{"paper":{"title":"Near-Optimal Bounds for Binary Embeddings of Arbitrary Sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS","math.FA"],"primary_cat":"cs.LG","authors_text":"Ben Recht, Samet Oymak","submitted_at":"2015-12-14T17:57:41Z","abstract_excerpt":"We study embedding a subset $K$ of the unit sphere to the Hamming cube $\\{-1,+1\\}^m$. We characterize the tradeoff between distortion and sample complexity $m$ in terms of the Gaussian width $\\omega(K)$ of the set. For subspaces and several structured sets we show that Gaussian maps provide the optimal tradeoff $m\\sim \\delta^{-2}\\omega^2(K)$, in particular for $\\delta$ distortion one needs $m\\approx\\delta^{-2}{d}$ where $d$ is the subspace dimension. For general sets, we provide sharp characterizations which reduces to $m\\approx{\\delta^{-4}}{\\omega^2(K)}$ after simplification. We provide impro"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.04433","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"}