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 derive a set of \(\tfrac{n(n+1)}{2}\) 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 \(\mathcal{O}(n\log n + LT)\) where \(T\leq n\) is a problem-dependent constant and \(L\) is the number of zeros identified by the tests. Numerical experiments confirm that, for a prescribed computational budget, the proposed methodology leads to significant improvements of the solving precision.
翻译:在本文中,我们提出了一个加速解决所谓的“ Sorted L- One 惩罚性估计” (SLOPE) 问题的方法。 我们的方法利用了“安全筛选”的概念, 在文献中仔细研究了 \ textit{ group- separable} corsity- 引导规范, 目的是确定 SLOPE 解决方案中的零。 更具体地说, 我们为 \\\ ( tfrac{n( n+1))\\\\\\\\ \ \ \ \ \ \ \ \ \ ( t\\ leq n\ \ ) 的每个元素提出了一套问题常数, 而\ ( L\ \ ) 是测试确定的零数, 证明如果这些不平等的某些子集得到核实, 后者是可以安全筛选的。 此外, 我们提出了一种高效的算法, 来将拟议的程序共同应用于所有原始变量。 我们的程序复杂 \ (\\\ macal{O} (n\ n\ n\ + LT)\\\\\\\\\ ) 。 我们的程序, 其中, 我们的程序是一个问题取决于的常数 。 我们的计算预算建议, 。 。 其中的问题是一个由测试确定的零数由测试确定的计算预算。