{"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-and-robust-fixed-rank-matrix-recovery","title":"Fast and Robust Fixed-Rank Matrix Recovery","arxiv_id":"1503.03004","date":"2015-03-10","proceeding":null,"authors":["German Ros","Julio Guerrero"],"abstract":"We address the problem of efficient sparse fixed-rank (S-FR) matrix\ndecomposition, i.e., splitting a corrupted matrix $M$ into an uncorrupted\nmatrix $L$ of rank $r$ and a sparse matrix of outliers $S$. Fixed-rank\nconstraints are usually imposed by the physical restrictions of the system\nunder study. Here we propose a method to perform accurate and very efficient\nS-FR decomposition that is more suitable for large-scale problems than existing\napproaches. Our method is a grateful combination of geometrical and algebraical\ntechniques, which avoids the bottleneck caused by the Truncated SVD (TSVD).\nInstead, a polar factorization is used to exploit the manifold structure of\nfixed-rank problems as the product of two Stiefel and an SPD manifold, leading\nto a better convergence and stability. Then, closed-form projectors help to\nspeed up each iteration of the method. We introduce a novel and fast projector\nfor the $\\text{SPD}$ manifold and a proof of its validity. Further acceleration\nis achieved using a Nystrom scheme. Extensive experiments with synthetic and\nreal data in the context of robust photometric stereo and spectral clustering\nshow that our proposals outperform the state of the art.","url_abs":"http://arxiv.org/abs/1503.03004v3","url_pdf":"http://arxiv.org/pdf/1503.03004v3.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-and-robust-fixed-rank-matrix-recovery","repo_url":"https://github.com/germanRos/FRADM","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[{"task_slug":"clustering","task_name":"Clustering"}],"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}