# Sparse and Robust Linear Regression: An Optimization Algorithm and Its Statistical Properties

arXiv: https://arxiv.org/abs/1505.05257 · HTML: https://mlsanity.com/paper/1505.05257/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 May 2015.

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

- [Adaptively Robust and Sparse K-means Clustering](https://mlsanity.com/paper/2407.06945/) (2024)
- [Multi-task learning via robust regularized clustering with non-convex group penalties](https://mlsanity.com/paper/2404.03250/) (2024)
- [An integrated perspective of robustness in regression through the lens of the bias-variance trade-off](https://mlsanity.com/paper/2407.10418/) (2024)

## Abstract (arXiv)

This paper studies sparse linear regression analysis with outliers in the responses. A parameter vector for modeling outliers is added to the standard linear regression model and then the sparse estimation problem for both coefficients and outliers is considered. The ℓ1 penalty is imposed for the coefficients, while various penalties including redescending type penalties are for the outliers. To solve the sparse estimation problem, we introduce an optimization algorithm. Under some conditions, we show the algorithmic and statistical convergence property for the coefficients obtained by the algorithm. Moreover, it is shown that the algorithm can recover the true support of the coefficients with probability going to one.