---
_id: '9111'
abstract:
- lang: eng
text: 'We study the probabilistic convergence between the mapper graph and the Reeb
graph of a topological space X equipped with a continuous function f:X→R. We first
give a categorification of the mapper graph and the Reeb graph by interpreting
them in terms of cosheaves and stratified covers of the real line R. We then introduce
a variant of the classic mapper graph of Singh et al. (in: Eurographics symposium
on point-based graphics, 2007), referred to as the enhanced mapper graph, and
demonstrate that such a construction approximates the Reeb graph of (X,f) when
it is applied to points randomly sampled from a probability density function concentrated
on (X,f). Our techniques are based on the interleaving distance of constructible
cosheaves and topological estimation via kernel density estimates. Following Munch
and Wang (In: 32nd international symposium on computational geometry, volume 51
of Leibniz international proceedings in informatics (LIPIcs), Dagstuhl, Germany,
pp 53:1–53:16, 2016), we first show that the mapper graph of (X,f), a constructible
R-space (with a fixed open cover), approximates the Reeb graph of the same space.
We then construct an isomorphism between the mapper of (X,f) to the mapper of
a super-level set of a probability density function concentrated on (X,f). Finally,
building on the approach of Bobrowski et al. (Bernoulli 23(1):288–328, 2017b),
we show that, with high probability, we can recover the mapper of the super-level
set given a sufficiently large sample. Our work is the first to consider the mapper
construction using the theory of cosheaves in a probabilistic setting. It is part
of an ongoing effort to combine sheaf theory, probability, and statistics, to
support topological data analysis with random data.'
acknowledgement: "AB was supported in part by the European Union’s Horizon 2020 research
and innovation\r\nprogramme under the Marie Sklodowska-Curie GrantAgreement No.
754411 and NSF IIS-1513616. OB was supported in part by the Israel Science Foundation,
Grant 1965/19. BW was supported in part by NSF IIS-1513616 and DBI-1661375. EM was
supported in part by NSF CMMI-1800466, DMS-1800446, and CCF-1907591.We would like
to thank the Institute for Mathematics and its Applications for hosting a workshop
titled Bridging Statistics and Sheaves in May 2018, where this work was conceived.\r\nOpen
Access funding provided by Institute of Science and Technology (IST Austria)."
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Adam
full_name: Brown, Adam
id: 70B7FDF6-608D-11E9-9333-8535E6697425
last_name: Brown
- first_name: Omer
full_name: Bobrowski, Omer
last_name: Bobrowski
- first_name: Elizabeth
full_name: Munch, Elizabeth
last_name: Munch
- first_name: Bei
full_name: Wang, Bei
last_name: Wang
citation:
ama: Brown A, Bobrowski O, Munch E, Wang B. Probabilistic convergence and stability
of random mapper graphs. Journal of Applied and Computational Topology.
2021;5(1):99-140. doi:10.1007/s41468-020-00063-x
apa: Brown, A., Bobrowski, O., Munch, E., & Wang, B. (2021). Probabilistic convergence
and stability of random mapper graphs. Journal of Applied and Computational
Topology. Springer Nature. https://doi.org/10.1007/s41468-020-00063-x
chicago: Brown, Adam, Omer Bobrowski, Elizabeth Munch, and Bei Wang. “Probabilistic
Convergence and Stability of Random Mapper Graphs.” Journal of Applied and
Computational Topology. Springer Nature, 2021. https://doi.org/10.1007/s41468-020-00063-x.
ieee: A. Brown, O. Bobrowski, E. Munch, and B. Wang, “Probabilistic convergence
and stability of random mapper graphs,” Journal of Applied and Computational
Topology, vol. 5, no. 1. Springer Nature, pp. 99–140, 2021.
ista: Brown A, Bobrowski O, Munch E, Wang B. 2021. Probabilistic convergence and
stability of random mapper graphs. Journal of Applied and Computational Topology.
5(1), 99–140.
mla: Brown, Adam, et al. “Probabilistic Convergence and Stability of Random Mapper
Graphs.” Journal of Applied and Computational Topology, vol. 5, no. 1,
Springer Nature, 2021, pp. 99–140, doi:10.1007/s41468-020-00063-x.
short: A. Brown, O. Bobrowski, E. Munch, B. Wang, Journal of Applied and Computational
Topology 5 (2021) 99–140.
date_created: 2021-02-11T14:41:02Z
date_published: 2021-03-01T00:00:00Z
date_updated: 2025-04-14T07:43:51Z
day: '01'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.1007/s41468-020-00063-x
ec_funded: 1
external_id:
arxiv:
- '1909.03488'
file:
- access_level: open_access
checksum: 3f02e9d47c428484733da0f588a3c069
content_type: application/pdf
creator: dernst
date_created: 2021-02-11T14:43:59Z
date_updated: 2021-02-11T14:43:59Z
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file_name: 2020_JourApplCompTopology_Brown.pdf
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has_accepted_license: '1'
intvolume: ' 5'
issue: '1'
language:
- iso: eng
license: https://creativecommons.org/licenses/by/4.0/
month: '03'
oa: 1
oa_version: Published Version
page: 99-140
project:
- _id: 260C2330-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '754411'
name: ISTplus - Postdoctoral Fellowships
publication: Journal of Applied and Computational Topology
publication_identifier:
eissn:
- 2367-1734
issn:
- 2367-1726
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Probabilistic convergence and stability of random mapper graphs
tmp:
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legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1
volume: 5
year: '2021'
...