# SLOPE - Adaptive variable selection via convex optimization

arXiv: https://arxiv.org/abs/1407.3824 · HTML: https://mlsanity.com/paper/1407.3824/

## Key facts

- Cited by 33 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Jul 2014.

## Built on by (most cited first, 30 of 33)

- [Sparse Neural Additive Model: Interpretable Deep Learning with Feature Selection via Group Sparsity](https://mlsanity.com/paper/2202.12482/) (2022)
- [You Are the Best Reviewer of Your Own Papers: An Owner-Assisted Scoring Mechanism](https://mlsanity.com/paper/2110.14802/) (2021)
- [High-probability minimax lower bounds](https://mlsanity.com/paper/2406.13447/) (2024)
- [Safe rules for the identification of zeros in the solutions of the SLOPE problem](https://mlsanity.com/paper/2110.11784/) (2021)
- [Generalization Error Bounds for Multiclass Sparse Linear Classifiers](https://mlsanity.com/paper/2204.06264/) (2022)
- [Exact Selective Inference with Randomization](https://mlsanity.com/paper/2212.12940/) (2022)
- [Scalable Spike-and-Slab](https://mlsanity.com/paper/2204.01668/) (2022)
- [Robust Methods for High-Dimensional Linear Learning](https://mlsanity.com/paper/2208.05447/) (2022)
- [Characterizing the SLOPE Trade-off: A Variational Perspective and the Donoho-Tanner Limit](https://mlsanity.com/paper/2105.13302/) (2021)
- [Coordinate Descent for SLOPE](https://mlsanity.com/paper/2210.14780/) (2022)
- [Efficient Designs of SLOPE Penalty Sequences in Finite Dimension](https://mlsanity.com/paper/2102.07211/) (2021)
- [The Hessian Screening Rule](https://mlsanity.com/paper/2104.13026/) (2021)
- [Distributionally robust risk evaluation with an isotonic constraint](https://mlsanity.com/paper/2407.06867/) (2024)
- [Efficient Algorithms for Regularized Nonnegative Scale-invariant Low-rank Approximation Models](https://mlsanity.com/paper/2403.18517/) (2024)
- [Strong Screening Rules for Group-based SLOPE Models](https://mlsanity.com/paper/2405.15357/) (2024)
- [Non-Asymptotic Uncertainty Quantification in High-Dimensional Learning](https://mlsanity.com/paper/2407.13666/) (2024)
- [Constrained Machine Learning: The Bagel Framework](https://mlsanity.com/paper/2112.01088/) (2021)
- [A Model-free Closeness-of-influence Test for Features in Supervised Learning](https://mlsanity.com/paper/2306.11855/) (2023)
- [Safe Screening Rules for Group OWL Models](https://mlsanity.com/paper/2504.03152/) (2025)
- [Dictionary-based Low-Rank Approximations and the Mixed Sparse Coding problem](https://mlsanity.com/paper/2111.12399/) (2021)
- [MISNN: Multiple Imputation via Semi-parametric Neural Networks](https://mlsanity.com/paper/2305.01794/) (2023)
- [A Unified Framework for Pattern Recovery in Penalized and Thresholded Estimation and its Geometry](https://mlsanity.com/paper/2307.10158/) (2023)
- [Robust penalized least squares of depth trimmed residuals regression for high-dimensional data](https://mlsanity.com/paper/2309.01666/) (2023)
- [Dual Feature Reduction for the Sparse-group Lasso and its Adaptive Variant](https://mlsanity.com/paper/2405.17094/) (2024)
- [Low Complexity Regularized Phase Retrieval](https://mlsanity.com/paper/2407.16413/) (2024)
- [Variable Selection Methods for Multivariate, Functional, and Complex Biomedical Data in the AI Age](https://mlsanity.com/paper/2501.06868/) (2025)
- [Safe Screening Rules for Group SLOPE](https://mlsanity.com/paper/2506.09451/) (2025)
- [Proximal Operators of Sorted Nonconvex Penalties](https://mlsanity.com/paper/2506.15315/) (2025)
- [Sharp Trade-Offs in High-Dimensional Inference via 2-Level SLOPE](https://mlsanity.com/paper/2507.09110/) (2025)
- [Hypergraphs on high dimensional time series sets using signature transform](https://mlsanity.com/paper/2507.15802/) (2025)

## Abstract (arXiv)

We introduce a new estimator for the vector of coefficients β in the linear model y=Xβ+z, where X has dimensions n× p with p possibly larger than n. SLOPE, short for Sorted L-One Penalized Estimation, is the solution to [minb∈ℝpfrac12Vert y-XbVert ℓ22+λ1vert bvert (1)+λ2vert bvert(2)+cdots+λpvert bvert(p),] where λ1≥λ2≥cdots≥λp≥0 and vert bvert(1)≥vert bvert(2)≥cdots≥vert bvert(p) are the decreasing absolute values of the entries of b. This is a convex program and we demonstrate a solution algorithm whose computational complexity is roughly comparable to that of classical ℓ1 procedures such as the Lasso. Here, the regularizer is a sorted ℓ1 norm, which penalizes the regression coefficients according to their rank: the higher the rank - that is, stronger the signal - the larger the penalty. This is similar to the Benjamini and Hochberg [J. Roy. Statist. Soc. Ser. B 57 (1995) 289-300] procedure (BH) which compares more significant p-values with more stringent thresholds. One notable choice of the sequence λi is given by the BH critical values λ_mathrm BH(i)=z(1-i· q/2p), where q∈(0,1) and z(α) is the quantile of a standard normal distribution. SLOPE aims to provide finite sample guarantees on the selected model; of special interest is the false discovery rate (FDR), defined as the expected proportion of irrelevant regressors among all selected predictors. Under orthogonal designs, SLOPE with λBH provably controls FDR at level q. Moreover, it also appears to have appreciable inferential properties under more general designs X while having substantial power, as demonstrated in a series of experiments running on both simulated and real data.