{"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/sensitivity-analysis-for-mirror-stratifiable","title":"Sensitivity Analysis for Mirror-Stratifiable Convex Functions","arxiv_id":"1707.03194","date":"2017-07-11","proceeding":null,"authors":["Jalal Fadili","Jérôme Malick","Gabriel Peyré"],"abstract":"This paper provides a set of sensitivity analysis and activity identification\nresults for a class of convex functions with a strong geometric structure, that\nwe coined \"mirror-stratifiable\". These functions are such that there is a\nbijection between a primal and a dual stratification of the space into\npartitioning sets, called strata. This pairing is crucial to track the strata\nthat are identifiable by solutions of parametrized optimization problems or by\niterates of optimization algorithms. This class of functions encompasses all\nregularizers routinely used in signal and image processing, machine learning,\nand statistics. We show that this \"mirror-stratifiable\" structure enjoys a nice\nsensitivity theory, allowing us to study stability of solutions of optimization\nproblems to small perturbations, as well as activity identification of\nfirst-order proximal splitting-type algorithms. Existing results in the\nliterature typically assume that, under a non-degeneracy condition, the active\nset associated to a minimizer is stable to small perturbations and is\nidentified in finite time by optimization schemes. In contrast, our results do\nnot require any non-degeneracy assumption: in consequence, the optimal active\nset is not necessarily stable anymore, but we are able to track precisely the\nset of identifiable strata.We show that these results have crucial implications\nwhen solving challenging ill-posed inverse problems via regularization, a\ntypical scenario where the non-degeneracy condition is not fulfilled. Our\ntheoretical results, illustrated by numerical simulations, allow to\ncharacterize the instability behaviour of the regularized solutions, by\nlocating the set of all low-dimensional strata that can be potentially\nidentified by these solutions.","url_abs":"http://arxiv.org/abs/1707.03194v3","url_pdf":"http://arxiv.org/pdf/1707.03194v3.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":"sensitivity-analysis-for-mirror-stratifiable","repo_url":"https://github.com/gpeyre/2017-SIOPT-stratification","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[{"task_slug":"sensitivity","task_name":"Sensitivity"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}