{"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/towards-a-zero-one-law-for-entrywise-low-rank","title":"Towards a Zero-One Law for Column Subset Selection","arxiv_id":"1811.01442","date":"2018-11-04","proceeding":"NeurIPS 2019 12","authors":["Zhao Song","David P. Woodruff","Peilin Zhong"],"abstract":"There are a number of approximation algorithms for NP-hard versions of low rank approximation, such as finding a rank-$k$ matrix $B$ minimizing the sum of absolute values of differences to a given $n$-by-$n$ matrix $A$, $\\min_{\\textrm{rank-}k~B}\\|A-B\\|_1$, or more generally finding a rank-$k$ matrix $B$ which minimizes the sum of $p$-th powers of absolute values of differences, $\\min_{\\textrm{rank-}k~B}\\|A-B\\|_p^p$. Many of these algorithms are linear time columns subset selection algorithms, returning a subset of $\\mathrm{poly}(k \\log n)$ columns whose cost is no more than a $\\mathrm{poly}(k)$ factor larger than the cost of the best rank-$k$ matrix. The above error measures are special cases of the following general entrywise low rank approximation problem: given an arbitrary function $g:\\mathbb{R} \\rightarrow \\mathbb{R}_{\\geq 0}$, find a rank-$k$ matrix $B$ which minimizes $\\|A-B\\|_g = \\sum_{i,j}g(A_{i,j}-B_{i,j})$. A natural question is which functions $g$ admit efficient approximation algorithms? Indeed, this is a central question of recent work studying generalized low rank models. In this work we give approximation algorithms for $\\textit{every}$ function $g$ which is approximately monotone and satisfies an approximate triangle inequality, and we show both of these conditions are necessary. Further, our algorithm is efficient if the function $g$ admits an efficient approximate regression algorithm. Our approximation algorithms handle functions which are not even scale-invariant, such as the Huber loss function, which we show have very different structural properties than $\\ell_p$-norms, e.g., one can show the lack of scale-invariance causes any column subset selection algorithm to provably require a $\\sqrt{\\log n}$ factor larger number of columns than $\\ell_p$-norms; nevertheless we design the first efficient column subset selection algorithms for such error measures.","url_abs":"https://arxiv.org/abs/1811.01442v2","url_pdf":"https://arxiv.org/pdf/1811.01442v2.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":"towards-a-zero-one-law-for-entrywise-low-rank","repo_url":"https://github.com/zpl7840/general_loss_column_subset_selection","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":{"status":"ok"}}],"tasks":[],"methods":[{"method_slug":"huber-loss","method_name":"Huber loss"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":"https://app.syntology.ai/?focus=1811.01442","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}