{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/equispaced-fourier-representations-for","title":"Equispaced Fourier representations for efficient Gaussian process regression from a billion data points","arxiv_id":"2210.10210","date":"2022-10-18","proceeding":null,"authors":["Philip Greengard","Manas Rachh","Alex Barnett"],"abstract":"We introduce a Fourier-based fast algorithm for Gaussian process regression in low dimensions. It approximates a translationally-invariant covariance kernel by complex exponentials on an equispaced Cartesian frequency grid of $M$ nodes. This results in a weight-space $M\\times M$ system matrix with Toeplitz structure, which can thus be applied to a vector in ${\\mathcal O}(M \\log{M})$ operations via the fast Fourier transform (FFT), independent of the number of data points $N$. The linear system can be set up in ${\\mathcal O}(N + M \\log{M})$ operations using nonuniform FFTs. This enables efficient massive-scale regression via an iterative solver, even for kernels with fat-tailed spectral densities (large $M$). We provide bounds on both kernel approximation and posterior mean errors. Numerical experiments for squared-exponential and Mat\\'ern kernels in one, two and three dimensions often show 1-2 orders of magnitude acceleration over state-of-the-art rank-structured solvers at comparable accuracy. Our method allows 2D Mat\\'ern-$\\mbox{$\\frac{3}{2}$}$ regression from $N=10^9$ data points to be performed in 2 minutes on a standard desktop, with posterior mean accuracy $10^{-3}$. This opens up spatial statistics applications 100 times larger than previously possible.","url_abs":"https://arxiv.org/abs/2210.10210v2","url_pdf":"https://arxiv.org/pdf/2210.10210v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"equispaced-fourier-representations-for","repo_url":"https://github.com/flatironinstitute/gp-shootout","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"pytorch","reach":{"status":"ok","spdx":"NOASSERTION"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=2210.10210","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}