{"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/a-riemannian-low-rank-method-for-optimization","title":"A Riemannian low-rank method for optimization over semidefinite matrices with block-diagonal constraints","arxiv_id":"1506.00575","date":"2015-06-01","proceeding":null,"authors":["Nicolas Boumal"],"abstract":"We propose a new algorithm to solve optimization problems of the form $\\min\nf(X)$ for a smooth function $f$ under the constraints that $X$ is positive\nsemidefinite and the diagonal blocks of $X$ are small identity matrices. Such\nproblems often arise as the result of relaxing a rank constraint (lifting). In\nparticular, many estimation tasks involving phases, rotations, orthonormal\nbases or permutations fit in this framework, and so do certain relaxations of\ncombinatorial problems such as Max-Cut. The proposed algorithm exploits the\nfacts that (1) such formulations admit low-rank solutions, and (2) their\nrank-restricted versions are smooth optimization problems on a Riemannian\nmanifold. Combining insights from both the Riemannian and the convex geometries\nof the problem, we characterize when second-order critical points of the smooth\nproblem reveal KKT points of the semidefinite problem. We compare against state\nof the art, mature software and find that, on certain interesting problem\ninstances, what we call the staircase method is orders of magnitude faster, is\nmore accurate and scales better. Code is available.","url_abs":"http://arxiv.org/abs/1506.00575v2","url_pdf":"http://arxiv.org/pdf/1506.00575v2.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":"a-riemannian-low-rank-method-for-optimization","repo_url":"https://github.com/pandrey-fr/maxcut","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"GPL-3.0"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":"https://syntology.ai/paper/1506.00575","atlas_url":"https://app.syntology.ai/?focus=1506.00575","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}