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Taking advantage of the natural grouping structure inside data, group sparse optimization can effectively improve the efficiency and stability of high-dimensional data analysis, and it has wide applications in a variety of fields such as machine learning, signal processing, and bioinformatics. Although there has been a lot of progress, it is still a challenge to construct a group sparse-inducing function with good properties and to identify significant groups. This article aims to address the group-sparsity-constrained minimization problem. We convert the problem to an equivalent weighted l p,q -norm ( , ) constrained optimization model, instead of its relaxation or approximation problem. Then, by applying the proximal gradient method, a solution method with theoretical convergence analysis is developed. Moreover, based on the properties proved in the Lagrangian dual framework, the homotopy technique is employed to cope with the parameter tuning task and to ensure that the output of the proposed homotopy algorithm is an L -stationary point of the original problem. The proposed weighted framework, with the central idea of identifying important groups, is compatible with a wide range of support set identification strategies, which can better meet the needs of different applications and improve the robustness of the model in practice. Both simulated and real data experiments demonstrate the superiority of the proposed method in terms of group feature selection accuracy and computational efficiency. Extensive experimental results in application areas such as compressed sensing, image recognition, and classifier design show that our method has great potential in a wide range of applications. Our codes will be available at https://github.com/jianglanfan/HIWT-GSC.

This article proposes a novel iterative weighted group thresholding method for group sparse recovery of signals from underdetermined linear systems. Based on an equivalent weighted group minimization problem with l p p -norm ( ), we derive closed-form solutions for a subproblem with respect to some specific values of p when using the proximal gradient method. Then, we design the corresponding algorithmic framework, including stopping criterion and the method of nonmonotone line search, and prove that the solution sequence generated by the proposed algorithm converges under some mild conditions. Moreover, based on the proposed algorithm, we develop a homotopy algorithm with an adaptively updated group threshold. Extensive computational experiments on the simulated and real data show that our approach is competitive with state-of-the-art methods in terms of exact group selection, estimation accuracy, and computation time.