{"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/sparse-filtered-nerves","title":"Sparse Filtered Nerves","arxiv_id":"1810.02149","date":"2018-10-04","proceeding":null,"authors":["Nello Blaser","Morten Brun"],"abstract":"Given a point cloud $P$ in Euclidean space and a positive parameter $t$ we can consider the $t$-neighborhood $P^{t}$ of $P$ consisting of points at distance less than $t$ to $P$. Homology of $P^{t}$ gives information about components, holes, voids etc. in $P^{t}$. The idea of persistent homology is that it may happen that we are interested in some of holes in the spaces $P^t$ that are not detected simultaneously in homology for a single value of $t$, but where each of these holes is detected for $t$ in a wide range. When the dimension of the ambient Euclidean space is small, persistent homology is efficiently computed by the $\\alpha$-complex. For dimension bigger than three this becomes resource consuming. Don Sheehy discovered that there exists a filtered simplicial complex whose size depends linearly on the cardinality of $P$ and whose persistent homology is an approximation of the persistent homology of the filtered topological space $\\{P^{t}\\}_{t \\ge 0}$. In this paper we pursue Sheehy's sparsification approach and give a more general approach to sparsification of filtered simplicial complexes computing the homology of filtered spaces of the form $\\{P^{t}\\}_{t \\ge 0}$ and more generally to sparsification of filtered Dowker nerves. To our best knowledge, this is the first approach to sparsification of general Dowker nerves.","url_abs":"http://arxiv.org/abs/1810.02149v2","url_pdf":"http://arxiv.org/pdf/1810.02149v2.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":"sparse-filtered-nerves","repo_url":"https://github.com/mbr085/Sparse-Dowker-Nerves","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":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}