Safe rules for the identification of zeros in the solutions of the SLOPE problem

In this paper we propose a methodology to accelerate the resolution of the so-called ``Sorted L-One Penalized Estimation'' (SLOPE) problem. Our method leverages the concept of ``safe screening'', well-studied in the literature for \textit{group-separable} sparsity-inducing norms, and aims at identifying the zeros in the solution of SLOPE. More specifically, we introduce a family of \(n!\) safe screening rules for this problem, where \(n\) is the dimension of the primal variable, and propose a tractable procedure to verify if one of these tests is passed. Our procedure has a complexity \(\mathcal{O}(n\log n + LT)\) where \(T\leq n\) is a problem-dependent constant and \(L\) is the number of zeros identified by the tests. We assess the performance of our proposed method on a numerical benchmark and emphasize that it leads to significant computational savings in many setups.