{"paper":{"title":"Low Rank Matrix Approximation in Linear Time","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":[],"primary_cat":"cs.CG","authors_text":"Sariel Har-Peled","submitted_at":"2014-10-31T16:46:36Z","abstract_excerpt":"$\\newcommand{\\MatA}{\\mathcal{M}}$ $\\newcommand{\\eps}{\\varepsilon}$ $\\newcommand{\\NSize}{\\mathsf{N}{}}$ $\\newcommand{\\MatB}{\\mathcal{B}}$ $\\newcommand{\\Fnorm}[1]{\\left\\| {#1} \\right\\|_F}$ $\\newcommand{\\PrcOpt}[2]{\\mu_{\\mathrm{opt}}\\pth{#1, #2}}$ $\\newcommand{\\pth}[1]{\\left(#1\\right)}$\n  Given a matrix $\\MatA$ with $n$ rows and $d$ columns, and fixed $k$ and $\\eps$, we present an algorithm that in linear time (i.e., $O(\\NSize )$) computes a $k$-rank matrix $\\MatB$ with approximation error $\\Fnorm{\\MatA - \\MatB}^2 \\leq (1+\\eps) \\PrcOpt{\\MatA}{k}$, where $\\NSize = n d$ is the input size, and $\\Prc"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.8802","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"}