New Restricted Isometry Property Analysis for ℓ1−ℓ2 Minimization Methods

Abstract

The ℓ1−ℓ2 regularization is a popular nonconvex yet Lipschitz continuous metric, which has been widely used in signal and image processing. The theory for the ℓ1−ℓ2 minimization method shows that it has superior sparse recovery performance over the classical ℓ1 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 ℓ1−ℓ2 minimization method. The novel technique used in our analysis of the ℓ1−ℓ2 minimization method is a crucial sparse representation adapted to the ℓ1−ℓ2 metric which is different from the other state-of-the-art works in the context of the ℓ1−ℓ2 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.