{"paper":{"title":"Optimal Oblivious Subspace Embeddings with Near-optimal Sparsity","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.LG","cs.NA","math.NA","math.PR","stat.ML"],"primary_cat":"cs.DS","authors_text":"Micha{\\l} Derezi\\'nski, Shabarish Chenakkod, Xiaoyu Dong","submitted_at":"2024-11-13T16:58:51Z","abstract_excerpt":"An oblivious subspace embedding is a random $m\\times n$ matrix $\\Pi$ such that, for any $d$-dimensional subspace, with high probability $\\Pi$ preserves the norms of all vectors in that subspace within a $1\\pm\\epsilon$ factor. In this work, we give an oblivious subspace embedding with the optimal dimension $m=\\Theta(d/\\epsilon^2)$ that has a near-optimal sparsity of $\\tilde O(1/\\epsilon)$ non-zero entries per column of $\\Pi$. This is the first result to nearly match the conjecture of Nelson and Nguyen [FOCS 2013] in terms of the best sparsity attainable by an optimal oblivious subspace embeddin"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.08773","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2411.08773/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}