# Robust and Sparse Estimation of Linear Regression Coefficients with Heavy-tailed Noises and Covariates

arXiv: https://arxiv.org/abs/2206.07594 · HTML: https://mlsanity.com/paper/2206.07594/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 65 references that are in the map.
- First posted on arXiv 15 Jun 2022.

## Builds on (most cited first, 30 of 32)

- [Robust Estimators in High Dimensions without the Computational Intractability](https://mlsanity.com/paper/1604.06443/) (2016)
- [Agnostic Estimation of Mean and Covariance](https://mlsanity.com/paper/1604.06968/) (2016)
- [Recent Advances in Algorithmic High-Dimensional Robust Statistics](https://mlsanity.com/paper/1911.05911/) (2019)
- [Being Robust (in High Dimensions) Can Be Practical](https://mlsanity.com/paper/1703.00893/) (2017)
- [Robust Estimation via Robust Gradient Estimation](https://mlsanity.com/paper/1802.06485/) (2018)
- [Efficient Algorithms and Lower Bounds for Robust Linear Regression](https://mlsanity.com/paper/1806.00040/) (2018)
- [Distribution-Independent PAC Learning of Halfspaces with Massart Noise](https://mlsanity.com/paper/1906.10075/) (2019)
- [Robust regression with covariate filtering: Heavy tails and adversarial contamination](https://mlsanity.com/paper/2009.12976/) (2020)
- [Robust Covariance and Scatter Matrix Estimation under Huber's Contamination Model](https://mlsanity.com/paper/1506.00691/) (2015)
- [Learning Halfspaces with Massart Noise Under Structured Distributions](https://mlsanity.com/paper/2002.05632/) (2020)
- [Robust Linear Regression: Optimal Rates in Polynomial Time](https://mlsanity.com/paper/2007.01394/) (2020)
- [Quantum Entropy Scoring for Fast Robust Mean Estimation and Improved Outlier Detection](https://mlsanity.com/paper/1906.11366/) (2019)
- [Robustly Learning a Gaussian: Getting Optimal Error, Efficiently](https://mlsanity.com/paper/1704.03866/) (2017)
- [High-Dimensional Robust Mean Estimation in Nearly-Linear Time](https://mlsanity.com/paper/1811.09380/) (2018)
- [Robust subgaussian estimation of a mean vector in nearly linear time](https://mlsanity.com/paper/1906.03058/) (2019)
- [Mean Estimation with Sub-Gaussian Rates in Polynomial Time](https://mlsanity.com/paper/1809.07425/) (2018)
- [Optimal Robust Linear Regression in Nearly Linear Time](https://mlsanity.com/paper/2007.08137/) (2020)
- [Outlier-robust estimation of a sparse linear model using ℓ1-penalized Huber's M-estimator](https://mlsanity.com/paper/1904.06288/) (2019)
- [High Dimensional Robust Sparse Regression](https://mlsanity.com/paper/1805.11643/) (2018)
- [Faster Algorithms for High-Dimensional Robust Covariance Estimation](https://mlsanity.com/paper/1906.04661/) (2019)
- [Statistical and computational trade-offs in estimation of sparse principal components](https://mlsanity.com/paper/1408.5369/) (2014)
- [A Shrinkage Principle for Heavy-Tailed Data: High-Dimensional Robust Low-Rank Matrix Recovery](https://mlsanity.com/paper/1603.08315/) (2016)
- [Outlier-Robust High-Dimensional Sparse Estimation via Iterative Filtering](https://mlsanity.com/paper/1911.08085/) (2019)
- [Efficiently Learning Adversarially Robust Halfspaces with Noise](https://mlsanity.com/paper/2005.07652/) (2020)
- [Robust Regression via Mutivariate Regression Depth](https://mlsanity.com/paper/1702.04656/) (2017)
- [Fast Mean Estimation with Sub-Gaussian Rates](https://mlsanity.com/paper/1902.01998/) (2019)
- [The Complexity of Adversarially Robust Proper Learning of Halfspaces with Agnostic Noise](https://mlsanity.com/paper/2007.15220/) (2020)
- [A Fast Spectral Algorithm for Mean Estimation with Sub-Gaussian Rates](https://mlsanity.com/paper/1908.04468/) (2019)
- [Sparse Representation of a Polytope and Recovery of Sparse Signals and Low-rank Matrices](https://mlsanity.com/paper/1306.1154/) (2013)
- [Robust Inference via Multiplier Bootstrap](https://mlsanity.com/paper/1903.07208/) (2019)

## Built on by (most cited first, 6 of 6)

- [High-dimensional robust regression under heavy-tailed data: Asymptotics and Universality](https://mlsanity.com/paper/2309.16476/) (2023)
- [Robust Methods for High-Dimensional Linear Learning](https://mlsanity.com/paper/2208.05447/) (2022)
- [Outlier Robust and Sparse Estimation of Linear Regression Coefficients](https://mlsanity.com/paper/2208.11592/) (2022)
- [Robust Sparse Regression with Non-Isotropic Designs](https://mlsanity.com/paper/2410.23937/) (2024)
- [Tractable Gaussian Phase Retrieval with Heavy Tails and Adversarial Corruption with Near-Linear Sample Complexity](https://mlsanity.com/paper/2601.18245/) (2026)
- [RobustLDS: Learning linear dynamical systems under adversarial corruptions](https://mlsanity.com/paper/2610.11906/) (2026)

## Abstract (arXiv)

Robust and sparse estimation of linear regression coefficients is investigated. The situation addressed by the present paper is that covariates and noises are sampled from heavy-tailed distributions, and the covariates and noises are contaminated by malicious outliers. Our estimator can be computed efficiently. Further, the error bound of the estimator is nearly optimal.