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A Theory of Sub-Barcodes

Authors Oliver A. Chubet , Kirk P. Gardner , Donald R. Sheehy

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ACM Subject Classification
  • Mathematics of computing → Algebraic topology
  • Mathematics of computing → Geometric topology
  • Theory of computation → Computational geometry
  • Computing methodologies → Algebraic algorithms
Keywords
  • Topology
  • Topological Data Analysis
  • Persistent Homology
  • Persistence Modules
  • Barcodes
  • Sub-barcodes
  • Factorizations
  • Lipschitz Extensions

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Abstract

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.

Cite As Get BibTex

Oliver A. Chubet, Kirk P. Gardner, and Donald R. Sheehy. A Theory of Sub-Barcodes. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 35:1-35:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.SoCG.2025.35

Author Details

Oliver A. Chubet
  • Department of Computer Science, North Carolina State University, Raleigh, NC, USA
Kirk P. Gardner
  • Department of Computer Science, North Carolina State University, Raleigh, NC, USA
Donald R. Sheehy
  • Department of Computer Science, North Carolina State University, Raleigh, NC, USA

Funding

This work was partially funded by the NSF under grant CCF-2017980.

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