{"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/when-are-nonconvex-problems-not-scary","title":"When Are Nonconvex Problems Not Scary?","arxiv_id":"1510.06096","date":"2015-10-21","proceeding":null,"authors":["Ju Sun","Qing Qu","John Wright"],"abstract":"In this note, we focus on smooth nonconvex optimization problems that obey:\n(1) all local minimizers are also global; and (2) around any saddle point or\nlocal maximizer, the objective has a negative directional curvature. Concrete\napplications such as dictionary learning, generalized phase retrieval, and\northogonal tensor decomposition are known to induce such structures. We\ndescribe a second-order trust-region algorithm that provably converges to a\nglobal minimizer efficiently, without special initializations. Finally we\nhighlight alternatives, and open problems in this direction.","url_abs":"http://arxiv.org/abs/1510.06096v2","url_pdf":"http://arxiv.org/pdf/1510.06096v2.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":"when-are-nonconvex-problems-not-scary","repo_url":"https://github.com/SKorablyov/MaxProblem","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"tf","reach":{"status":"ok"}},{"paper_slug":"when-are-nonconvex-problems-not-scary","repo_url":"https://github.com/SKorablyov/Pranam","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"tf","reach":{"status":"ok"}},{"paper_slug":"when-are-nonconvex-problems-not-scary","repo_url":"https://github.com/sunju/pr_plain","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok"}}],"tasks":[{"task_slug":"dictionary-learning","task_name":"Dictionary Learning"},{"task_slug":"retrieval","task_name":"Retrieval"},{"task_slug":"tensor-decomposition","task_name":"Tensor Decomposition"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":"https://app.syntology.ai/?focus=1510.06096","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}