Robust Decoding from 1Bit Compressive Sampling with Least Squares
Abstract
In 1bit compressive sensing (1bit CS) where target signal is coded into a binary measurement, one goal is to recover the signal from noisy and quantized samples. Mathematically, the 1bit CS model reads: $y = \eta \odot\textrm{sign} (\Psi x^* + \epsilon)$, where $x^{*}\in \mathcal{R}^{n}, y\in \mathcal{R}^{m}$, $\Psi \in \mathcal{R}^{m\times n}$, and $\epsilon$ is the random error before quantization and $\eta\in \mathcal{R}^{n}$ is a random vector modeling the sign flips. Due to the presence of nonlinearity, noise and sign flips, it is quite challenging to decode from the 1bit CS. In this paper, we consider least squares approach under the overdetermined and underdetermined settings. For $m>n$, we show that, up to a constant $c$, with high probability, the least squares solution $x_{\textrm{ls}}$ approximates $ x^*$ with precision $\delta$ as long as $m \geq\widetilde{\mathcal{O}}(\frac{n}{\delta^2})$. For $m< n$, we prove that, up to a constant $c$, with high probability, the $\ell_1$regularized leastsquares solution $x_{\ell_1}$ lies in the ball with center $x^*$ and radius $\delta$ provided that $m \geq \mathcal{O}( \frac{s\log n}{\delta^2})$ and $\x^*\_0 := s < m$. We introduce a Newton type method, the socalled primal and dual active set (PDAS) algorithm, to solve the nonsmooth optimization problem. The PDAS possesses the property of onestep convergence. It only requires to solve a small least squares problem on the active set. Therefore, the PDAS is extremely efficient for recovering sparse signals through continuation. We propose a novel regularization parameter selection rule which does not introduce any extra computational overhead. Extensive numerical experiments are presented to illustrate the robustness of our proposed model and the efficiency of our algorithm.
 Publication:

arXiv eprints
 Pub Date:
 November 2017
 DOI:
 10.48550/arXiv.1711.01206
 arXiv:
 arXiv:1711.01206
 Bibcode:
 2017arXiv171101206H
 Keywords:

 Mathematics  Numerical Analysis;
 Statistics  Computation