Sparse Recovery via $\ell_1^2-η\ell_2^2$ Minimization

Abstract

The weighted difference of squared norms (WDSN) penalty $\ell_1^2-η\ell_2^2$ with $0\leq η\leq 1$ has attracted considerable attention due to its strong sparsity-promoting ability and favorable reconstruction performance in compressed sensing and inverse problems. However, exact recovery guarantees and restricted isometry property (RIP) analysis for WDSN minimization have not yet been established. In this paper, we address this gap. First, we establish sufficient conditions for the exact recovery of $k$-sparse signals based on the null space property (NSP). Then, under the $δ_{2k}$-RIP condition, we derive stable recovery guarantees for both $k$-sparse signals and general signals, and characterize upper bounds on the reconstruction error. Furthermore, we propose a WDSN-based regularized model to handle both noiseless and noisy observations in a unified framework. To design an efficient algorithm, we derive an explicit formula for the proximal operator of the WDSN functional. Based on this proximal solver, we develop a suitable variable-splitting scheme within the alternating direction method of multipliers (ADMM) and establish its global convergence under some mild conditions. Finally, numerical experiments show that the proposed method outperforms the iterative half variation method in both noiseless and noisy sparse recovery tasks.