# The Geometry of Uniqueness, Sparsity and Clustering in Penalized Estimation

arXiv: https://arxiv.org/abs/2004.09106 · HTML: https://mlsanity.com/paper/2004.09106/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Apr 2020.

## Built on by (most cited first, 9 of 9)

- [Foolish Crowds Support Benign Overfitting](https://mlsanity.com/paper/2110.02914/) (2021)
- [Safe rules for the identification of zeros in the solutions of the SLOPE problem](https://mlsanity.com/paper/2110.11784/) (2021)
- [Coordinate Descent for SLOPE](https://mlsanity.com/paper/2210.14780/) (2022)
- [Strong Screening Rules for Group-based SLOPE Models](https://mlsanity.com/paper/2405.15357/) (2024)
- [A Unified Framework for Pattern Recovery in Penalized and Thresholded Estimation and its Geometry](https://mlsanity.com/paper/2307.10158/) (2023)
- [Dual Feature Reduction for the Sparse-group Lasso and its Adaptive Variant](https://mlsanity.com/paper/2405.17094/) (2024)
- [Isometry pursuit](https://mlsanity.com/paper/2411.18502/) (2024)
- [Proximal Operators of Sorted Nonconvex Penalties](https://mlsanity.com/paper/2506.15315/) (2025)
- [Quantization through Piecewise-Affine Regularization: Optimization and Statistical Guarantees](https://mlsanity.com/paper/2508.11112/) (2025)

## Abstract (arXiv)

We provide a necessary and sufficient condition for the uniqueness of penalized least-squares estimators whose penalty term is given by a norm with a polytope unit ball, covering a wide range of methods including SLOPE, PACS, fused, clustered and classical LASSO as well as the related method of basis pursuit. We consider a strong type of uniqueness that is relevant for statistical problems. The uniqueness condition is geometric and involves how the row span of the design matrix intersects the faces of the dual norm unit ball, which for SLOPE is given by the signed permutahedron. Further considerations based this condition also allow to derive results on sparsity and clustering features. In particular, we define the notion of a SLOPE pattern to describe both sparsity and clustering properties of this method and also provide a geometric characterization of accessible SLOPE patterns.