{"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/on-approximation-of-2-d-persistence-modules","title":"On Approximation of $2$D Persistence Modules by Interval-decomposables","arxiv_id":"1911.01637","date":"2019-11-05","proceeding":null,"authors":["Hideto Asashiba","Emerson G. Escolar","Ken Nakashima","Michio Yoshiwaki"],"abstract":"In this work, we propose a new invariant for $2$D persistence modules called the compressed multiplicity and show that it generalizes the notions of the dimension vector and the rank invariant. In addition, for a $2$D persistence module $M$, we propose an \"interval-decomposable replacement\" $\\delta^{\\ast}(M)$ (in the split Grothendieck group of the category of persistence modules), which is expressed by a pair of interval-decomposable modules, that is, its positive and negative parts. We show that $M$ is interval-decomposable if and only if $\\delta^{\\ast}(M)$ is equal to $M$ in the split Grothendieck group. Furthermore, even for modules $M$ not necessarily interval-decomposable, $\\delta^{\\ast}(M)$ preserves the dimension vector and the rank invariant of $M$. In addition, we provide an algorithm to compute $\\delta^{\\ast}(M)$ (a high-level algorithm in the general case, and a detailed algorithm for the size $2\\times n$ case).","url_abs":"https://arxiv.org/abs/1911.01637v4","url_pdf":"https://arxiv.org/pdf/1911.01637v4.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":"on-approximation-of-2-d-persistence-modules","repo_url":"https://github.com/emerson-escolar/pmgap","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"GPL-3.0"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1911.01637","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}