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Persistence and Sheaves: from Theory to Applications

Nicolas Berkouk 1, 2 
2 DATASHAPE - Understanding the Shape of Data
CRISAM - Inria Sophia Antipolis - Méditerranée , Inria Saclay - Ile de France
Abstract : Topological data analysis is a recent field of research aiming at using techniques coming from algebraic topology to define descriptors of datasets. To be useful in practice, these descriptors must be computable, and coming with a notion of metric, in order to express their stability properties with respect to the noise that always comes with real world data. Persistence theory was elaborated in the early 2000’s as a first theoretical setting to define such descriptors - the now famous so-called barcodes. However very well suited to be implemented in a computer, persistence theory has certain limitations. In this manuscript, we establish explicit links between the theory of derived sheaves equipped with the convolution distance (after Kashiwara-Schapira) and persistence theory. We start by showing a derived isometry theorem for constructible sheaves over R, that is, we express the convolution distance between two sheaves as a matching distance between their graded barcodes. This enables us to conclude in this setting that the convolution distance is closed, and that the collection of constructible sheaves over R equipped with the convolution distance is locally path-connected. Then, we observe that the collection of zig-zag/level sets persistence modules associated to a real valued function carry extra structure, which we call Mayer-Vietoris systems. We classify all Mayer-Vietoris systems under finiteness assumptions. This allows us to establish a functorial isometric correspondence between the derived category of constructible sheaves over R equipped with the convolution distance, and the category of strongly pfd Mayer-Vietoris systems endowed with the interleaving distance. We deduce from this result a way to compute barcodes of sheaves from already existing software. Finally, we give a purely sheaf theoretic definition of the notion of ephemeral persistence module. We prove that the observable category of persistence modules (the quotient category of persistence modules by the sub-category of ephemeral ones) is equivalent to the well-known category of gamma-sheaves.
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Submitted on : Sunday, October 18, 2020 - 3:40:52 PM
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Nicolas Berkouk. Persistence and Sheaves: from Theory to Applications. Algebraic Topology [math.AT]. École polytechnique; INRIA Saclay, 2020. English. ⟨tel-02970509v1⟩

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