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Level-sets persistence and sheaf theory

Nicolas Berkouk 1 Grégory Ginot 2 Steve Oudot 1
1 DATASHAPE - Understanding the Shape of Data
CRISAM - Inria Sophia Antipolis - Méditerranée , Inria Saclay - Ile de France
Abstract : In this paper we provide an explicit connection between level-sets persistence and derived sheaf theory over the real line. In particular we construct a functor from 2-parameter persistence modules to sheaves over $\mathbb{R}$, as well as a functor in the other direction. We also observe that the 2-parameter persistence modules arising from the level sets of Morse functions carry extra structure that we call a Mayer-Vietoris system. We prove classification, barcode decomposition, and stability theorems for these Mayer-Vietoris systems, and we show that the aforementioned functors establish a pseudo-isometric equivalence of categories between derived constructible sheaves with the convolution or (derived) bottleneck distance and the interleaving distance of strictly pointwise finite-dimensional Mayer-Vietoris systems. Ultimately, our results provide a functorial equivalence between level-sets persistence and derived pushforward for continuous real-valued functions.
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Preprints, Working Papers, ...
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https://hal.inria.fr/hal-02425597
Contributor : Steve Oudot <>
Submitted on : Monday, December 30, 2019 - 7:20:42 PM
Last modification on : Saturday, May 1, 2021 - 3:46:14 AM

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  • HAL Id : hal-02425597, version 1
  • ARXIV : 1907.09759

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Nicolas Berkouk, Grégory Ginot, Steve Oudot. Level-sets persistence and sheaf theory. 2019. ⟨hal-02425597⟩

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