Robust Sparse Recovery in Impulsive Noise via $\ell _p$ -$\ell _1$ Optimization

Abstract

This paper addresses the issue of robust sparse recovery in compressive sensing (CS) in the presence of impulsive measurement noise. Recently, robust data-fitting models, such as ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -norm, Lorentzian-norm, and Huber penalty function, have been employed to replace the popular ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -norm loss model to gain more robust performance. In this paper, we propose a robust formulation for sparse recovery using the generalized ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> -norm with 0 ≤ p <; 2 as the metric for the residual error. To solve this formulation efficiently, we develop an alternating direction method (ADM) via incorporating the proximity operator of ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> -norm functions into the framework of augmented Lagrangian methods. Furthermore, to derive a convergent method for the nonconvex case of p <; 1, a smoothing strategy has been employed. The convergence conditions of the proposed algorithm have been analyzed for both the convex and nonconvex cases. The new algorithm has been compared with some state-of-the-art robust algorithms via numerical simulations to show its improved performance in highly impulsive noise.