The Dantzig selector: Recovery of Signal via $\ell_1-\alpha \ell_2$ Minimization

TL;DR

This paper introduces a novel Dantzig selector based on $\,\ell_1-\alpha\ell_2$ minimization for improved signal recovery, providing theoretical guarantees and demonstrating superior performance over existing methods through extensive experiments.

Contribution

The paper proposes a new Dantzig selector using $\,\ell_1-\alpha\ell_2$ minimization, with theoretical recovery guarantees and an effective algorithm, outperforming existing methods.

Findings

The proposed Dantzig selector achieves better recovery accuracy.

The method performs well under Gaussian, impulsive, and uniform noise.

Theoretical guarantees are established based on RIP.

Abstract

In the paper, we proposed the Dantzig selector based on the $\ell_{1}-\alpha \ell_{2}$~$(0< \alpha \leq1)$ minimization for the signal recovery. In the Dantzig selector, the constraint $\|{\bf A}^{\top}({\bf b}-{\bf A}{\bf x})\|_\infty \leq \eta$ for some small constant $\eta>0$ means the columns of ${\bf A}$ has very weakly correlated with the error vector ${\bf e}={\bf A}{\bf x}-{\bf b}$. First, recovery guarantees based on the restricted isometry property (RIP) are established for signals. Next, we propose the effective algorithm to solve the proposed Dantzig selector. Last, we illustrate the proposed model and algorithm by extensive numerical experiments for the recovery of signals in the cases of Gaussian, impulsive and uniform noise. And the performance of the proposed Dantzig selector is better than that of the existing methods.