{"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/convex-total-least-squares","title":"Convex Total Least Squares","arxiv_id":"1406.0189","date":"2014-06-01","proceeding":null,"authors":["Dmitry Malioutov","Nikolai Slavov"],"abstract":"We study the total least squares (TLS) problem that generalizes least squares\nregression by allowing measurement errors in both dependent and independent\nvariables. TLS is widely used in applied fields including computer vision,\nsystem identification and econometrics. The special case when all dependent and\nindependent variables have the same level of uncorrelated Gaussian noise, known\nas ordinary TLS, can be solved by singular value decomposition (SVD). However,\nSVD cannot solve many important practical TLS problems with realistic noise\nstructure, such as having varying measurement noise, known structure on the\nerrors, or large outliers requiring robust error-norms. To solve such problems,\nwe develop convex relaxation approaches for a general class of structured TLS\n(STLS). We show both theoretically and experimentally, that while the plain\nnuclear norm relaxation incurs large approximation errors for STLS, the\nre-weighted nuclear norm approach is very effective, and achieves better\naccuracy on challenging STLS problems than popular non-convex solvers. We\ndescribe a fast solution based on augmented Lagrangian formulation, and apply\nour approach to an important class of biological problems that use population\naverage measurements to infer cell-type and physiological-state specific\nexpression levels that are very hard to measure directly.","url_abs":"http://arxiv.org/abs/1406.0189v1","url_pdf":"http://arxiv.org/pdf/1406.0189v1.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":"convex-total-least-squares","repo_url":"https://github.com/SlavovLab/STLS","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"convex-total-least-squares","repo_url":"https://github.com/nslavov/RCweb","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[{"task_slug":"econometrics","task_name":"Econometrics"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}