Robust Decoding from 1-Bit Compressive Sampling with Least Squares
For researchers in compressive sensing, this work provides theoretical guarantees and an efficient algorithm for robust 1-bit CS recovery under noise and sign flips, though the approach is incremental (combining existing least squares and ℓ1 regularization with a known active set method).
This paper proposes least squares and ℓ1-regularized least squares methods for robust signal recovery from 1-bit compressive sensing with noise and sign flips, achieving recovery precision δ with sample complexity O(n/δ²) for over-determined and O(s log n/δ²) for under-determined settings. A primal-dual active set algorithm with one-step convergence is introduced for efficient sparse recovery.
In 1-bit compressive sensing (1-bit CS) where target signal is coded into a binary measurement, one goal is to recover the signal from noisy and quantized samples. Mathematically, the 1-bit CS model reads: $y = η\odot\textrm{sign} (Ψx^* + ε)$, where $x^{*}\in \mathcal{R}^{n}, y\in \mathcal{R}^{m}$, $Ψ\in \mathcal{R}^{m\times n}$, and $ε$ is the random error before quantization and $η\in \mathcal{R}^{n}$ 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 1-bit CS. In this paper, we consider least squares approach under the over-determined and under-determined 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 $δ$ as long as $m \geq\widetilde{\mathcal{O}}(\frac{n}{δ^2})$. For $m< n$, we prove that, up to a constant $c$, with high probability, the $\ell_1$-regularized least-squares solution $x_{\ell_1}$ lies in the ball with center $x^*$ and radius $δ$ provided that $m \geq \mathcal{O}( \frac{s\log n}{δ^2})$ and $\|x^*\|_0 := s < m$. We introduce a Newton type method, the so-called primal and dual active set (PDAS) algorithm, to solve the nonsmooth optimization problem. The PDAS possesses the property of one-step 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.