A robust recovery algorithm with smoothing strategies

Abstract

This paper addresses the robust sparse recovery problem in the presence of impulsive measurement noise. In order to overcome the poor performance of ℓ2-norm loss function with the outliers under the impulsive noise, we employ the ℓ1-norm as the loss function for the residual error, which is less sensitive to outliers in the measurements than the popular ℓ2-loss. To rise to the challenges introduced by the non-smooth problem, we first employ two smoothing strategies to approximate the ℓ1-norm loss function: one introduces a relaxation factor in the ℓ1-norm and the other uses the infimal convolution smoothing technique to transform it into a smooth counterpart. Both of them can approximate the ℓ1-norm with arbitrary degree of accuracy and provide a Lipschitz continuous gradient loss function. Then, we employ the accelerated proximal gradient (APG) and monotone APG (mAPG) frameworks for the convex and non-convex regularization functions, respectively. The convergence performance is discussed for generalized regularization penalty. The simulation result demonstrates our conclusions and indicates that the algorithm proposed in this paper can improve the reconstruction quality.