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 group-separable sparsity-inducing norms, and aims ato identify the zeros in the solution of SLOPE. More specifically, we derive a set of n(n+1) 2 inequalities for each element of the n-dimensional primal vector and prove that the latter can be safely screened if some subsets of these inequalities are verified. We propose moreover an efficient algorithm to jointly apply the proposed procedure to all the primal variables. Our procedure has a complexity O(n log n + LT), where T < n is a problem-dependent constant and L is the number of zeros identified by the test. Numerical experiments confirm that, for a prescribed computational budget, the proposed methodology leads to significant improvements in the solving precision.