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Tracking the Persistence of Harmonic Chains: Barcode and Stability

Authors Tao Hou , Salman Parsa , Bei Wang

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ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Mathematics of computing → Topology
Keywords
  • Persistent homology
  • harmonic chains
  • topological data analysis

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Abstract

The persistence barcode is a topological descriptor of data that plays a fundamental role in topological data analysis. Given a filtration of data, the persistence barcode tracks the evolution of its homology groups. In this paper, we introduce a new type of barcode, called the harmonic chain barcode, which tracks the evolution of harmonic chains. In addition, we show that the harmonic chain barcode is stable. Given a filtration of a simplicial complex of size m, we present an algorithm to compute its harmonic chain barcode in O(m³) time. Consequently, the harmonic chain barcode can enrich the family of topological descriptors in applications where a persistence barcode is applicable, such as feature vectorization and machine learning.

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Tao Hou, Salman Parsa, and Bei Wang. Tracking the Persistence of Harmonic Chains: Barcode and Stability. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 58:1-58:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.SoCG.2025.58

Author Details

Tao Hou
  • University of Oregon, Eugene, OR, USA
Salman Parsa
  • DePaul University, Chicago, IL, USA
Bei Wang
  • University of Utah, Salt Lake City, UT, USA

Funding

  • Hou, Tao: Supported by NSF CCF 2439255.
  • Parsa, Salman: Supported by DOE DE-SC0021015.
  • Wang, Bei: Supported by DOE DE-SC0021015, NSF IIS-2145499 and NSF DMS-2301361.

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