One-bit compressive sampling via ℓ₀minimization

The problem of 1-bit compressive sampling is addressed in this paper. We introduce an optimization model for reconstruction of sparse signals from 1-bit measurements. The model targets a solution that has the least ℓ₀-norm among all signals satisfying consistency constraints stemming from the 1-bit measurements. An algorithm for solving the model is developed. Convergence analysis of the algorithm is presented. Our approach is to obtain a sequence of optimization problems by successively approximating the ℓ₀-norm and to solve resulting problems by exploiting the proximity operator. We examine the performance of our proposed algorithm and compare it with the renormalized fixed point iteration (RFPI) (Boufounos and Baraniuk, 1-bit compressive sensing, 2008; Movahed et al., A robust RFPI-based 1-bit compressive sensing reconstruction algorithm, 2012), the generalized approximate message passing (GAMP) (Kamilov et al., IEEE Signal Process. Lett. 19(10):607–610, 2012), the linear programming (LP) (Plan and Vershynin, Commun. Pure Appl. Math. 66:1275–1297, 2013), and the binary iterative hard thresholding (BIHT) (Jacques et al., IEEE Trans. Inf. Theory 59:2082–2102, 2013) state-of-the-art algorithms for 1-bit compressive sampling reconstruction.