In the field of sparse signal recovery, the existence of group structures within signals provides an inherent advantage for signal recovery. One classical model that exploits such group sparsity is the group sparse least absolute deviation (SGLAD) model. In this research, we aim to enhance the SGLAD model by reducing the influence of regulatory parameters, thereby achieving model simplicity. To this end, we introduce a novel extension known as the Group \(L_1\) -LAD model. By employing the restricted isometry property (RIP) theory from compressed sensing, we investigate the recovery bounds of this model and demonstrate that it outperforms the conventional SGLAD model. Additionally, we propose an efficient solution approach based on the alternating direction method of multipliers (ADMM) and analyze its convergence properties in solving the Group \(L_1\) -LAD model. Finally, we evaluate the effectiveness of both the proposed model and algorithm through comprehensive numerical experiments. The results confirm the superior performance of the model in accurately recovering sparse signals with group structures. Quantitatively, under appropriate conditions, the proposed model reduces the relative recovery error by approximately 83% compared to the SGLAD model.
