Characterization of ℓ1 minimizer in one-bit compressed sensing

Abstract

The conception of one-bit compressive sensing (one-bit CS) was first introduced in 2008 by Boufounos and Baraniuk [1-Bit compressive sensing, in Proc. Conf. on Information Science and Systems (CISS, Princeton, NJ, 2008)]. Since then, many efficient algorithms have been developed for dealing with the one-bit CS problem. However, few theoretical results are available on one-bit CS. In this paper, we focus on one-bit CS theory with its relaxation model [Formula: see text] and present a necessary and sufficient condition such that the signal [Formula: see text] is the unique [Formula: see text] minimizer in noiseless one-bit CS (Theorem 3). Moreover, by using the improved separation theorem of convex sets (Theorem 6), we completely characterize the [Formula: see text] minimizer in one-bit CS (Theorem 2). Finally, as an application of Theorem 2, the [Formula: see text] minimizer for the considered model can, in general, be a non-sparse vector.