# Scalable Gaussian Processes with Grid-Structured Eigenfunctions (GP-GRIEF)

arXiv: https://arxiv.org/abs/1807.02125 · HTML: https://mlsanity.com/paper/1807.02125/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Jul 2018.

## Built on by (most cited first, 9 of 9)

- [Posterior and Computational Uncertainty in Gaussian Processes](https://mlsanity.com/paper/2205.15449/) (2022)
- [Kernel Interpolation for Scalable Online Gaussian Processes](https://mlsanity.com/paper/2103.01454/) (2021)
- [Data-Driven Design for Metamaterials and Multiscale Systems: A Review](https://mlsanity.com/paper/2307.05506/) (2023)
- [Scalable Variational Gaussian Processes via Harmonic Kernel Decomposition](https://mlsanity.com/paper/2106.05992/) (2021)
- [Hierarchical Inducing Point Gaussian Process for Inter-domain Observations](https://mlsanity.com/paper/2103.00393/) (2021)
- [Semi-Parametric Inducing Point Networks and Neural Processes](https://mlsanity.com/paper/2205.11718/) (2022)
- [Input Dependent Sparse Gaussian Processes](https://mlsanity.com/paper/2107.07281/) (2021)
- [Adaptive Sparse Gaussian Process](https://mlsanity.com/paper/2302.10325/) (2023)
- [New Bounds for Sparse Variational Gaussian Processes](https://mlsanity.com/paper/2502.08730/) (2025)

## Abstract (arXiv)

We introduce a kernel approximation strategy that enables computation of the Gaussian process log marginal likelihood and all hyperparameter derivatives in O(p) time. Our GRIEF kernel consists of p eigenfunctions found using a Nystrom approximation from a dense Cartesian product grid of inducing points. By exploiting algebraic properties of Kronecker and Khatri-Rao tensor products, computational complexity of the training procedure can be practically independent of the number of inducing points. This allows us to use arbitrarily many inducing points to achieve a globally accurate kernel approximation, even in high-dimensional problems. The fast likelihood evaluation enables type-I or II Bayesian inference on large-scale datasets. We benchmark our algorithms on real-world problems with up to two-million training points and 1033 inducing points.