One-Bit Compressive Sensing via Schur-Concave Function Minimization

Abstract

Much effort has been devoted to recovering sparse signals from one-bit measurements in recent years. However, it is still quite challenging to recover signals with high fidelity, which is desired in practical one-bit compressive sensing (1-bit CS) applications. We introduce the notion of Schur-concavity in this paper and propose to construct signals by taking advantage of Schur-Concave functions , which are capable of enhancing sparsity. Specifically, the Schur-concave functions can be employed to measure the degree of concentration, and the sparse solutions are obtained at the minima. As a representative of the Schur-concave family, the normalized $\ell _1$ Shannon entropy function ( $\ell _1$ -SEF) is exploited. The resulting optimization problem is nonconvex. Hence, we convert it into a series of weighted ${\ell _1}$ -norm subproblems, which are solved iteratively by a generalized fixed-point continuation algorithm. Numerical results are provided to illustrate the effectiveness and superiority of the proposed 1-bit CS algorithm.