{"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/fast-direct-methods-for-gaussian-processes","title":"Fast Direct Methods for Gaussian Processes","arxiv_id":"1403.6015","date":"2014-03-24","proceeding":null,"authors":["Sivaram Ambikasaran","Daniel Foreman-Mackey","Leslie Greengard","David W. Hogg","Michael O'Neil"],"abstract":"A number of problems in probability and statistics can be addressed using the multivariate normal (Gaussian) distribution. In the one-dimensional case, computing the probability for a given mean and variance simply requires the evaluation of the corresponding Gaussian density. In the $n$-dimensional setting, however, it requires the inversion of an $n \\times n$ covariance matrix, $C$, as well as the evaluation of its determinant, $\\det(C)$. In many cases, such as regression using Gaussian processes, the covariance matrix is of the form $C = \\sigma^2 I + K$, where $K$ is computed using a specified covariance kernel which depends on the data and additional parameters (hyperparameters). The matrix $C$ is typically dense, causing standard direct methods for inversion and determinant evaluation to require $\\mathcal O(n^3)$ work. This cost is prohibitive for large-scale modeling. Here, we show that for the most commonly used covariance functions, the matrix $C$ can be hierarchically factored into a product of block low-rank updates of the identity matrix, yielding an $\\mathcal O (n\\log^2 n) $ algorithm for inversion. More importantly, we show that this factorization enables the evaluation of the determinant $\\det(C)$, permitting the direct calculation of probabilities in high dimensions under fairly broad assumptions on the kernel defining $K$. Our fast algorithm brings many problems in marginalization and the adaptation of hyperparameters within practical reach using a single CPU core. The combination of nearly optimal scaling in terms of problem size with high-performance computing resources will permit the modeling of previously intractable problems. We illustrate the performance of the scheme on standard covariance kernels.","url_abs":"http://arxiv.org/abs/1403.6015v2","url_pdf":"http://arxiv.org/pdf/1403.6015v2.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":"fast-direct-methods-for-gaussian-processes","repo_url":"https://github.com/sivaramambikasaran/HODLR","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"NOASSERTION"}},{"paper_slug":"fast-direct-methods-for-gaussian-processes","repo_url":"https://github.com/transientlunatic/heron","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"pytorch","reach":{"status":"ok","spdx":"ISC"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1403.6015","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"1403.6015"}},"developers":"https://syntology.ai/developers","read_at":"2026-09-24T18:15:14+00:00","read_at_is":"when the build read Syntology's graph, not when any sample ran","claim":"Per-sample execution status on synthesized fixtures; not a correctness claim about the paper. 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