标题： 精确恢复在双重稀疏模型中：充分必要信号条件 显示英文标题

标题： Exact recovery in the double sparse model: sufficient and necessary signal conditions

摘要： 具有回归系数组稀疏性和元素稀疏性的双稀疏线性模型最近引起了广泛关注。 本文建立了双稀疏模型中精确支持恢复与最优最小信号条件之间的充要关系。 具体而言，在所提出的信号条件下，一个两阶段的双稀疏迭代硬阈值过程通过适当地选择阈值参数可以实现精确的支持恢复。 此外，该过程保持渐近正态性，与给定真实支持的OLS估计量一致，因此具备oracle性质。 反之，我们证明如果违反这些信号条件，则没有方法能够实现精确的支持恢复。 这填补了双稀疏模型支持恢复的minimax最优性理论中的关键空白。 最后，提供了数值实验来支持我们的理论发现。 显示英文摘要

摘要： The double sparse linear model, which has both group-wise and element-wise sparsity in regression coefficients, has attracted lots of attention recently. This paper establishes the sufficient and necessary relationship between the exact support recovery and the optimal minimum signal conditions in the double sparse model. Specifically, sharply under the proposed signal conditions, a two-stage double sparse iterative hard thresholding procedure achieves exact support recovery with a suitably chosen threshold parameter. Also, this procedure maintains asymptotic normality aligning with an OLS estimator given true support, hence holding the oracle properties. Conversely, we prove that no method can achieve exact support recovery if these signal conditions are violated. This fills a critical gap in the minimax optimality theory on support recovery of the double sparse model. Finally, numerical experiments are provided to support our theoretical findings.