{"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/matrix-completion-via-nonconvex","title":"Matrix Completion via Nonconvex Regularization: Convergence of the Proximal Gradient Algorithm","arxiv_id":"1903.00702","date":"2019-03-02","proceeding":null,"authors":["Fei Wen","Rendong Ying","Peilin Liu","Trieu-Kien Truong"],"abstract":"Matrix completion has attracted much interest in the past decade in machine\nlearning and computer vision. For low-rank promotion in matrix completion, the\nnuclear norm penalty is convenient due to its convexity but has a bias problem.\nRecently, various algorithms using nonconvex penalties have been proposed,\namong which the proximal gradient descent (PGD) algorithm is one of the most\nefficient and effective. For the nonconvex PGD algorithm, whether it converges\nto a local minimizer and its convergence rate are still unclear. This work\nprovides a nontrivial analysis on the PGD algorithm in the nonconvex case.\nBesides the convergence to a stationary point for a generalized nonconvex\npenalty, we provide more deep analysis on a popular and important class of\nnonconvex penalties which have discontinuous thresholding functions. For such\npenalties, we establish the finite rank convergence, convergence to restricted\nstrictly local minimizer and eventually linear convergence rate of the PGD\nalgorithm. Meanwhile, convergence to a local minimizer has been proved for the\nhard-thresholding penalty. Our result is the first shows that, nonconvex\nregularized matrix completion only has restricted strictly local minimizers,\nand the PGD algorithm can converge to such minimizers with eventually linear\nrate under certain conditions. Illustration of the PGD algorithm via\nexperiments has also been provided. Code is available at\nhttps://github.com/FWen/nmc.","url_abs":"http://arxiv.org/abs/1903.00702v1","url_pdf":"http://arxiv.org/pdf/1903.00702v1.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":"matrix-completion-via-nonconvex","repo_url":"https://github.com/FWen/nmc","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}