Abstract
We interpret some results of persistent homology and barcodes (in any dimension) with the language of microlocal sheaf theory. For that purpose we study the derived category of sheaves on a real finite-dimensional vector space \(\mathbb {V}\). By using the operation of convolution, we introduce a pseudo-distance on this category and prove in particular a stability result for direct images. Then we assume that \(\mathbb {V}\) is endowed with a closed convex proper cone \(\gamma \) with non empty interior and study \(\gamma \)-sheaves, that is, constructible sheaves with microsupport contained in the antipodal to the polar cone (equivalently, constructible sheaves for the \(\gamma \)-topology). We prove that such sheaves may be approximated (for the pseudo-distance) by “piecewise linear” \(\gamma \)-sheaves. Finally we show that these last sheaves are constant on stratifications by \(\gamma \)-locally closed sets, an analogue of barcodes in higher dimension.
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Notes
\(\mu {\mathrm{supp}}(F)\) was denoted by \({\text {SS}}(F)\) in Kashiwara and Schapira (1990).
See Sect. 2.4 for related constructions.
In practice the cone \(\gamma \) will be polyhedral.
As already mentioned, these results were clarified during discussions of the second named author with Nicolas Berkouk.
This statement is due to Curry (2013, Th.4.2.10).
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Acknowledgements
The second named author warmly thanks Gregory Ginot for having organized a seminar on persistent homology, at the origin of this paper, and Benoît Jubin for fruitful discussions on this subject. In this seminar, Nicolas Berkouk and Steve Oudot pointed out the problem of approximating constructible sheaves with objects which would be similar to higher dimensional barcodes, what we do, in some sense, here. Moreover, the links between \(\gamma \)-sheaves and persistent modules (Proposition 2.15) were clarified during discussions with Nicolas Berkouk.
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The research of M.K was supported by Grant-in-Aid for Scientific Research (B) 15H03608, Japan Society for the Promotion of Science. The research of P.S was supported by the ANR-15-CE40-0007 “MICROLOCAL”.
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Kashiwara, M., Schapira, P. Persistent homology and microlocal sheaf theory. J Appl. and Comput. Topology 2, 83–113 (2018). https://doi.org/10.1007/s41468-018-0019-z
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DOI: https://doi.org/10.1007/s41468-018-0019-z