{"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/morse-theory-for-chromatic-delaunay","title":"Morse Theory for Chromatic Delaunay Triangulations","arxiv_id":"2405.19303","date":"2024-05-29","proceeding":null,"authors":["Abhinav Natarajan","Thomas Chaplin","Adam Brown","Maria-Jose Jimenez"],"abstract":"The chromatic alpha filtration is a generalization of the alpha filtration that can encode spatial relationships among classes of labelled point cloud data, and has applications in topological data analysis of multi-species data. In this paper we introduce the chromatic Delaunay-\\v{C}ech and chromatic Delaunay-Rips filtrations, which are computationally favourable alternatives to the chromatic alpha filtration. We use generalized discrete Morse theory to show that the \\v{C}ech, chromatic Delaunay-\\v{C}ech, and chromatic alpha filtrations are related by simplicial collapses. Our result generalizes a result of Bauer and Edelsbrunner from the non-chromatic to the chromatic setting. We also show that the chromatic Delaunay-Rips filtration is locally stable to perturbations of the underlying point cloud. Our results provide theoretical justification for the use of chromatic Delaunay-\\v{C}ech and chromatic Delaunay-Rips filtrations in applications, and we demonstrate their computational advantage with numerical experiments.","url_abs":"https://arxiv.org/abs/2405.19303v2","url_pdf":"https://arxiv.org/pdf/2405.19303v2.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":"morse-theory-for-chromatic-delaunay","repo_url":"https://github.com/abhinavnatarajan/chalc","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"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}