Nonconvex Penalized Regularization for Robust Sparse Recovery in the Presence of $S\alpha S$ Noise

Nonconvex penalties have recently received considerable attention in sparse recovery based on Gaussian assumptions. However, many sparse recovery problems occur in the presence of impulsive noises. This paper is concerned with the analysis and comparison of different sparsity-inducing penalties for <inline-formula> <tex-math notation="LaTeX">$L_{1}$ </tex-math></inline-formula>-loss function-based robust sparse recovery. To solve these nonconvex and nonsmooth optimization problems, we use the alternating direction method of multipliers framework to split this difficult problem into tractable sub-problems in combination with corresponding iterative proximal operators. This paper employs different nonconvex penalties and compares the performances, advantages, and properties and provides guidance for the choice of the best regularizer for sparse recovery with different levels of impulsive noise. Experimental results indicate that convex lasso (<inline-formula> <tex-math notation="LaTeX">$L_{1}$ </tex-math></inline-formula>-norm) penalty is more effective for the suppression of highly impulsive noise than nonconvex penalties, while the nonconvex penalties show the potential to improve the performance in low and medium level noise. Moreover, among these nonconvex penalties, <inline-formula> <tex-math notation="LaTeX">$L_{p}$ </tex-math></inline-formula> norm can often obtain better recovery performance.