# Integrated Variational Fourier Features for Fast Spatial Modelling with Gaussian Processes

arXiv: https://arxiv.org/abs/2308.14142 · HTML: https://mlsanity.com/paper/2308.14142/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 31 references that are in the map.
- First posted on arXiv 27 Aug 2023.

## Builds on (most cited first, 19 of 19)

- [GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Acceleration](https://mlsanity.com/paper/1809.11165/) (2018)
- [Scalable Variational Gaussian Process Classification](https://mlsanity.com/paper/1411.2005/) (2014)
- [Kernel Interpolation for Scalable Structured Gaussian Processes (KISS-GP)](https://mlsanity.com/paper/1503.01057/) (2015)
- [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)
- [Hilbert Space Methods for Reduced-Rank Gaussian Process Regression](https://mlsanity.com/paper/1401.5508/) (2014)
- [Variational Fourier features for Gaussian processes](https://mlsanity.com/paper/1611.06740/) (2016)
- [Rates of Convergence for Sparse Variational Gaussian Process Regression](https://mlsanity.com/paper/1903.03571/) (2019)
- [On Sparse variational methods and the Kullback-Leibler divergence between stochastic processes](https://mlsanity.com/paper/1504.07027/) (2015)
- [Convergence of Sparse Variational Inference in Gaussian Processes Regression](https://mlsanity.com/paper/2008.00323/) (2020)
- [Constant-Time Predictive Distributions for Gaussian Processes](https://mlsanity.com/paper/1803.06058/) (2018)
- [Sparse Gaussian Processes with Spherical Harmonic Features](https://mlsanity.com/paper/2006.16649/) (2020)
- [Product Kernel Interpolation for Scalable Gaussian Processes](https://mlsanity.com/paper/1802.08903/) (2018)
- [Sparse within Sparse Gaussian Processes using Neighbor Information](https://mlsanity.com/paper/2011.05041/) (2020)
- [Improving the Gaussian Process Sparse Spectrum Approximation by Representing Uncertainty in Frequency Inputs](https://mlsanity.com/paper/1503.02424/) (2015)
- [Variational Orthogonal Features](https://mlsanity.com/paper/2006.13170/) (2020)
- [Log-Linear-Time Gaussian Processes Using Binary Tree Kernels](https://mlsanity.com/paper/2210.01633/) (2022)
- [Actually Sparse Variational Gaussian Processes](https://mlsanity.com/paper/2304.05091/) (2023)
- [B'ezier Gaussian Processes for Tall and Wide Data](https://mlsanity.com/paper/2209.00343/) (2022)

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

- [Scalable Random Wavelet Features: Efficient Non-Stationary Kernel Approximation with Convergence Guarantees](https://mlsanity.com/paper/2602.00987/) (2026)
- [Adaptive RKHS Fourier Features for Compositional Gaussian Process Models](https://mlsanity.com/paper/2407.01856/) (2024)
- [Stein Random Feature Regression](https://mlsanity.com/paper/2406.00438/) (2024)

## Abstract (arXiv)

Sparse variational approximations are popular methods for scaling up inference and learning in Gaussian processes to larger datasets. For N training points, exact inference has O(N3) cost; with M ll N features, state of the art sparse variational methods have O(NM2) cost. Recently, methods have been proposed using more sophisticated features; these promise O(M3) cost, with good performance in low dimensional tasks such as spatial modelling, but they only work with a very limited class of kernels, excluding some of the most commonly used. In this work, we propose integrated Fourier features, which extends these performance benefits to a very broad class of stationary covariance functions. We motivate the method and choice of parameters from a convergence analysis and empirical exploration, and show practical speedup in synthetic and real world spatial regression tasks.