{"paper":{"title":"Phase Transition in the One-bit Johnson-Lindenstrauss Lemma","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.FA","authors_text":"Amadou Bah, Bryson Kagy, Emily Smith","submitted_at":"2019-03-06T00:15:49Z","abstract_excerpt":"The Johnson-Lindenstrauss Lemma (J-L Lemma) is a cornerstone of dimension reduction techniques. We study it in the one-bit context, namely we consider the unit sphere $ \\mathbb S ^{N-1}$, with normalized geodesic metric, and map a finite set $ \\mathbf{X} \\subset \\mathbb{S}^{N-1}$ into the Hamming cube $\\mathbb{H}_m = \\{0,1\\}^m$, with normalized Hamming metric. We find that for $ 0< \\delta <1$, and $m>\\frac{\\ln n}{2\\delta^2}$ there is a $\\delta$-RIP from $\\mathbf{X}$ into $\\mathbb{H}_m$. This is surprising as the value of $ m$ is virtually identical to best known bound linear J-L Lemma. In both"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.02123","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"}