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The space of persistence diagrams on $n$ points coarsely embeds into Hilbert space


Authors: Atish Mitra and Žiga Virk
Journal: Proc. Amer. Math. Soc. 149 (2021), 2693-2703
MSC (2020): Primary 54F45, 46C05; Secondary 55M10
DOI: https://doi.org/10.1090/proc/15363
Published electronically: March 22, 2021
MathSciNet review: 4246818
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Abstract: We prove that the space of persistence diagrams on $n$ points (with the bottleneck or a Wasserstein distance) coarsely embeds into Hilbert space by showing it is of asymptotic dimension $2n$. Such an embedding enables utilisation of Hilbert space techniques on the space of persistence diagrams. We also prove that when the number of points is not bounded, the corresponding spaces of persistence diagrams do not have finite asymptotic dimension. Furthermore, in the case of the bottleneck distance, the corresponding space does not coarsely embed into Hilbert space.


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Additional Information

Atish Mitra
Affiliation: Department of Mathematical Sciences, Montana Technological University, Butte, Montana 59701
MR Author ID: 819244
Email: atish.mitra@gmail.com

Žiga Virk
Affiliation: Faculty of Computer and Information Science, University of Ljubljana, Slovenia, 1000
Email: ziga.virk@fri.uni-lj.si

Received by editor(s): November 19, 2019
Received by editor(s) in revised form: September 18, 2020, September 24, 2020, and September 25, 2020
Published electronically: March 22, 2021
Additional Notes: This research was partially supported by a bilateral grant BI-US/18-20-060 of ARRS. The second author was supported by Slovenian Research Agency grants N1-0114, P1-0292, J1-8131, and N1-0064.
Communicated by: Kenneth Bromberg
Article copyright: © Copyright 2021 American Mathematical Society