{"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/approximate-decomposable-submodular-function","title":"Approximate Decomposable Submodular Function Minimization for Cardinality-Based Components","arxiv_id":"2110.14859","date":"2021-10-28","proceeding":"NeurIPS 2021 12","authors":["Nate Veldt","Austin R. Benson","Jon Kleinberg"],"abstract":"Minimizing a sum of simple submodular functions of limited support is a special case of general submodular function minimization that has seen numerous applications in machine learning. We develop fast techniques for instances where components in the sum are cardinality-based, meaning they depend only on the size of the input set. This variant is one of the most widely applied in practice, encompassing, e.g., common energy functions arising in image segmentation and recent generalized hypergraph cut functions. We develop the first approximation algorithms for this problem, where the approximations can be quickly computed via reduction to a sparse graph cut problem, with graph sparsity controlled by the desired approximation factor. Our method relies on a new connection between sparse graph reduction techniques and piecewise linear approximations to concave functions. Our sparse reduction technique leads to significant improvements in theoretical runtimes, as well as substantial practical gains in problems ranging from benchmark image segmentation tasks to hypergraph clustering problems.","url_abs":"https://arxiv.org/abs/2110.14859v1","url_pdf":"https://arxiv.org/pdf/2110.14859v1.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":"approximate-decomposable-submodular-function","repo_url":"https://github.com/nveldt/SparseCardDSFM","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":"segmentation","task_name":"Segmentation"},{"task_slug":"semantic-segmentation","task_name":"Semantic Segmentation"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}