Robust Decoding from Binary Measurements with Cardinality Constraint Least Squares

Abstract

The main goal of 1-bit compressive sampling is to decode \(n\) dimensional signals with sparsity level \(s\) from \(m\) binary measurements. This is a challenging task due to the presence of nonlinearity, noises and sign flips. In this paper, the cardinality constraint least square is proposed as a desired decoder. We prove that, up to a constant \(c\), with high probability, the proposed decoder achieves a minimax estimation error as long as \(m \geq \mathcal{O}( s\log n)\). Computationally, we utilize a generalized Newton algorithm (GNA) to solve the cardinality constraint minimization problem with the cost of solving a least squares problem with small size at each iteration. We prove that, with high probability, the \(\ell_{\infty}\) norm of the estimation error between the output of GNA and the underlying target decays to \(\mathcal{O}(\sqrt{\frac{\log n }{m}}) \) after at most \(\mathcal{O}(\log s)\) iterations. Moreover, the underlying support can be recovered with high probability in \(\mathcal{O}(\log s)\) steps provided that the target signal is detectable. Extensive numerical simulations and comparisons with state-of-the-art methods are presented to illustrate the robustness of our proposed decoder and the efficiency of the GNA algorithm.