reformulated equivalently as an $\ell_0$-minimization problem with linear constraints. This reformulation naturally leads to a new linear-program-based decoding method, referred to as the 1-bit basis pursuit, which is remarkably different from existing formulations. It turns out that the uniqueness condition for the solution of the 1-bit basis pursuit yields the so-called restricted range space property (RRSP) of the transposed sensing matrix. This concept provides a basis to develop sign recovery conditions for sparse signals through 1-bit measurements. We prove that if the sign of a sparse signal can be exactly recovered from 1-bit measurements with 1-bit 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 1-bit basis pursuit. Keywords: 1-bit compressive sensing, restricted range space property, 1-bit basis pursuit, linear program, $\ell_0$-minimization, sparse signal recovery. Category 1: Convex and Nonsmooth Optimization (Convex Optimization ) Category 2: Applications -- Science and Engineering (Data-Mining ) Citation: Download: [PDF]Entry Submitted: 07/03/2016Entry Accepted: 07/04/2016Entry Last Modified: 07/03/2016Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Optmization Society.