SIAM Journal on Imaging Sciences, Volume 14, Issue 2, Page 530-557, January 2021.
The $\ell_1-\ell_2$ regularization is a popular nonconvex yet Lipschitz continuous metric, which has been widely used in signal and image processing. The theory for the $\ell_1-\ell_2$ minimization method shows that it has superior sparse recovery performance over the classical $\ell_1$ minimization method. The motivation and major contribution of this paper is to provide a positive answer to the open problem posed in [T.-H. Ma, Y. Lou, and T.-Z. Huang, SIAM J. Imaging Sci., 10 (2017), pp. 1346--1380] about the sufficient conditions that can be sharpened for the $\ell_1-\ell_2$ minimization method. The novel technique used in our analysis of the $\ell_1-\ell_2$ minimization method is a crucial sparse representation adapted to the $\ell_1-\ell_2$ metric which is different from the other state-of-the-art works in the context of the $\ell_1-\ell_2$ minimization method. The new restricted isometry property (RIP) analysis is better than the existing RIP based conditions to guarantee the exact and stable recovery of signals.

中文翻译：

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$ \ ell_1- \ ell_2 $最小化方法的新受限等距特性分析

SIAM影像科学杂志，第14卷，第2期，第530-557页，2021年1月。
$ \ ell_1- \ ell_2 $正则化是一种流行的非凸但Lipschitz连续度量，已广泛用于信号和图像处理中。$ \ ell_1- \ ell_2 $最小化方法的理论表明，与传统的$ \ ell_1 $最小化方法相比，它具有更出色的稀疏恢复性能。本文的动机和主要贡献是对[T.-H. S. H.]提出的开放性问题提供积极的答案。马，楼露和T.-Z。Huang，SIAM J. Imaging Sci。，第10卷，（2017），第1346--1380页]，介绍了对于$ \ ell_1- \ ell_2 $最小化方法可以改善的充分条件。在我们对$ \ ell_1- \ ell_2 $最小化方法的分析中使用的新颖技术是一种适应于$ \ ell_1- \ ell_2 $指标的关键稀疏表示，这与$ \ ell_1- \ ell_2 $最小化方法的上下文。新的受限等距特性（RIP）分析优于现有的基于RIP的条件，可确保信号的准确和稳定恢复。