{"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/fast-low-rank-matrix-learning-with-nonconvex","title":"Fast Low-Rank Matrix Learning with Nonconvex Regularization","arxiv_id":"1512.00984","date":"2015-12-03","proceeding":null,"authors":["Quanming Yao","James T. Kwok","Wenliang Zhong"],"abstract":"Low-rank modeling has a lot of important applications in machine learning,\ncomputer vision and social network analysis. While the matrix rank is often\napproximated by the convex nuclear norm, the use of nonconvex low-rank\nregularizers has demonstrated better recovery performance. However, the\nresultant optimization problem is much more challenging. A very recent\nstate-of-the-art is based on the proximal gradient algorithm. However, it\nrequires an expensive full SVD in each proximal step. In this paper, we show\nthat for many commonly-used nonconvex low-rank regularizers, a cutoff can be\nderived to automatically threshold the singular values obtained from the\nproximal operator. This allows the use of power method to approximate the SVD\nefficiently. Besides, the proximal operator can be reduced to that of a much\nsmaller matrix projected onto this leading subspace. Convergence, with a rate\nof O(1/T) where T is the number of iterations, can be guaranteed. Extensive\nexperiments are performed on matrix completion and robust principal component\nanalysis. The proposed method achieves significant speedup over the\nstate-of-the-art. Moreover, the matrix solution obtained is more accurate and\nhas a lower rank than that of the traditional nuclear norm regularizer.","url_abs":"http://arxiv.org/abs/1512.00984v1","url_pdf":"http://arxiv.org/pdf/1512.00984v1.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":"fast-low-rank-matrix-learning-with-nonconvex","repo_url":"https://github.com/quanmingyao/FaNCL","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[{"task_slug":"matrix-completion","task_name":"Matrix Completion"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}