{"paper":{"title":"Optimal CUR Matrix Decompositions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG","math.NA"],"primary_cat":"cs.DS","authors_text":"Christos Boutsidis, David P. Woodruff","submitted_at":"2014-05-30T16:44:06Z","abstract_excerpt":"The CUR decomposition of an $m \\times n$ matrix $A$ finds an $m \\times c$ matrix $C$ with a subset of $c < n$ columns of $A,$ together with an $r \\times n$ matrix $R$ with a subset of $r < m$ rows of $A,$ as well as a $c \\times r$ low-rank matrix $U$ such that the matrix $C U R$ approximates the matrix $A,$ that is, $ || A - CUR ||_F^2 \\le (1+\\epsilon) || A - A_k||_F^2$, where $||.||_F$ denotes the Frobenius norm and $A_k$ is the best $m \\times n$ matrix of rank $k$ constructed via the SVD. We present input-sparsity-time and deterministic algorithms for constructing such a CUR decomposition wh"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.7910","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"}