# On the Approximation of Kernel functions

arXiv: https://arxiv.org/abs/2403.06731 · HTML: https://mlsanity.com/paper/2403.06731/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 25 references that are in the map.
- First posted on arXiv 11 Mar 2024.

## Builds on (most cited first, 5 of 5)

- [Sobolev Norm Learning Rates for Regularized Least-Squares Algorithm](https://mlsanity.com/paper/1702.07254/) (2017)
- [Less is More: Nystr"om Computational Regularization](https://mlsanity.com/paper/1507.04717/) (2015)
- [FALKON: An Optimal Large Scale Kernel Method](https://mlsanity.com/paper/1705.10958/) (2017)
- [Statistical Optimality and Computational Efficiency of Nystr"om Kernel PCA](https://mlsanity.com/paper/2105.08875/) (2021)
- [A Bound on the Maximal Marginal Degrees of Freedom](https://mlsanity.com/paper/2402.12885/) (2024)

## Built on by (most cited first, 3 of 3)

- [Fixed-Gaussian Spectral Algorithms: Minimax Optimal Rates for Misspecified Learning and Transfer](https://mlsanity.com/paper/2501.10870/) (2025)
- [Uniform convergence for Gaussian kernel ridge regression](https://mlsanity.com/paper/2508.11274/) (2025)
- [Generalized nonparametric regression in reproducing kernel Hilbert spaces: Consistency and rates of convergence](https://mlsanity.com/paper/2606.22993/) (2026)

## Abstract (arXiv)

Various methods in statistical learning build on kernels considered in reproducing kernel Hilbert spaces. In applications, the kernel is often selected based on characteristics of the problem and the data. This kernel is then employed to infer response variables at points, where no explanatory data were observed. The data considered here are located in compact sets in higher dimensions and the paper addresses approximations of the kernel itself. The new approach considers Taylor series approximations of radial kernel functions. For the Gauss kernel on the unit cube, the paper establishes an upper bound of the associated eigenfunctions, which grows only polynomially with respect to the index. The novel approach substantiates smaller regularization parameters than considered in the literature, overall leading to better approximations. This improvement confirms low rank approximation methods such as the Nystr"om method.