# A Unified Framework for Pattern Recovery in Penalized and Thresholded Estimation and its Geometry

arXiv: https://arxiv.org/abs/2307.10158 · HTML: https://mlsanity.com/paper/2307.10158/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 47 references that are in the map.
- First posted on arXiv 19 Jul 2023.

## Builds on (most cited first, 15 of 15)

- [Exact post-selection inference, with application to the lasso](https://mlsanity.com/paper/1311.6238/) (2013)
- [SLOPE - Adaptive variable selection via convex optimization](https://mlsanity.com/paper/1407.3824/) (2014)
- [The Geometry of Uniqueness, Sparsity and Clustering in Penalized Estimation](https://mlsanity.com/paper/2004.09106/) (2020)
- [Model Consistency of Partly Smooth Regularizers](https://mlsanity.com/paper/1405.1004/) (2014)
- [Sensitivity Analysis for Mirror-Stratifiable Convex Functions](https://mlsanity.com/paper/1707.03194/) (2017)
- [A theory of optimal convex regularization for low-dimensional recovery](https://mlsanity.com/paper/2112.03540/) (2021)
- [Model Selection with Low Complexity Priors](https://mlsanity.com/paper/1307.2342/) (2013)
- [An Exact Solution Path Algorithm for SLOPE and Quasi-Spherical OSCAR](https://mlsanity.com/paper/2010.15511/) (2020)
- [On the Distribution, Model Selection Properties and Uniqueness of the Lasso Estimator in Low and High Dimensions](https://mlsanity.com/paper/1708.09608/) (2017)
- [Solution Uniqueness of Convex Piecewise Affine Functions Based Optimization with Applications to Constrained ℓ1 Minimization](https://mlsanity.com/paper/1711.05882/) (2017)
- [The Geometry of Sparse Analysis Regularization](https://mlsanity.com/paper/1907.01769/) (2019)
- [Maximal Solutions of Sparse Analysis Regularization](https://mlsanity.com/paper/1703.00192/) (2017)
- [Model Consistency for Learning with Mirror-Stratifiable Regularizers](https://mlsanity.com/paper/1803.08381/) (2018)
- [On the sign recovery by LASSO, thresholded LASSO and thresholded Basis Pursuit Denoising](https://mlsanity.com/paper/1812.05723/) (2018)
- [Improving Lasso for model selection and prediction](https://mlsanity.com/paper/1907.03025/) (2019)

## Abstract (arXiv)

We consider the framework of penalized estimation where the penalty term is given by a real-valued polyhedral gauge, which encompasses methods such as LASSO, generalized LASSO, SLOPE, OSCAR, PACS and others. Each of these estimators is defined through an optimization problem and can uncover a different structure or ``pattern'' of the unknown parameter vector. We define a novel and general notion of patterns based on subdifferentials and formalize an approach to measure pattern complexity. For pattern recovery, we provide a minimal condition for a particular pattern to be detected by the procedure with positive probability, the so-called accessibility condition. Using our approach, we also introduce the stronger noiseless recovery condition. For the LASSO, it is well known that the irrepresentability condition is necessary for pattern recovery with probability larger than 1/2 and we show that the noiseless recovery plays exactly the same role in our general framework, thereby unifying and extending the irrepresentability condition to a broad class of penalized estimators. We also show that the noiseless recovery condition can be relaxed when turning to so-called thresholded penalized estimators: we prove that the necessary condition of accessibility is already sufficient for sure pattern recovery by thresholded penalized estimation provided that the noise is small enough. Throughout the article, we demonstrate how our findings can be interpreted through a geometrical lens.