# Relating Leverage Scores and Density using Regularized Christoffel Functions

arXiv: https://arxiv.org/abs/1805.07943 · HTML: https://mlsanity.com/paper/1805.07943/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 May 2018.

## Built on by (most cited first, 10 of 10)

- [Information Theory with Kernel Methods](https://mlsanity.com/paper/2202.08545/) (2022)
- [Fast Statistical Leverage Score Approximation in Kernel Ridge Regression](https://mlsanity.com/paper/2103.05238/) (2021)
- [Chameleon: A Flexible Data-mixing Framework for Language Model Pretraining and Finetuning](https://mlsanity.com/paper/2505.24844/) (2025)
- [On regularized Radon-Nikodym differentiation](https://mlsanity.com/paper/2308.07887/) (2023)
- [Efficient Numerical Integration in Reproducing Kernel Hilbert Spaces via Leverage Scores Sampling](https://mlsanity.com/paper/2311.13548/) (2023)
- [A Scalable Nystrom-Based Kernel Two-Sample Test with Permutations](https://mlsanity.com/paper/2502.13570/) (2025)
- [On the relationship between multivariate splines and infinitely-wide neural networks](https://mlsanity.com/paper/2302.03459/) (2023)
- [Optimal Learning Rates for Regularized Least-Squares with a Fourier Capacity Condition](https://mlsanity.com/paper/2204.07856/) (2022)
- [General regularization in covariate shift adaptation](https://mlsanity.com/paper/2307.11503/) (2023)
- [Signal reconstruction using determinantal sampling](https://mlsanity.com/paper/2310.09437/) (2023)

## Abstract (arXiv)

Statistical leverage scores emerged as a fundamental tool for matrix sketching and column sampling with applications to low rank approximation, regression, random feature learning and quadrature. Yet, the very nature of this quantity is barely understood. Borrowing ideas from the orthogonal polynomial literature, we introduce the regularized Christoffel function associated to a positive definite kernel. This uncovers a variational formulation for leverage scores for kernel methods and allows to elucidate their relationships with the chosen kernel as well as population density. Our main result quantitatively describes a decreasing relation between leverage score and population density for a broad class of kernels on Euclidean spaces. Numerical simulations support our findings.