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Persistence and Sheaves : from Theory to Applications

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 res-pect 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 des-criptors - the now famous so-called barcodes. Howe-ver very well suited to be implemented in a compu-ter, persistence theory has certain limitations. In this manuscript, we establish explicit links between the theory of derived sheaves equipped with the convolu-tion distance (after Kashiwara-Schapira) and persis-tence 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 collec-tion 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 sys-tems under finiteness assumptions. This allows us to establish a functorial isometric correspondence bet-ween 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 mo-dules (the quotient category of persistence modules by the sub-category of ephemeral ones) is equivalent to the well-known category of -sheaves.
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Nicolas Berkouk. Persistence and Sheaves : from Theory to Applications. Algebraic Topology [math.AT]. Institut Polytechnique de Paris, 2020. English. ⟨NNT : 2020IPPAX032⟩. ⟨tel-02976867⟩

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