In this paper, we focus our attention on the high-dimensional double sparse linear regression, that is, a combination of element-wise and group-wise sparsity.To address this problem, we propose an IHT-style (iterative hard thresholding) procedure that dynamically updates the threshold at each step. We establish the matching upper and lower bounds for parameter estimation, showing the optimality of our proposal in the minimax sense. Coupled with a novel sparse group information criterion, we develop a fully adaptive procedure to handle unknown group sparsity and noise levels.We show that our adaptive procedure achieves optimal statistical accuracy with fast convergence. Finally, we demonstrate the superiority of our method by comparing it with several state-of-the-art algorithms on both synthetic and real-world datasets.

翻译：本文关注高维双重稀疏线性回归问题，即元素稀疏性与组稀疏性的结合。为解决该问题，我们提出一种迭代硬阈值（IHT）类方法，其每一步动态更新阈值。我们建立了参数估计的匹配上下界，证明了所提方法在极小极大意义下的最优性。结合一种新型稀疏组信息准则，我们开发了完全自适应的过程以处理未知的组稀疏性和噪声水平。结果表明，该自适应方法能够以快速收敛实现最优统计精度。最后，通过在合成数据集和真实世界数据集上与多种最先进算法进行比较，验证了我们方法的优越性。