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

arXiv: https://arxiv.org/abs/2110.11784 · HTML: https://mlsanity.com/paper/2110.11784/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 52 references that are in the map.
- First posted on arXiv 22 Oct 2021.

## Builds on (most cited first, 18 of 18)

- [Gap Safe screening rules for sparsity enforcing penalties](https://mlsanity.com/paper/1611.05780/) (2016)
- [SLOPE - Adaptive variable selection via convex optimization](https://mlsanity.com/paper/1407.3824/) (2014)
- [Mind the duality gap: safer rules for the Lasso](https://mlsanity.com/paper/1505.03410/) (2015)
- [Fast OSCAR and OWL Regression via Safe Screening Rules](https://mlsanity.com/paper/2006.16433/) (2020)
- [Screening Tests for Lasso Problems](https://mlsanity.com/paper/1405.4897/) (2014)
- [Algorithmic Analysis and Statistical Estimation of SLOPE via Approximate Message Passing](https://mlsanity.com/paper/1907.07502/) (2019)
- [Group SLOPE - adaptive selection of groups of predictors](https://mlsanity.com/paper/1610.04960/) (2016)
- [Safe Screening With Variational Inequalities and Its Application to LASSO](https://mlsanity.com/paper/1307.7577/) (2013)
- [The Alternating Descent Conditional Gradient Method for Sparse Inverse Problems](https://mlsanity.com/paper/1507.01562/) (2015)
- [The Geometry of Uniqueness, Sparsity and Clustering in Penalized Estimation](https://mlsanity.com/paper/2004.09106/) (2020)
- [The Strong Screening Rule for SLOPE](https://mlsanity.com/paper/2005.03730/) (2020)
- [Screening Rules and its Complexity for Active Set Identification](https://mlsanity.com/paper/2009.02709/) (2020)
- [Expanding boundaries of Gap Safe screening](https://mlsanity.com/paper/2102.10846/) (2021)
- [Safe squeezing for antisparse coding](https://mlsanity.com/paper/1911.07508/) (2019)
- [Solving the OSCAR and SLOPE Models Using a Semismooth Newton-Based Augmented Lagrangian Method](https://mlsanity.com/paper/1803.10740/) (2018)
- [Screen & Relax: Accelerating the resolution of Elastic-net by safe identification of the solution support](https://mlsanity.com/paper/2110.07281/) (2021)
- [An O(nlog(n)) Algorithm for Projecting Onto the Ordered Weighted ℓ1 Norm Ball](https://mlsanity.com/paper/1505.00870/) (2015)
- [Scalable Sparse Subspace Clustering via Ordered Weighted ℓ1 Regression](https://mlsanity.com/paper/1807.03746/) (2018)

## Built on by (most cited first, 6 of 6)

- [Coordinate Descent for SLOPE](https://mlsanity.com/paper/2210.14780/) (2022)
- [Beyond GAP screening for Lasso by exploiting new dual cutting half-spaces with supplementary material](https://mlsanity.com/paper/2203.00987/) (2022)
- [One to beat them all: "RYU" -- a unifying framework for the construction of safe balls](https://mlsanity.com/paper/2312.00640/) (2023)
- [Strong Screening Rules for Group-based SLOPE Models](https://mlsanity.com/paper/2405.15357/) (2024)
- [Proximal Operators of Sorted Nonconvex Penalties](https://mlsanity.com/paper/2506.15315/) (2025)
- [Efficient Solvers for SLOPE in R, Python, Julia, and C++](https://mlsanity.com/paper/2511.02430/) (2025)

## Abstract (arXiv)

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 textitgroup-separable sparsity-inducing norms, and aims at identifying the zeros in the solution of SLOPE. More specifically, we derive a set of (tfracn(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(nlog n + LT)) where (T≤ 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.