{"paper":{"title":"The Johnson-Lindenstrauss lemma is optimal for linear dimensionality reduction","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG","cs.DS","math.FA","math.IT"],"primary_cat":"cs.IT","authors_text":"Jelani Nelson, Kasper Green Larsen","submitted_at":"2014-11-10T12:53:41Z","abstract_excerpt":"For any $n>1$ and $0<\\varepsilon<1/2$, we show the existence of an $n^{O(1)}$-point subset $X$ of $\\mathbb{R}^n$ such that any linear map from $(X,\\ell_2)$ to $\\ell_2^m$ with distortion at most $1+\\varepsilon$ must have $m = \\Omega(\\min\\{n, \\varepsilon^{-2}\\log n\\})$. Our lower bound matches the upper bounds provided by the identity matrix and the Johnson-Lindenstrauss lemma, improving the previous lower bound of Alon by a $\\log(1/\\varepsilon)$ factor."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.2404","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}