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Foundations of Computational Mathematics

, Volume 17, Issue 6, pp 1367–1406 | Cite as

Multidimensional Persistence and Noise

  • Martina Scolamiero
  • Wojciech ChachólskiEmail author
  • Anders Lundman
  • Ryan Ramanujam
  • Sebastian Öberg
Article

Abstract

In this paper, we study multidimensional persistence modules (Carlsson and Zomorodian in Discrete Comput Geom 42(1):71–93, 2009; Lesnick in Found Comput Math 15(3):613–650, 2015) via what we call tame functors and noise systems. A noise system leads to a pseudometric topology on the category of tame functors. We show how this pseudometric can be used to identify persistent features of compact multidimensional persistence modules. To count such features, we introduce the feature counting invariant and prove that assigning this invariant to compact tame functors is a 1-Lipschitz operation. For one-dimensional persistence, we explain how, by choosing an appropriate noise system, the feature counting invariant identifies the same persistent features as the classical barcode construction.

Keywords

Multidimensional persistence Persistence modules Noise systems Stable invariants 

Mathematics Subject Classification

Primary: 55 18 68 

Notes

Acknowledgments

We would like to thank Claudia Landi for inspiring discussions about stability.

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Copyright information

© SFoCM 2016

Authors and Affiliations

  • Martina Scolamiero
    • 1
  • Wojciech Chachólski
    • 2
    Email author
  • Anders Lundman
    • 2
  • Ryan Ramanujam
    • 2
  • Sebastian Öberg
    • 2
  1. 1.EPFL SV BMI UPHESSLausanneSwitzerland
  2. 2.Department of MathematicsKTH StockholmStockholmSweden

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