{"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/the-proxy-step-size-technique-for-regularized","title":"The Proxy Step-size Technique for Regularized Optimization on the Sphere Manifold","arxiv_id":"2209.01812","date":"2022-09-05","proceeding":null,"authors":["Fang Bai","Adrien Bartoli"],"abstract":"We give an effective solution to the regularized optimization problem $g (\\boldsymbol{x}) + h (\\boldsymbol{x})$, where $\\boldsymbol{x}$ is constrained on the unit sphere $\\Vert \\boldsymbol{x} \\Vert_2 = 1$. Here $g (\\cdot)$ is a smooth cost with Lipschitz continuous gradient within the unit ball $\\{\\boldsymbol{x} : \\Vert \\boldsymbol{x} \\Vert_2 \\le 1 \\}$ whereas $h (\\cdot)$ is typically non-smooth but convex and absolutely homogeneous, \\textit{e.g.,}~norm regularizers and their combinations. Our solution is based on the Riemannian proximal gradient, using an idea we call \\textit{proxy step-size} -- a scalar variable which we prove is monotone with respect to the actual step-size within an interval. The proxy step-size exists ubiquitously for convex and absolutely homogeneous $h(\\cdot)$, and decides the actual step-size and the tangent update in closed-form, thus the complete proximal gradient iteration. Based on these insights, we design a Riemannian proximal gradient method using the proxy step-size. We prove that our method converges to a critical point, guided by a line-search technique based on the $g(\\cdot)$ cost only. The proposed method can be implemented in a couple of lines of code. We show its usefulness by applying nuclear norm, $\\ell_1$ norm, and nuclear-spectral norm regularization to three classical computer vision problems. The improvements are consistent and backed by numerical experiments.","url_abs":"https://arxiv.org/abs/2209.01812v1","url_pdf":"https://arxiv.org/pdf/2209.01812v1.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":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"the-proxy-step-size-technique-for-regularized","repo_url":"https://bitbucket.org/fangbai/proxystepsize-pgs","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}