{"paper":{"title":"Low Rank Approximation and Regression in Input Sparsity Time","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"David P. Woodruff, Kenneth L. Clarkson","submitted_at":"2012-07-26T18:50:00Z","abstract_excerpt":"We design a new distribution over $\\poly(r \\eps^{-1}) \\times n$ matrices $S$ so that for any fixed $n \\times d$ matrix $A$ of rank $r$, with probability at least 9/10, $\\norm{SAx}_2 = (1 \\pm \\eps)\\norm{Ax}_2$ simultaneously for all $x \\in \\mathbb{R}^d$. Such a matrix $S$ is called a \\emph{subspace embedding}. Furthermore, $SA$ can be computed in $\\nnz(A) + \\poly(d \\eps^{-1})$ time, where $\\nnz(A)$ is the number of non-zero entries of $A$. This improves over all previous subspace embeddings, which required at least $\\Omega(nd \\log d)$ time to achieve this property. We call our matrices $S$ \\emp"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.6365","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"}