{"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/total-least-squares-regression-in-input","title":"Total Least Squares Regression in Input Sparsity Time","arxiv_id":"1909.12441","date":"2019-09-27","proceeding":"NeurIPS 2019 12","authors":["Huaian Diao","Zhao Song","David P. Woodruff","Xin Yang"],"abstract":"In the total least squares problem, one is given an $m \\times n$ matrix $A$, and an $m \\times d$ matrix $B$, and one seeks to \"correct\" both $A$ and $B$, obtaining matrices $\\hat{A}$ and $\\hat{B}$, so that there exists an $X$ satisfying the equation $\\hat{A}X = \\hat{B}$. Typically the problem is overconstrained, meaning that $m \\gg \\max(n,d)$. The cost of the solution $\\hat{A}, \\hat{B}$ is given by $\\|A-\\hat{A}\\|_F^2 + \\|B - \\hat{B}\\|_F^2$. We give an algorithm for finding a solution $X$ to the linear system $\\hat{A}X=\\hat{B}$ for which the cost $\\|A-\\hat{A}\\|_F^2 + \\|B-\\hat{B}\\|_F^2$ is at most a multiplicative $(1+\\epsilon)$ factor times the optimal cost, up to an additive error $\\eta$ that may be an arbitrarily small function of $n$. Importantly, our running time is $\\tilde{O}( \\mathrm{nnz}(A) + \\mathrm{nnz}(B) ) + \\mathrm{poly}(n/\\epsilon) \\cdot d$, where for a matrix $C$, $\\mathrm{nnz}(C)$ denotes its number of non-zero entries. Importantly, our running time does not directly depend on the large parameter $m$. As total least squares regression is known to be solvable via low rank approximation, a natural approach is to invoke fast algorithms for approximate low rank approximation, obtaining matrices $\\hat{A}$ and $\\hat{B}$ from this low rank approximation, and then solving for $X$ so that $\\hat{A}X = \\hat{B}$. However, existing algorithms do not apply since in total least squares the rank of the low rank approximation needs to be $n$, and so the running time of known methods would be at least $mn^2$. In contrast, we are able to achieve a much faster running time for finding $X$ by never explicitly forming the equation $\\hat{A} X = \\hat{B}$, but instead solving for an $X$ which is a solution to an implicit such equation. Finally, we generalize our algorithm to the total least squares problem with regularization.","url_abs":"https://arxiv.org/abs/1909.12441v1","url_pdf":"https://arxiv.org/pdf/1909.12441v1.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":"total-least-squares-regression-in-input","repo_url":"https://github.com/yangxinuw/total_least_squares_code","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[{"task_slug":"regression-1","task_name":"regression"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}