{"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/the-epsilon-alternating-least-squares-for","title":"The Epsilon-Alternating Least Squares for Orthogonal Low-Rank Tensor Approximation and Its Global Convergence","arxiv_id":"1911.10921","date":"2019-11-25","proceeding":null,"authors":["Yuning Yang"],"abstract":"The epsilon alternating least squares ($\\epsilon$-ALS) is developed and analyzed for canonical polyadic decomposition (approximation) of a higher-order tensor where one or more of the factor matrices are assumed to be columnwisely orthonormal. It is shown that the algorithm globally converges to a KKT point for all tensors without any assumption. For the original ALS, by further studying the properties of the polar decomposition, we also establish its global convergence under a reality assumption not stronger than those in the literature. These results completely address a question concerning the global convergence raised in [L. Wang, M. T. Chu and B. Yu, \\emph{SIAM J. Matrix Anal. Appl.}, 36 (2015), pp. 1--19]. In addition, an initialization procedure is proposed, which possesses a provable lower bound when the number of columnwisely orthonormal factors is one. Armed with this initialization procedure, numerical experiments show that the $\\epsilon$-ALS exhibits a promising performance in terms of efficiency and effectiveness.","url_abs":"https://arxiv.org/abs/1911.10921v3","url_pdf":"https://arxiv.org/pdf/1911.10921v3.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":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"the-epsilon-alternating-least-squares-for","repo_url":"https://github.com/yuningyang19/epsilon-ALS","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}