{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/average-case-column-subset-selection-for","title":"Average Case Column Subset Selection for Entrywise \\ell_1-Norm Loss","arxiv_id":null,"date":"2019-12-01","proceeding":"NeurIPS 2019 12","authors":["Zhao Song","David Woodruff","Peilin Zhong"],"abstract":"We study the column subset selection problem with respect to the entrywise $\\ell_1$-norm loss. It is known that in the worst case, to obtain a good rank-$k$ approximation to a matrix, one needs an arbitrarily large $n^{\\Omega(1)}$ number of columns to obtain a $(1+\\epsilon)$-approximation to an $n \\times n$ matrix. Nevertheless, we show that under certain minimal and realistic distributional settings, it is possible to obtain a $(1+\\epsilon)$-approximation with a nearly linear running time and poly$(k/\\epsilon)+O(k\\log n)$ columns. Namely, we show that if the input matrix $A$ has the form $A = B + E$, where $B$ is an arbitrary rank-$k$ matrix, and $E$ is a matrix with i.i.d. entries drawn from any distribution $\\mu$ for which the $(1+\\gamma)$-th moment exists, for an arbitrarily small constant $\\gamma > 0$, then it is possible to obtain a $(1+\\epsilon)$-approximate column subset selection to the entrywise $\\ell_1$-norm in nearly linear time. Conversely we show that if the first moment does not exist, then it is not possible to obtain a $(1+\\epsilon)$-approximate subset selection algorithm even if one chooses any $n^{o(1)}$ columns. This is the first algorithm of any kind for achieving a $(1+\\epsilon)$-approximation for entrywise $\\ell_1$-norm loss low rank approximation.","url_abs":"http://papers.nips.cc/paper/9201-average-case-column-subset-selection-for-entrywise-ell_1-norm-loss","url_pdf":"http://papers.nips.cc/paper/9201-average-case-column-subset-selection-for-entrywise-ell_1-norm-loss.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"average-case-column-subset-selection-for","repo_url":"https://github.com/zpl7840/noise_l1_low_rank_approximation","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}