{"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/minimal-lipschitz-and-infty-harmonic","title":"Minimal Lipschitz and $\\infty$-Harmonic Extensions of Vector-Valued Functions on Finite Graphs","arxiv_id":"1903.04873","date":"2019-03-12","proceeding":null,"authors":["Miroslav Bačák","Johannes Hertrich","Sebastian Neumayer","Gabriele Steidl"],"abstract":"This paper deals with extensions of vector-valued functions on finite graphs fulfilling distinguished minimality properties. We show that so-called lex and L-lex minimal extensions are actually the same and call them minimal Lipschitz extensions. Then we prove that the solution of the graph $p$-Laplacians converge to these extensions as $p\\to \\infty$. Furthermore, we examine the relation between minimal Lipschitz extensions and iterated weighted midrange filters and address their connection to $\\infty$-Laplacians for scalar-valued functions. A convergence proof for an iterative algorithm proposed by Elmoataz et al.~(2014) for finding the zero of the $\\infty$-Laplacian is given. Finally, we present applications in image inpainting.","url_abs":"http://arxiv.org/abs/1903.04873v1","url_pdf":"http://arxiv.org/pdf/1903.04873v1.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":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"minimal-lipschitz-and-infty-harmonic","repo_url":"https://github.com/johertrich/Minimal_Lipschitz_Extensions","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}