{"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/slack-and-margin-rescaling-as-convex","title":"Slack and Margin Rescaling as Convex Extensions of Supermodular Functions","arxiv_id":"1606.05918","date":"2016-06-19","proceeding":null,"authors":["Matthew B. Blaschko"],"abstract":"Slack and margin rescaling are variants of the structured output SVM, which\nis frequently applied to problems in computer vision such as image\nsegmentation, object localization, and learning parts based object models. They\ndefine convex surrogates to task specific loss functions, which, when\nspecialized to non-additive loss functions for multi-label problems, yield\nextensions to increasing set functions. We demonstrate in this paper that we\nmay use these concepts to define polynomial time convex extensions of arbitrary\nsupermodular functions, providing an analysis framework for the tightness of\nthese surrogates. This analysis framework shows that, while neither margin nor\nslack rescaling dominate the other, known bounds on supermodular functions can\nbe used to derive extensions that dominate both of these, indicating possible\ndirections for defining novel structured output prediction surrogates. In\naddition to the analysis of structured prediction loss functions, these results\nimply an approach to supermodular minimization in which margin rescaling is\ncombined with non-polynomial time convex extensions to compute a sequence of LP\nrelaxations reminiscent of a cutting plane method. This approach is applied to\nthe problem of selecting representative exemplars from a set of images,\nvalidating our theoretical contributions.","url_abs":"http://arxiv.org/abs/1606.05918v2","url_pdf":"http://arxiv.org/pdf/1606.05918v2.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":"slack-and-margin-rescaling-as-convex","repo_url":"https://github.com/blaschko/supermodularLP","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[{"task_slug":"image-segmentation","task_name":"Image Segmentation"},{"task_slug":"object-localization","task_name":"Object Localization"},{"task_slug":"semantic-segmentation","task_name":"Semantic Segmentation"},{"task_slug":"structured-prediction","task_name":"Structured Prediction"}],"methods":[{"method_slug":"svm","method_name":"SVM"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}