# Gauss-Legendre Features for Gaussian Process Regression

arXiv: https://arxiv.org/abs/2101.01137 · HTML: https://mlsanity.com/paper/2101.01137/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 50 references that are in the map.
- First posted on arXiv 4 Jan 2021.

## Builds on (most cited first, 11 of 11)

- [Kernel Interpolation for Scalable Structured Gaussian Processes (KISS-GP)](https://mlsanity.com/paper/1503.01057/) (2015)
- [On the Equivalence between Kernel Quadrature Rules and Random Feature Expansions](https://mlsanity.com/paper/1502.06800/) (2015)
- [On the Error of Random Fourier Features](https://mlsanity.com/paper/1506.02785/) (2015)
- [Random Fourier Features for Kernel Ridge Regression: Approximation Bounds and Statistical Guarantees](https://mlsanity.com/paper/1804.09893/) (2018)
- [Quasi-Monte Carlo Feature Maps for Shift-Invariant Kernels](https://mlsanity.com/paper/1412.8293/) (2014)
- [Optimal Rates for Random Fourier Features](https://mlsanity.com/paper/1506.02155/) (2015)
- [Scalable Log Determinants for Gaussian Process Kernel Learning](https://mlsanity.com/paper/1711.03481/) (2017)
- [Quadrature-based features for kernel approximation](https://mlsanity.com/paper/1802.03832/) (2018)
- [Gaussian Quadrature for Kernel Features](https://mlsanity.com/paper/1709.02605/) (2017)
- [Spatial Mapping with Gaussian Processes and Nonstationary Fourier Features](https://mlsanity.com/paper/1711.05615/) (2017)
- [High-performance Kernel Machines with Implicit Distributed Optimization and Randomization](https://mlsanity.com/paper/1409.0940/) (2014)

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

- [Large-Scale Learning with Fourier Features and Tensor Decompositions](https://mlsanity.com/paper/2109.01545/) (2021)
- [Trigonometric Quadrature Fourier Features for Scalable Gaussian Process Regression](https://mlsanity.com/paper/2310.14544/) (2023)
- [On the design of scalable, high-precision spherical-radial Fourier features](https://mlsanity.com/paper/2408.13231/) (2024)
- [Exploiting Hankel-Toeplitz Structures for Fast Computation of Kernel Precision Matrices](https://mlsanity.com/paper/2408.02346/) (2024)
- [Unbiased Stochastic Optimization for Gaussian Processes on Finite Dimensional RKHS](https://mlsanity.com/paper/2508.20588/) (2025)

## Abstract (arXiv)

Gaussian processes provide a powerful probabilistic kernel learning framework, which allows learning high quality nonparametric regression models via methods such as Gaussian process regression. Nevertheless, the learning phase of Gaussian process regression requires massive computations which are not realistic for large datasets. In this paper, we present a Gauss-Legendre quadrature based approach for scaling up Gaussian process regression via a low rank approximation of the kernel matrix. We utilize the structure of the low rank approximation to achieve effective hyperparameter learning, training and prediction. Our method is very much inspired by the well-known random Fourier features approach, which also builds low-rank approximations via numerical integration. However, our method is capable of generating high quality approximation to the kernel using an amount of features which is poly-logarithmic in the number of training points, while similar guarantees will require an amount that is at the very least linear in the number of training points when random Fourier features. Furthermore, the structure of the low-rank approximation that our method builds is subtly different from the one generated by random Fourier features, and this enables much more efficient hyperparameter learning. The utility of our method for learning with low-dimensional datasets is demonstrated using numerical experiments.