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Persistent (Co)Homology in Matrix Multiplication Time

Authors Dmitriy Morozov , Primoz Skraba

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ACM Subject Classification
  • Computing methodologies → Algebraic algorithms
  • Mathematics of computing → Algebraic topology
  • Computing methodologies → Linear algebra algorithms
  • Theory of computation → Computational geometry
Keywords
  • persistent homology
  • matrix multiplication
  • cycle representatives

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Abstract

Most algorithms for computing persistent homology do so by tracking cycles that represent homology classes. There are many choices of such cycles, and specific choices have found different uses in applications. Although it is known that persistence diagrams can be computed in matrix multiplication time for the more general case of zigzag persistent homology [Milosavljević et al., 2011], it is not clear how to extract cycle representatives, especially if specific representatives are desired. In this paper, we provide the same matrix multiplication bound for computing representatives for the two choices common in applications in the case of ordinary persistent (co)homology. We first provide a fast version of the reduction algorithm, which is simpler than the algorithm in [Milosavljević et al., 2011], but returns a different set of representatives than the standard algorithm [Edelsbrunner et al., 2002]. We then give a fast version of a variant called the row algorithm [De Silva et al., 2011], which returns the same representatives as the standard algorithm.

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Dmitriy Morozov and Primoz Skraba. Persistent (Co)Homology in Matrix Multiplication Time. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 68:1-68:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.SoCG.2025.68

Author Details

Dmitriy Morozov
  • Lawrence Berkeley National Laboratory, Berkeley, CA, USA
Primoz Skraba
  • Queen Mary University of London, UK

Funding

  • Morozov, Dmitriy: supported by the Director, Office of Science, Office of Advanced Scientific Computing Research, of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.
  • Skraba, Primoz: supported by the EPSRC AI Hub on Mathematical Foundations of Intelligence: An "Erlangen Programme" for AI No. EP/Y028872/1.

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