1-Bit Compressive Sensing: Reformulation and RRSP-Based Recovery Theory

The 1-bit compressive sensing has been studied recently in the field of sparse signal recovery. Since the amplitude information of sparse signals in 1-bit models is not available, the solution to the 1-bit models is no longer unique in general. As a result, the aim of 1-bit compressive sensing is to recover the signal within a positive scalar factor by using some decoding methods. In this paper, we propose a general 1-bit compressive sensing model by taking into account the situations where the measurement vectors might include zero components which are excluded in existing 1-bit models. Such a 1-bit compressive sensing model can be reformulated equivalently as an l0-minimization problem with linear constraints. This observation leads naturally to a linear-program-based decoding method, referred to as the 1-bit basis pursuit. 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 transpose of sensing matrices. This concept provides an important basis to develop the uniform and nonuniform recovery theory for sparse signals via 1-bit measurements. One of the main results in this paper claims that if the transpose of the sensing matrix admits the RRSP of order k, then every k-sparse signal with a positive scalar factor can be exactly reconstructed via the 1-bit basis pursuit.