{"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/induced-matchings-and-the-algebraic-stability","title":"Induced Matchings and the Algebraic Stability of Persistence Barcodes","arxiv_id":"1311.3681","date":"2013-11-14","proceeding":null,"authors":["Ulrich Bauer","Michael Lesnick"],"abstract":"We define a simple, explicit map sending a morphism $f:M \\rightarrow N$ of pointwise finite dimensional persistence modules to a matching between the barcodes of $M$ and $N$. Our main result is that, in a precise sense, the quality of this matching is tightly controlled by the lengths of the longest intervals in the barcodes of $\\ker f$ and $\\mathop{\\mathrm{coker}} f$. As an immediate corollary, we obtain a new proof of the algebraic stability of persistence, a fundamental result in the theory of persistent homology. In contrast to previous proofs, ours shows explicitly how a $\\delta$-interleaving morphism between two persistence modules induces a $\\delta$-matching between the barcodes of the two modules. Our main result also specializes to a structure theorem for submodules and quotients of persistence modules, and yields a novel \"single-morphism\" characterization of the interleaving relation on persistence modules.","url_abs":"http://arxiv.org/abs/1311.3681v4","url_pdf":"http://arxiv.org/pdf/1311.3681v4.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":"induced-matchings-and-the-algebraic-stability","repo_url":"https://github.com/nstucki/betti-matching","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"pytorch","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}