  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.
