{"paper":{"title":"Toward a unified theory of sparse dimensionality reduction in Euclidean space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG","cs.IT","math.IT","math.PR","stat.ML"],"primary_cat":"cs.DS","authors_text":"Jean Bourgain, Jelani Nelson, Sjoerd Dirksen","submitted_at":"2013-11-11T19:30:30Z","abstract_excerpt":"Let $\\Phi\\in\\mathbb{R}^{m\\times n}$ be a sparse Johnson-Lindenstrauss transform [KN14] with $s$ non-zeroes per column. For a subset $T$ of the unit sphere, $\\varepsilon\\in(0,1/2)$ given, we study settings for $m,s$ required to ensure $$ \\mathop{\\mathbb{E}}_\\Phi \\sup_{x\\in T} \\left|\\|\\Phi x\\|_2^2 - 1 \\right| < \\varepsilon , $$ i.e. so that $\\Phi$ preserves the norm of every $x\\in T$ simultaneously and multiplicatively up to $1+\\varepsilon$. We introduce a new complexity parameter, which depends on the geometry of $T$, and show that it suffices to choose $s$ and $m$ such that this parameter is s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.2542","kind":"arxiv","version":4},"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"}