{"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/hypergraph-p-laplacian-a-differential","title":"Hypergraph $p$-Laplacian: A Differential Geometry View","arxiv_id":"1711.08171","date":"2017-11-22","proceeding":null,"authors":["Shota Saito","Danilo P. Mandic","Hideyuki Suzuki"],"abstract":"The graph Laplacian plays key roles in information processing of relational\ndata, and has analogies with the Laplacian in differential geometry. In this\npaper, we generalize the analogy between graph Laplacian and differential\ngeometry to the hypergraph setting, and propose a novel hypergraph\n$p$-Laplacian. Unlike the existing two-node graph Laplacians, this\ngeneralization makes it possible to analyze hypergraphs, where the edges are\nallowed to connect any number of nodes. Moreover, we propose a semi-supervised\nlearning method based on the proposed hypergraph $p$-Laplacian, and formalize\nthem as the analogue to the Dirichlet problem, which often appears in physics.\nWe further explore theoretical connections to normalized hypergraph cut on a\nhypergraph, and propose normalized cut corresponding to hypergraph\n$p$-Laplacian. The proposed $p$-Laplacian is shown to outperform standard\nhypergraph Laplacians in the experiment on a hypergraph semi-supervised\nlearning and normalized cut setting.","url_abs":"http://arxiv.org/abs/1711.08171v1","url_pdf":"http://arxiv.org/pdf/1711.08171v1.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":"hypergraph-p-laplacian-a-differential","repo_url":"https://github.com/ShotaSAITO/Hypergraph-Laplacian","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"hypergraph-p-laplacian-a-differential","repo_url":"https://github.com/KAIDI3270/hypergraph_laplacian_python3","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1711.08171","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}