{"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/spurious-local-minima-in-nonconvex-sum-of","title":"Spurious local minima in nonconvex sum-of-squares optimization","arxiv_id":"2411.02208","date":"2024-11-04","proceeding":null,"authors":["Grigoriy Blekherman","Rainer Sinn","Mauricio Velasco","Shixuan Zhang"],"abstract":"We study spurious second-order stationary points and local minima in a nonconvex low-rank formulation of sum-of-squares optimization on a real variety $X$. We reformulate the problem of finding a spurious local minimum in terms of syzygies of the underlying linear series, and also bring in topological tools to study this problem. When the variety $X$ is of minimal degree, there exist spurious second-order stationary points if and only if both the dimension and the codimension of the variety are greater than one, answering a question by Legat, Yuan, and Parrilo. Moreover, for surfaces of minimal degree, we provide sufficient conditions to exclude points from being spurious local minima. In particular, all second-order stationary points associated with infinite Gram matrices on the Veronese surface, corresponding to ternary quartics, lie on the boundary and can be written as a binary quartic, up to a linear change of coordinates, complementing work by Scheiderer on decompositions of ternary quartics as a sum of three squares. For general varieties of higher degree, we give examples and characterizations of spurious second-order stationary points in the interior, together with a restricted path algorithm that avoids such points with controlled step sizes, and numerical experiment results illustrating the empirical successes on plane cubic curves and Veronese varieties.","url_abs":"https://arxiv.org/abs/2411.02208v1","url_pdf":"https://arxiv.org/pdf/2411.02208v1.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":"spurious-local-minima-in-nonconvex-sum-of","repo_url":"https://github.com/shixuan-zhang/LowRankSOS.jl","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":1,"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}