A comparison framework for interleaved persistence modules

Abstract

We present a generalization of the induced matching theorem of as reported by Bauer and Lesnick (in: Proceedings of the thirtieth annual symposium computational geometry 2014) and use it to prove a generalization of the algebraic stability theorem for \({\mathbb {R}}\)-indexed pointwise finite-dimensional persistence modules. Via numerous examples, we show how the generalized algebraic stability theorem enables the computation of rigorous error bounds in the space of persistence diagrams that go beyond the typical formulation in terms of bottleneck (or log bottleneck) distance.

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Acknowledgements

R. L. would like to thank Charles Weibel, Michael Lesnick, and Ulrich Bauer for the many insightful discussions that led to the results presented in this paper. The authors also thank the anonymous reviewers for their suggested corrections. On behalf of all authors, the corresponding author states that there is no conflict of interest.

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Correspondence to Rachel Levanger.

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S. H. and K. M. were partially supported by Grants NSF-DMS-1125174, 1248071, 1521771, NIH 1R01GM126555-01 and DARPA contracts HR0011-16-2-0033, FA8750-17-C-0054. M. K. was supported by ERC Gudhi (ERC-2013-ADG-339025). R. L. was supported by DARPA contracts HR0011-17-1-0004 and HR0011-16-2-0033.

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Harker, S., Kramár, M., Levanger, R. et al. A comparison framework for interleaved persistence modules. J Appl. and Comput. Topology 3, 85–118 (2019). https://doi.org/10.1007/s41468-019-00026-x

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Keywords

  • Persistence module
  • Persistence diagram
  • Persistent homology
  • Error bounds
  • Topological data analysis

Mathematics Subject Classification

  • 55N35
  • 55U10
  • 65G99