{"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/guaranteed-rank-minimization-via-singular","title":"Guaranteed Rank Minimization via Singular Value Projection","arxiv_id":"0909.5457","date":"2009-09-30","proceeding":"NeurIPS 2010 12","authors":["Raghu Meka","Prateek Jain","Inderjit S. Dhillon"],"abstract":"Minimizing the rank of a matrix subject to affine constraints is a\nfundamental problem with many important applications in machine learning and\nstatistics. In this paper we propose a simple and fast algorithm SVP (Singular\nValue Projection) for rank minimization with affine constraints (ARMP) and show\nthat SVP recovers the minimum rank solution for affine constraints that satisfy\nthe \"restricted isometry property\" and show robustness of our method to noise.\nOur results improve upon a recent breakthrough by Recht, Fazel and Parillo\n(RFP07) and Lee and Bresler (LB09) in three significant ways:\n  1) our method (SVP) is significantly simpler to analyze and easier to\nimplement,\n  2) we give recovery guarantees under strictly weaker isometry assumptions\n  3) we give geometric convergence guarantees for SVP even in presense of noise\nand, as demonstrated empirically, SVP is significantly faster on real-world and\nsynthetic problems.\n  In addition, we address the practically important problem of low-rank matrix\ncompletion (MCP), which can be seen as a special case of ARMP. We empirically\ndemonstrate that our algorithm recovers low-rank incoherent matrices from an\nalmost optimal number of uniformly sampled entries. We make partial progress\ntowards proving exact recovery and provide some intuition for the strong\nperformance of SVP applied to matrix completion by showing a more restricted\nisometry property. Our algorithm outperforms existing methods, such as those of\n\\cite{RFP07,CR08,CT09,CCS08,KOM09,LB09}, for ARMP and the matrix-completion\nproblem by an order of magnitude and is also significantly more robust to\nnoise.","url_abs":"http://arxiv.org/abs/0909.5457v3","url_pdf":"http://arxiv.org/pdf/0909.5457v3.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":"guaranteed-rank-minimization-via-singular","repo_url":"https://github.com/HauLiang/Matrix-Completion-Methods","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"MIT"}}],"tasks":[{"task_slug":"low-rank-matrix-completion","task_name":"Low-Rank Matrix Completion"},{"task_slug":"matrix-completion","task_name":"Matrix Completion"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=0909.5457","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}