This paper introduces a rigorous approach to establish the sharp minimax optimalities of both LASSO and SLOPE within the framework of double sparse structures, notably without relying on RIP-type conditions. Crucially, our findings illuminate that the achievement of these optimalities is fundamentally anchored in a sparse group normalization condition, complemented by several novel sparse group restricted eigenvalue (RE)-type conditions introduced in this study. We further provide a comprehensive comparative analysis of these eigenvalue conditions. Furthermore, we demonstrate that these conditions hold with high probability across a wide range of random matrices. Our exploration extends to encompass the random design, where we prove the random design properties and optimal sample complexity under both weak moment distribution and sub-Gaussian distribution.

翻译：本文提出了一种严谨的方法，在双稀疏结构框架下建立了LASSO和SLOPE的尖锐极小极大最优性，且无需依赖RIP型条件。至关重要的是，我们的发现表明，这些最优性的实现本质上依赖于一种稀疏群归一化条件，并辅以本文引入的若干新型稀疏群限制特征值（RE）型条件。我们进一步对这些特征值条件进行了全面的比较分析。此外，我们证明这些条件在广泛的随机矩阵下以高概率成立。我们的研究还扩展至随机设计情形，在弱矩分布和次高斯分布下验证了随机设计性质及最优样本复杂度。