# Group sparse recovery in impulsive noise via alternating direction method of multipliers

In this paper, we consider the recovery of group sparse signals corrupted by impulsive noise. In some recent literature, researchers have utilized stable data fitting models, like $l_1$-norm, Huber penalty function and Lorentzian-norm, to substitute the $l_2$-norm data fidelity model to obtain more robust performance. In this paper, a stable model is developed, which exploits the generalized $l_p$-norm as the measure for the error for sparse reconstruction. In order to address this model, we propose an efficient alternative direction method of multipliers, which includes the proximity operator of $l_p$-norm functions to the framework of Lagrangian methods. Besides, to guarantee the convergence of the algorithm in the case of $0\leq p<1$ (nonconvex case), we took advantage of a smoothing strategy. For both $0\leq p<1$ (nonconvex case) and $1\leq p\leq2$ (convex case), we have derived the conditions of the convergence for the proposed algorithm. Moreover, under the block restricted isometry property with constant $\delta_{\tau k_0}<\tau/(4-\tau)$ for $0<\tau<4/3$ and $\delta_{\tau k_0}<\sqrt{(\tau-1)/\tau}$ for $\tau\geq4/3$, a sharp sufficient condition for group sparse recovery in the presence of impulsive noise and its associated error upper bound estimation are established. Numerical results based on the synthetic block sparse signals and the real-world FECG signals demonstrate the effectiveness and robustness of new algorithm in highly impulsive noise.

## Citation

Address: Tiansheng Road No. 2, Beibei district, Chongqing, 400715, China;December, 4, 2018