# Trigonometric Quadrature Fourier Features for Scalable Gaussian Process Regression

arXiv: https://arxiv.org/abs/2310.14544 · HTML: https://mlsanity.com/paper/2310.14544/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 63 references that are in the map.
- First posted on arXiv 23 Oct 2023.

## Builds on (most cited first, 26 of 26)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [GPflow: A Gaussian process library using TensorFlow](https://mlsanity.com/paper/1610.08733/) (2016)
- [Efficiently Sampling Functions from Gaussian Process Posteriors](https://mlsanity.com/paper/2002.09309/) (2020)
- [Orthogonal Random Features](https://mlsanity.com/paper/1610.09072/) (2016)
- [Distributed Gaussian Processes](https://mlsanity.com/paper/1502.02843/) (2015)
- [Federated Bayesian Optimization via Thompson Sampling](https://mlsanity.com/paper/2010.10154/) (2020)
- [A general framework for Vecchia approximations of Gaussian processes](https://mlsanity.com/paper/1708.06302/) (2017)
- [Pathwise Conditioning of Gaussian Processes](https://mlsanity.com/paper/2011.04026/) (2020)
- [Quasi-Monte Carlo Feature Maps for Shift-Invariant Kernels](https://mlsanity.com/paper/1412.8293/) (2014)
- [Random Feature Expansions for Deep Gaussian Processes](https://mlsanity.com/paper/1610.04386/) (2016)
- [Differentially Private Federated Bayesian Optimization with Distributed Exploration](https://mlsanity.com/paper/2110.14153/) (2021)
- [Weighted Gaussian Process Bandits for Non-stationary Environments](https://mlsanity.com/paper/2107.02371/) (2021)
- [Sampling from Gaussian Process Posteriors using Stochastic Gradient Descent](https://mlsanity.com/paper/2306.11589/) (2023)
- [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)
- [Bayesian Optimization under Heavy-tailed Payoffs](https://mlsanity.com/paper/1909.07040/) (2019)
- [Additive Gaussian Processes Revisited](https://mlsanity.com/paper/2206.09861/) (2022)
- [Spatial Mapping with Gaussian Processes and Nonstationary Fourier Features](https://mlsanity.com/paper/1711.05615/) (2017)
- [A hierarchical expected improvement method for Bayesian optimization](https://mlsanity.com/paper/1911.07285/) (2019)
- [No-Regret Algorithms for Private Gaussian Process Bandit Optimization](https://mlsanity.com/paper/2102.12467/) (2021)
- [Low-Precision Arithmetic for Fast Gaussian Processes](https://mlsanity.com/paper/2207.06856/) (2022)
- [Variational Nearest Neighbor Gaussian Process](https://mlsanity.com/paper/2202.01694/) (2022)
- [Additive Multi-Index Gaussian process modeling, with application to multi-physics surrogate modeling of the quark-gluon plasma](https://mlsanity.com/paper/2306.07299/) (2023)
- [Gauss-Legendre Features for Gaussian Process Regression](https://mlsanity.com/paper/2101.01137/) (2021)
- [Bayesian Non-linear Latent Variable Modeling via Random Fourier Features](https://mlsanity.com/paper/2306.08352/) (2023)

## Built on by (most cited first, 2 of 2)

- [Scalable Deep Basis Kernel Gaussian Processes](https://mlsanity.com/paper/2505.18526/) (2025)
- [Certifiable Safe Model-Based Reinforcement Learning with Control-Affine Dynamics Approximation](https://mlsanity.com/paper/2607.16501/) (2026)

## Abstract (arXiv)

Fourier feature approximations have been successfully applied in the literature for scalable Gaussian Process (GP) regression. In particular, Quadrature Fourier Features (QFF) derived from Gaussian quadrature rules have gained popularity in recent years due to their improved approximation accuracy and better calibrated uncertainty estimates compared to Random Fourier Feature (RFF) methods. However, a key limitation of QFF is that its performance can suffer from well-known pathologies related to highly oscillatory quadrature, resulting in mediocre approximation with limited features. We address this critical issue via a new Trigonometric Quadrature Fourier Feature (TQFF) method, which uses a novel non-Gaussian quadrature rule specifically tailored for the desired Fourier transform. We derive an exact quadrature rule for TQFF, along with kernel approximation error bounds for the resulting feature map. We then demonstrate the improved performance of our method over RFF and Gaussian QFF in a suite of numerical experiments and applications, and show the TQFF enjoys accurate GP approximations over a broad range of length-scales using fewer features.