For the linear inverse problem with sparsity constraints, the $l_0$ regularized problem is NP-hard, and existing approaches either utilize greedy algorithms to find almost-optimal solutions or to approximate the $l_0$ regularization with its convex counterparts. In this paper, we propose a novel and concise regularization, namely the sparse group $k$-max regularization, which can not only simultaneously enhance the group-wise and in-group sparsity, but also casts no additional restraints on the magnitude of variables in each group, which is especially important for variables at different scales, so that it approximate the $l_0$ norm more closely. We also establish an iterative soft thresholding algorithm with local optimality conditions and complexity analysis provided. Through numerical experiments on both synthetic and real-world datasets, we verify the effectiveness and flexibility of the proposed method.

翻译：针对具有稀疏性约束的线性逆问题，$l_0$正则化问题是NP难的，现有方法要么采用贪心算法寻找近似最优解，要么用其凸松弛形式近似$l_0$正则化。本文提出一种新颖且简洁的正则化方法——稀疏群组$k$-最大正则化，该方法不仅能同时增强群组内和群组间的稀疏性，而且不对每个群组中变量的大小施加额外约束，这对于不同尺度的变量尤为重要，从而能够更精确地逼近$l_0$范数。我们还建立了一种迭代软阈值算法，并给出了局部最优性条件和复杂度分析。通过在合成数据集和真实数据集上的数值实验，我们验证了所提方法的有效性和灵活性。