Robust Decoding from Binary Measurements with Cardinality Constraint Least Squares

TL;DR

This paper introduces a robust decoding method for 1-bit compressive sampling using a cardinality constraint least squares approach, achieving near-optimal error bounds and efficient support recovery with a generalized Newton algorithm.

Contribution

It proposes a novel cardinality constraint least squares decoder and a generalized Newton algorithm for efficient, robust decoding in 1-bit compressive sampling, with theoretical guarantees.

Findings

Achieves minimax estimation error of order √(log n / m).

Recovers signal support with high probability in O(log s) steps.

Demonstrates robustness and efficiency through extensive simulations.

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.