Robust Sparse Recovery in Impulsive Noise via Continuous Mixed Norm
This letter investigates the problem of sparse signal recovery in the presence of additive impulsive noise. The heavy-tailed impulsive noise is well modeled with stable distributions. Since there is no explicit formula for the probability density function of <inline-formula><tex-math notation="LaTeX">$S\alpha S$</tex-math></inline-formula> distribution, alternative approximations are used, such as, generalized Gaussian distribution, which imposes <inline-formula><tex-math notation="LaTeX">$\ell _{p}$</tex-math></inline-formula>-norm fidelity on the residual error. In this letter, we exploit a continuous mixed norm (CMN) for robust sparse recovery instead of <inline-formula> <tex-math notation="LaTeX">$\ell _{p}$</tex-math></inline-formula>-norm. We show that in blind conditions, i.e., in the case where the parameters of the noise distribution are unknown, incorporating CMN can lead to near-optimal recovery. We apply alternating direction method of multipliers for solving the problem induced by utilizing CMN for robust sparse recovery. In this approach, CMN is replaced with a surrogate function and the majorization–minimization technique is incorporated to solve the problem. Simulation results confirm the efficiency of the proposed method compared to some recent algorithms for robust sparse recovery in impulsive noise.