# Simultaneous Feature Selection and Outlier Detection with Optimality Guarantees

arXiv: https://arxiv.org/abs/2007.06114 · HTML: https://mlsanity.com/paper/2007.06114/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Jul 2020.

## Built on by (most cited first, 3 of 3)

- [Outlier detection in regression: conic quadratic formulations](https://mlsanity.com/paper/2307.05975/) (2023)
- [Robust Variable Selection and Estimation Via Adaptive Elastic Net S-Estimators for Linear Regression](https://mlsanity.com/paper/2107.03325/) (2021)
- [Computation of Least Trimmed Squares: A Branch-and-Bound framework with Hyperplane Arrangement Enhancements](https://mlsanity.com/paper/2604.11584/) (2026)

## Abstract (arXiv)

Sparse estimation methods capable of tolerating outliers have been broadly investigated in the last decade. We contribute to this research considering high-dimensional regression problems contaminated by multiple mean-shift outliers which affect both the response and the design matrix. We develop a general framework for this class of problems and propose the use of mixed-integer programming to simultaneously perform feature selection and outlier detection with provably optimal guarantees. We characterize the theoretical properties of our approach, i.e. a necessary and sufficient condition for the robustly strong oracle property, which allows the number of features to exponentially increase with the sample size; the optimal estimation of the parameters; and the breakdown point of the resulting estimates. Moreover, we provide computationally efficient procedures to tune integer constraints and to warm-start the algorithm. We show the superior performance of our proposal compared to existing heuristic methods through numerical simulations and an application investigating the relationships between the human microbiome and childhood obesity.