# Estimation of sparse linear regression coefficients under L-subexponential covariates

arXiv: https://arxiv.org/abs/2304.11958 · HTML: https://mlsanity.com/paper/2304.11958/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 25 references that are in the map.
- First posted on arXiv 24 Apr 2023.

## Builds on (most cited first, 8 of 8)

- [CVXPY: A Python-Embedded Modeling Language for Convex Optimization](https://mlsanity.com/paper/1603.00943/) (2016)
- [Adaptive Huber Regression](https://mlsanity.com/paper/1706.06991/) (2017)
- [Moving Beyond Sub-Gaussianity in High-Dimensional Statistics: Applications in Covariance Estimation and Linear Regression](https://mlsanity.com/paper/1804.02605/) (2018)
- [Outlier-robust estimation of a sparse linear model using ℓ1-penalized Huber's M-estimator](https://mlsanity.com/paper/1904.06288/) (2019)
- [A Shrinkage Principle for Heavy-Tailed Data: High-Dimensional Robust Low-Rank Matrix Recovery](https://mlsanity.com/paper/1603.08315/) (2016)
- [Learning Compact Neural Networks with Regularization](https://mlsanity.com/paper/1802.01223/) (2018)
- [Robust Inference via Multiplier Bootstrap](https://mlsanity.com/paper/1903.07208/) (2019)
- [Generic Error Bounds for the Generalized Lasso with Sub-Exponential Data](https://mlsanity.com/paper/2004.05361/) (2020)

## Abstract (arXiv)

We tackle estimating sparse coefficients in a linear regression when the covariates are sampled from an L-subexponential random vector. This vector belongs to a class of distributions that exhibit heavier tails than Gaussian random vector. Previous studies have established error bounds similar to those derived for Gaussian random vectors. However, these methods require stronger conditions than those used for Gaussian random vectors to derive the error bounds. In this study, we present an error bound identical to the one obtained for Gaussian random vectors up to constant factors without imposing stronger conditions, when the covariates are drawn from an L-subexponential random vector. Interestingly, we employ an ℓ1-penalized Huber regression, which is known for its robustness against heavy-tailed random noises rather than covariates. We believe that this study uncovers a new aspect of the ℓ1-penalized Huber regression method.