{"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/low-rank-inducing-norms-with-optimality","title":"Low-Rank Inducing Norms with Optimality Interpretations","arxiv_id":"1612.03186","date":"2016-12-09","proceeding":null,"authors":["Christian Grussler","Pontus Giselsson"],"abstract":"Optimization problems with rank constraints appear in many diverse fields\nsuch as control, machine learning and image analysis. Since the rank constraint\nis non-convex, these problems are often approximately solved via convex\nrelaxations. Nuclear norm regularization is the prevailing convexifying\ntechnique for dealing with these types of problem. This paper introduces a\nfamily of low-rank inducing norms and regularizers which includes the nuclear\nnorm as a special case. A posteriori guarantees on solving an underlying rank\nconstrained optimization problem with these convex relaxations are provided. We\nevaluate the performance of the low-rank inducing norms on three matrix\ncompletion problems. In all examples, the nuclear norm heuristic is\noutperformed by convex relaxations based on other low-rank inducing norms. For\ntwo of the problems there exist low-rank inducing norms that succeed in\nrecovering the partially unknown matrix, while the nuclear norm fails. These\nlow-rank inducing norms are shown to be representable as semi-definite\nprograms. Moreover, these norms have cheaply computable proximal mappings,\nwhich makes it possible to also solve problems of large size using first-order\nmethods.","url_abs":"http://arxiv.org/abs/1612.03186v2","url_pdf":"http://arxiv.org/pdf/1612.03186v2.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":"low-rank-inducing-norms-with-optimality","repo_url":"https://github.com/LowRankOpt/LRINorm","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"low-rank-inducing-norms-with-optimality","repo_url":"https://github.com/LowRankOpt/LRIPy","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"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}