# Online and Offline Robust Multivariate Linear Regression

arXiv: https://arxiv.org/abs/2404.19496 · HTML: https://mlsanity.com/paper/2404.19496/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 51 references that are in the map.
- First posted on arXiv 30 Apr 2024.

## Builds on (most cited first, 6 of 6)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Non-strongly-convex smooth stochastic approximation with convergence rate O(1/n)](https://mlsanity.com/paper/1306.2119/) (2013)
- [Online Robust Regression via SGD on the l1 loss](https://mlsanity.com/paper/2007.00399/) (2020)
- [The shooting S-estimator for robust regression](https://mlsanity.com/paper/1506.01223/) (2015)
- [Estimation of large block structured covariance matrices: Application to "multi-omic" approaches to study seed quality](https://mlsanity.com/paper/1806.10093/) (2018)
- [A robust model-based clustering based on the geometric median and the Median Covariation Matrix](https://mlsanity.com/paper/2211.08131/) (2022)

## Abstract (arXiv)

We consider the robust estimation of the parameters of multivariate Gaussian linear regression models. To this aim we consider robust version of the usual (Mahalanobis) least-square criterion, with or without Ridge regularization. We introduce two methods each considered contrast: (i) online stochastic gradient descent algorithms and their averaged versions and (ii) offline fix-point algorithms. Under weak assumptions, we prove the asymptotic normality of the resulting estimates. Because the variance matrix of the noise is usually unknown, we propose to plug a robust estimate of it in the Mahalanobis-based stochastic gradient descent algorithms. We show, on synthetic data, the dramatic gain in terms of robustness of the proposed estimates as compared to the classical least-square ones. Well also show the computational efficiency of the online versions of the proposed algorithms. All the proposed algorithms are implemented in the R package RobRegression available on CRAN.