{"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/the-fast-johnson-lindenstrauss-transform-is","title":"The Fast Johnson-Lindenstrauss Transform is Even Faster","arxiv_id":"2204.01800","date":"2022-04-04","proceeding":null,"authors":["Ora Nova Fandina","Mikael Møller Høgsgaard","Kasper Green Larsen"],"abstract":"The seminal Fast Johnson-Lindenstrauss (Fast JL) transform by Ailon and Chazelle (SICOMP'09) embeds a set of $n$ points in $d$-dimensional Euclidean space into optimal $k=O(\\varepsilon^{-2} \\ln n)$ dimensions, while preserving all pairwise distances to within a factor $(1 \\pm \\varepsilon)$. The Fast JL transform supports computing the embedding of a data point in $O(d \\ln d +k \\ln^2 n)$ time, where the $d \\ln d$ term comes from multiplication with a $d \\times d$ Hadamard matrix and the $k \\ln^2 n$ term comes from multiplication with a sparse $k \\times d$ matrix. Despite the Fast JL transform being more than a decade old, it is one of the fastest dimensionality reduction techniques for many tradeoffs between $\\varepsilon, d$ and $n$. In this work, we give a surprising new analysis of the Fast JL transform, showing that the $k \\ln^2 n$ term in the embedding time can be improved to $(k \\ln^2 n)/\\alpha$ for an $\\alpha = \\Omega(\\min\\{\\varepsilon^{-1}\\ln(1/\\varepsilon), \\ln n\\})$. The improvement follows by using an even sparser matrix. We also complement our improved analysis with a lower bound showing that our new analysis is in fact tight.","url_abs":"https://arxiv.org/abs/2204.01800v1","url_pdf":"https://arxiv.org/pdf/2204.01800v1.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":"abstracts"},"code_links":[{"paper_slug":"the-fast-johnson-lindenstrauss-transform-is","repo_url":"https://github.com/AntoAndGar/Intrinsic-Dimension","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"pytorch","reach":{"status":"ok"}}],"tasks":[{"task_slug":"dimensionality-reduction","task_name":"Dimensionality Reduction"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=2204.01800","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}