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# Robust Decoding from Binary Measurements with Cardinality Constraint Least Squares

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### 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.

### Author and article information

###### Journal
03 June 2020
###### Article
2006.02890