# Linear predictor on linearly-generated data with missing values: non consistency and solutions

arXiv: https://arxiv.org/abs/2002.00658 · HTML: https://mlsanity.com/paper/2002.00658/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Feb 2020.

## Built on by (most cited first, 20 of 20)

- [What's a good imputation to predict with missing values?](https://mlsanity.com/paper/2106.00311/) (2021)
- [Conformal Prediction with Missing Values](https://mlsanity.com/paper/2306.02732/) (2023)
- [Benchmarking missing-values approaches for predictive models on health databases](https://mlsanity.com/paper/2202.10580/) (2022)
- [The Missing Indicator Method: From Low to High Dimensions](https://mlsanity.com/paper/2211.09259/) (2022)
- [Domain Adaptation under Missingness Shift](https://mlsanity.com/paper/2211.02093/) (2022)
- [Sharing pattern submodels for prediction with missing values](https://mlsanity.com/paper/2206.11161/) (2022)
- [Random features models: a way to study the success of naive imputation](https://mlsanity.com/paper/2402.03839/) (2024)
- [Interpretable Generalized Additive Models for Datasets with Missing Values](https://mlsanity.com/paper/2412.02646/) (2024)
- [Are labels informative in semi-supervised learning? -- Estimating and leveraging the missing-data mechanism](https://mlsanity.com/paper/2302.07540/) (2023)
- [Adapting Fairness Interventions to Missing Values](https://mlsanity.com/paper/2305.19429/) (2023)
- [Knockout: A simple way to handle missing inputs](https://mlsanity.com/paper/2405.20448/) (2024)
- [MINTY: Rule-based Models that Minimize the Need for Imputing Features with Missing Values](https://mlsanity.com/paper/2311.14108/) (2023)
- [Simple Imputation Rules for Prediction with Missing Data: Contrasting Theoretical Guarantees with Empirical Performance](https://mlsanity.com/paper/2104.03158/) (2021)
- [Adaptive Optimization for Prediction with Missing Data](https://mlsanity.com/paper/2402.01543/) (2024)
- [Minimax rate of consistency for linear models with missing values](https://mlsanity.com/paper/2202.01463/) (2022)
- [When Pattern-by-Pattern Works: Theoretical and Empirical Insights for Logistic Models with Missing Values](https://mlsanity.com/paper/2507.13024/) (2025)
- [Linear Regression under Missing or Corrupted Coordinates](https://mlsanity.com/paper/2509.19242/) (2025)
- [Enhancing Mortality Prediction in Heart Failure Patients: Exploring Preprocessing Methods for Imbalanced Clinical Datasets](https://mlsanity.com/paper/2310.00457/) (2023)
- [Learning from Complementary Features](https://mlsanity.com/paper/2408.14788/) (2024)
- [Handling Covariate Mismatch in Collaborative Linear Prediction](https://mlsanity.com/paper/2602.02083/) (2026)

## Abstract (arXiv)

We consider building predictors when the data have missing values. We study the seemingly-simple case where the target to predict is a linear function of the fully-observed data and we show that, in the presence of missing values, the optimal predictor may not be linear. In the particular Gaussian case, it can be written as a linear function of multiway interactions between the observed data and the various missing-value indicators. Due to its intrinsic complexity, we study a simple approximation and prove generalization bounds with finite samples, highlighting regimes for which each method performs best. We then show that multilayer perceptrons with ReLU activation functions can be consistent, and can explore good trade-offs between the true model and approximations. Our study highlights the interesting family of models that are beneficial to fit with missing values depending on the amount of data available.