# A statistical perspective of sampling scores for linear regression

arXiv: https://arxiv.org/abs/1507.05870 · HTML: https://mlsanity.com/paper/1507.05870/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Jul 2015.

## Built on by (most cited first, 5 of 5)

- [CS4ML: A general framework for active learning with arbitrary data based on Christoffel functions](https://mlsanity.com/paper/2306.00945/) (2023)
- [Optimal sampling for least-squares approximation](https://mlsanity.com/paper/2409.02342/) (2024)
- [A unified framework for learning with nonlinear model classes from arbitrary linear samples](https://mlsanity.com/paper/2311.14886/) (2023)
- [Active Learning for Conditional Generative Compressed Sensing](https://mlsanity.com/paper/2605.05435/) (2026)
- [Christoffel-DPS: Optimal sensor placement in diffusion posterior sampling for arbitrary distributions](https://mlsanity.com/paper/2605.06861/) (2026)

## Abstract (arXiv)

In this paper, we consider a statistical problem of learning a linear model from noisy samples. Existing work has focused on approximating the least squares solution by using leverage-based scores as an importance sampling distribution. However, no finite sample statistical guarantees and no computationally efficient optimal sampling strategies have been proposed. To evaluate the statistical properties of different sampling strategies, we propose a simple yet effective estimator, which is easy for theoretical analysis and is useful in multitask linear regression. We derive the exact mean square error of the proposed estimator for any given sampling scores. Based on minimizing the mean square error, we propose the optimal sampling scores for both estimator and predictor, and show that they are influenced by the noise-to-signal ratio. Numerical simulations match the theoretical analysis well.