{"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-optimization-with-convex-constraints","title":"Low-rank Optimization with Convex Constraints","arxiv_id":"1606.01793","date":"2016-06-06","proceeding":null,"authors":["Christian Grussler","Anders Rantzer","Pontus Giselsson"],"abstract":"The problem of low-rank approximation with convex constraints, which appears\nin data analysis, system identification, model order reduction, low-order\ncontroller design and low-complexity modelling is considered. Given a matrix,\nthe objective is to find a low-rank approximation that meets rank and convex\nconstraints, while minimizing the distance to the matrix in the squared\nFrobenius norm. In many situations, this non-convex problem is convexified by\nnuclear norm regularization. However, we will see that the approximations\nobtained by this method may be far from optimal. In this paper, we propose an\nalternative convex relaxation that uses the convex envelope of the squared\nFrobenius norm and the rank constraint. With this approach, easily verifiable\nconditions are obtained under which the solutions to the convex relaxation and\nthe original non-convex problem coincide. An SDP representation of the convex\nenvelope is derived, which allows us to apply this approach to several known\nproblems. Our example on optimal low-rank Hankel approximation/model reduction\nillustrates that the proposed convex relaxation performs consistently better\nthan nuclear norm regularization and may outperform balanced truncation.","url_abs":"http://arxiv.org/abs/1606.01793v3","url_pdf":"http://arxiv.org/pdf/1606.01793v3.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-optimization-with-convex-constraints","repo_url":"https://github.com/LowRankOpt/LRINorm","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"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}