Robust recovery in 1-bit compressive sensing via ℓq-constrained least squares

Abstract

In this paper, we propose using ℓq-constrained least-squares to decode n dimensional signals with sparsity level s from m noisy and sign flipped 1-bit quantized measurements. We prove that the solution of the proposed decoder approximates the target signals with the precision δ up to a positive constant with high probability as long as m≥O(s2/q−1lognδ2). A weighted primal-dual active set algorithm with continuation is utilized for computing the proposed estimator by combining the data driven majority vote tuning parameter selection rule. Comprehensive numerical simulations indicate that our proposed decoder is robust to noise and sign flips and performs better than state-of-the-art methods.