2026-07-21T20:02:53Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:23187 2025-10-02T12:42:22Z ddc:004 open_access

A Theory of Sub-Barcodes Chubet, Oliver A. Gardner, Kirk P. Sheehy, Donald R. Topology Topological Data Analysis Persistent Homology Persistence Modules Barcodes Sub-barcodes Factorizations Lipschitz Extensions The primary tool in topological data analysis (TDA) is persistent homology, which involves computing a barcode - often from point-cloud or scalar field data - that serves as a topological signature for the underlying function. In this work, we introduce sub-barcodes and show how they arise naturally from factorizations of persistence module homomorphisms. We show that, as a partial order induced by factorizations, the relation of being a sub-barcode is strictly stronger than the rank invariant, and we apply sub-barcode theory to the problem of inferring information about the barcode of an unknown Lipschitz function from samples. The advantage of this approach is that it permits strong guarantees - with no noise - while requiring no sampling assumptions, and the resulting barcode is guaranteed to be a sub-barcode of every Lipschitz function that agrees with the data. We also present an algorithmic theory that allows for the efficient approximation of sub-barcodes using filtered Delaunay triangulations for Euclidean inputs. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Oliver A. Chubet and Kirk P. Gardner and Donald R. Sheehy 2025 Is Part Of LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2025.35 urn:nbn:de:0030-drops-231873 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.35 eng https://creativecommons.org/licenses/by/4.0/legalcode