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This paper introduces a rigorous approach to establish the sharp minimaxoptimalities of both LASSO and SLOPE within the framework of double sparsestructures, notably without relying on RIP-type conditions. Crucially, ourfindings illuminate that the achievement of these optimalities is fundamentallyanchored in a sparse group normalization condition, complemented by severalnovel sparse group restricted eigenvalue (RE)-type conditions introduced inthis study. We further provide a comprehensive comparative analysis of theseeigenvalue conditions. Furthermore, we demonstrate that these conditions holdwith high probability across a wide range of random matrices. Our explorationextends to encompass the random design, where we prove the random designproperties and optimal sample complexity under both weak moment distributionand sub-Gaussian distribution.

Sharp minimax optimality of LASSO and SLOPE under double sparsity assumption