{"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/riemannian-optimization-on-relaxed-indicator","title":"Riemannian Optimization on Relaxed Indicator Matrix Manifold","arxiv_id":"2503.20505","date":"2025-03-26","proceeding":null,"authors":["Jinghui Yuan","Fangyuan Xie","Feiping Nie","Xuelong Li"],"abstract":"The indicator matrix plays an important role in machine learning, but optimizing it is an NP-hard problem. We propose a new relaxation of the indicator matrix and prove that this relaxation forms a manifold, which we call the Relaxed Indicator Matrix Manifold (RIM manifold). Based on Riemannian geometry, we develop a Riemannian toolbox for optimization on the RIM manifold. Specifically, we provide several methods of Retraction, including a fast Retraction method to obtain geodesics. We point out that the RIM manifold is a generalization of the double stochastic manifold, and it is much faster than existing methods on the double stochastic manifold, which has a complexity of \\( \\mathcal{O}(n^3) \\), while RIM manifold optimization is \\( \\mathcal{O}(n) \\) and often yields better results. We conducted extensive experiments, including image denoising, with millions of variables to support our conclusion, and applied the RIM manifold to Ratio Cut, we provide a rigorous convergence proof and achieve clustering results that outperform the state-of-the-art methods. Our Code in \\href{https://github.com/Yuan-Jinghui/Riemannian-Optimization-on-Relaxed-Indicator-Matrix-Manifold}{here}.","url_abs":"https://arxiv.org/abs/2503.20505v2","url_pdf":"https://arxiv.org/pdf/2503.20505v2.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":"riemannian-optimization-on-relaxed-indicator","repo_url":"https://github.com/Yuan-Jinghui/Riemannian-Optimization-on-Relaxed-Indicator-Matrix-Manifold","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"MIT"}}],"tasks":[{"task_slug":"denoising","task_name":"Denoising"},{"task_slug":"graph-clustering","task_name":"Graph Clustering"},{"task_slug":"image-denoising","task_name":"Image Denoising"},{"task_slug":"riemannian-optimization","task_name":"Riemannian optimization"},{"task_slug":"global-optimization","task_name":"global-optimization"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":"https://syntology.ai/paper/2503.20505","atlas_url":"https://app.syntology.ai/?focus=2503.20505","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}