# RIGID: Robust Linear Regression with Missing Data

arXiv: https://arxiv.org/abs/2205.13635 · HTML: https://mlsanity.com/paper/2205.13635/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 43 references that are in the map.
- First posted on arXiv 26 May 2022.

## Builds on (most cited first, 13 of 13)

- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Missing Data Imputation using Optimal Transport](https://mlsanity.com/paper/2002.03860/) (2020)
- [Sparse DNNs with Improved Adversarial Robustness](https://mlsanity.com/paper/1810.09619/) (2018)
- [What's a good imputation to predict with missing values?](https://mlsanity.com/paper/2106.00311/) (2021)
- [On the consistency of supervised learning with missing values](https://mlsanity.com/paper/1902.06931/) (2019)
- [Imputation for High-Dimensional Linear Regression](https://mlsanity.com/paper/2001.09180/) (2020)
- [Combating Adversarial Attacks Using Sparse Representations](https://mlsanity.com/paper/1803.03880/) (2018)
- [Fast Convex Pruning of Deep Neural Networks](https://mlsanity.com/paper/1806.06457/) (2018)
- [Robust Linear Regression: A Review and Comparison](https://mlsanity.com/paper/1404.6274/) (2014)
- [Sparse spectral estimation with missing and corrupted measurements](https://mlsanity.com/paper/1811.10443/) (2018)
- [Minimax Rate-optimal Estimation of High-dimensional Covariance Matrices with Incomplete Data](https://mlsanity.com/paper/1605.04358/) (2016)
- [Precision Matrix Estimation with Noisy and Missing Data](https://mlsanity.com/paper/1904.03548/) (2019)

## Built on by (most cited first, 2 of 2)

- [Specifying and Solving Robust Empirical Risk Minimization Problems Using CVXPY](https://mlsanity.com/paper/2306.05649/) (2023)
- [Learning Over Dirty Data with Minimal Repairs](https://mlsanity.com/paper/2503.13921/) (2025)

## Abstract (arXiv)

We present a robust framework to perform linear regression with missing entries in the features. By considering an elliptical data distribution, and specifically a multivariate normal model, we are able to conditionally formulate a distribution for the missing entries and present a robust framework, which minimizes the worst case error caused by the uncertainty about the missing data. We show that the proposed formulation, which naturally takes into account the dependency between different variables, ultimately reduces to a convex program, for which a customized and scalable solver can be delivered. In addition to a detailed analysis to deliver such solver, we also asymptoticly analyze the behavior of the proposed framework, and present technical discussions to estimate the required input parameters. We complement our analysis with experiments performed on synthetic, semi-synthetic, and real data, and show how the proposed formulation improves the prediction accuracy and robustness, and outperforms the competing techniques. Missing data is a common problem associated with many datasets in machine learning. With the significant increase in using robust optimization techniques to train machine learning models, this paper presents a novel robust regression framework that operates by minimizing the uncertainty associated with missing data. The proposed approach allows training models with incomplete data, while minimizing the impact of uncertainty associated with the unavailable data. The ideas developed in this paper can be generalized beyond linear models and elliptical data distributions.