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Algebraic Homotopy Interleaving Distance

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Part of the Lecture Notes in Computer Science book series (LNIP,volume 12829)

Abstract

The theory of persistence, which arises from topological data analysis, has been intensively studied in the one-parameter case both theoretically and in its applications. However, its extension to the multi-parameter case raises numerous difficulties, where it has been proven that no barcode-like decomposition exists. To tackle this problem, algebraic invariants have been proposed to summarize multi-parameter persistence modules, adapting classical ideas from commutative algebra and algebraic geometry to this context. Nevertheless, the crucial question of their stability has raised little attention so far, and many of the proposed invariants do not satisfy a naive form of stability.

In this paper, we equip the homotopy and the derived category of multi-parameter persistence modules with an appropriate interleaving distance. We prove that resolution functors are always isometric with respect to this distance. As an application, this explains why the graded-Betti numbers of a persistence module do not satisfy a naive form of stability. This opens the door to performing homological algebra operations while keeping track of stability. We believe this approach can lead to the definition of new stable invariants for multi-parameter persistence, and to new computable lower bounds for the interleaving distance (which has been recently shown to be NP-hard to compute in [2]).

Supported by Innosuisse grant 45665.1 IP-ICT.

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Notes

  1. 1.

    Where \(F_a^{\oplus \xi (F)(a)}\) is the direct sum of \(\xi (F)(a)\) copies of \(F_a\).

References

  1. Bjerkevik, H.B.: Stability of higher-dimensional interval decomposable persistence modules (2016)

    Google Scholar 

  2. Bjerkevik, H.B., Botnan, M.B., Kerber, M.: Computing the interleaving distance is NP-hard. In: Foundations of Computational Mathematics (2018)

    Google Scholar 

  3. Blumberg, A.J., Lesnick, M.: Universality of the homotopy interleaving distance (2017)

    Google Scholar 

  4. Carlsson, G.: Zomorodian. The theory of multidimensional persistence. Discrete and Computational Geometry, Afra (2009)

    Google Scholar 

  5. Chacholski, W., Scolamiero, M., Vaccarino, F.: Combinatorial presentation of multidimensional persistent homology. J. Pure Appl. Algebra 221, 1055–1075 (2017)

    MathSciNet  CrossRef  Google Scholar 

  6. Chuang, J., Lazarev, A.: Rank functions on triangulated categories (2021)

    Google Scholar 

  7. Heather, A.H., Schenck, H., Tillmann, U.: Stratifying multiparameter persistent homology, Nina Otter (2017)

    Google Scholar 

  8. Hiraoka, Y., Ike, Y., Yoshiwaki, M.: Algebraic stability theorem for derived categories of zigzag persistence modules (2020)

    Google Scholar 

  9. Kashiwara, M., Schapira, P.: Sheaves on Manifolds. Springer, Berlin (1990). https://doi.org/10.1007/978-3-662-02661-8

  10. Lesnick, M.: The theory of the interleaving distance on multidimensional persistence modules. Foundations of Computational Mathematics (2015)

    Google Scholar 

  11. Lesnick, M., Wright, M.: Interactive visualization of 2-D persistence modules. arXiv:1512.00180

  12. Milicevic, N.: Convolution of persistence modules (2020)

    Google Scholar 

  13. Miller, E.: Modules over posets: commutative and homological algebra (2019)

    Google Scholar 

  14. Steve, Y.: Oudot. American Mathematical Society, Persistence theory, From quiver representations to data analysis (2015)

    Google Scholar 

  15. Peeva, I.: Graded Syzygies. AA, vol. 16, 1st edn. Springer-Verlag, London (2011). https://doi.org/10.1007/978-0-85729-177-6

    CrossRef  MATH  Google Scholar 

  16. Stacks Project, Resolution functors. https://stacks.math.columbia.edu/tag/013U

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Correspondence to Nicolas Berkouk .

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Berkouk, N. (2021). Algebraic Homotopy Interleaving Distance. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2021. Lecture Notes in Computer Science(), vol 12829. Springer, Cham. https://doi.org/10.1007/978-3-030-80209-7_70

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  • DOI: https://doi.org/10.1007/978-3-030-80209-7_70

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-80208-0

  • Online ISBN: 978-3-030-80209-7

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