{"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-ordered-weighted-ell_1-norm-atomic","title":"The Ordered Weighted $\\ell_1$ Norm: Atomic Formulation, Projections, and Algorithms","arxiv_id":"1409.4271","date":"2014-09-15","proceeding":null,"authors":["Xiangrong Zeng","Mário A. T. Figueiredo"],"abstract":"The ordered weighted $\\ell_1$ norm (OWL) was recently proposed, with two\ndifferent motivations: its good statistical properties as a sparsity promoting\nregularizer; the fact that it generalizes the so-called {\\it octagonal\nshrinkage and clustering algorithm for regression} (OSCAR), which has the\nability to cluster/group regression variables that are highly correlated. This\npaper contains several contributions to the study and application of OWL\nregularization: the derivation of the atomic formulation of the OWL norm; the\nderivation of the dual of the OWL norm, based on its atomic formulation; a new\nand simpler derivation of the proximity operator of the OWL norm; an efficient\nscheme to compute the Euclidean projection onto an OWL ball; the instantiation\nof the conditional gradient (CG, also known as Frank-Wolfe) algorithm for\nlinear regression problems under OWL regularization; the instantiation of\naccelerated projected gradient algorithms for the same class of problems.\nFinally, a set of experiments give evidence that accelerated projected gradient\nalgorithms are considerably faster than CG, for the class of problems\nconsidered.","url_abs":"http://arxiv.org/abs/1409.4271v5","url_pdf":"http://arxiv.org/pdf/1409.4271v5.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":"the-ordered-weighted-ell_1-norm-atomic","repo_url":"https://github.com/dhruvdcoder/sparse-structured-attention","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"pytorch","reach":null},{"paper_slug":"the-ordered-weighted-ell_1-norm-atomic","repo_url":"https://github.com/vene/sparse-structured-attention","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"pytorch","reach":{"status":"unanswered"}},{"paper_slug":"the-ordered-weighted-ell_1-norm-atomic","repo_url":"https://github.com/weiwang2330/sparse-structured-attention","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"pytorch","reach":null}],"tasks":[{"task_slug":"clustering","task_name":"Clustering"},{"task_slug":"regression-1","task_name":"regression"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1409.4271","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}