{"paper":{"title":"Tight Bounds for $\\ell_p$ Oblivious Subspace Embeddings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"David P. Woodruff, Ruosong Wang","submitted_at":"2018-01-13T10:40:05Z","abstract_excerpt":"An $\\ell_p$ oblivious subspace embedding is a distribution over $r \\times n$ matrices $\\Pi$ such that for any fixed $n \\times d$ matrix $A$, $$\\Pr_{\\Pi}[\\textrm{for all }x, \\ \\|Ax\\|_p \\leq \\|\\Pi Ax\\|_p \\leq \\kappa \\|Ax\\|_p] \\geq 9/10,$$ where $r$ is the dimension of the embedding, $\\kappa$ is the distortion of the embedding, and for an $n$-dimensional vector $y$, $\\|y\\|_p$ is the $\\ell_p$-norm. Another important property is the sparsity of $\\Pi$, that is, the maximum number of non-zero entries per column, as this determines the running time of computing $\\Pi \\cdot A$. While for $p = 2$ there a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.04414","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":""},"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"}