2026-07-25T02:44:17Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:17683 2024-03-06T11:00:18Z ddc:004 open_access

Barriers for Faster Dimensionality Reduction Nova Fandina, Ora Møller Høgsgaard, Mikael Green Larsen, Kasper Dimensional reduction Lower bound Linear Circuits The Johnson-Lindenstrauss transform allows one to embed a dataset of n points in ℝ^d into ℝ^m, while preserving the pairwise distance between any pair of points up to a factor (1 ± ε), provided that m = Ω(ε^{-2} lg n). The transform has found an overwhelming number of algorithmic applications, allowing to speed up algorithms and reducing memory consumption at the price of a small loss in accuracy. A central line of research on such transforms, focus on developing fast embedding algorithms, with the classic example being the Fast JL transform by Ailon and Chazelle. All known such algorithms have an embedding time of Ω(d lg d), but no lower bounds rule out a clean O(d) embedding time. In this work, we establish the first non-trivial lower bounds (of magnitude Ω(m lg m)) for a large class of embedding algorithms, including in particular most known upper bounds. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Ora Nova Fandina and Mikael Møller Høgsgaard and Kasper Green Larsen 2023 Is Part Of LIPIcs, Volume 254, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.STACS.2023.31 urn:nbn:de:0030-drops-176838 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023.31 eng https://creativecommons.org/licenses/by/4.0/legalcode