Basic metric geometry of the bottleneck distance

Che, Mauricio; Galaz-García, Fernando; Guijarro, Luis; Membrillo Solis, Ingrid; Valiunas, Motiejus

doi:10.1090/proc/16776

Basic metric geometry of the bottleneck distance
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by Mauricio Che, Fernando Galaz-García, Luis Guijarro, Ingrid Membrillo Solis and Motiejus Valiunas;

Proc. Amer. Math. Soc. 152 (2024), 3575-3591

DOI: https://doi.org/10.1090/proc/16776

Published electronically: June 12, 2024

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Abstract:

Given a metric pair $(X,A)$, i.e. a metric space $X$ and a distinguished closed set $A\subset X$, one may construct in a functorial way a pointed pseudometric space $\mathcal {D}_\infty (X,A)$ of persistence diagrams equipped with the bottleneck distance. We investigate the basic metric properties of the spaces $\mathcal {D}_\infty (X,A)$ and obtain characterizations of their metrizability, completeness, separability, and geodesicity.

References

Hideto Asashiba, Mickaël Buchet, Emerson G. Escolar, Ken Nakashima, and Michio Yoshiwaki, On interval decomposability of 2D persistence modules, Comput. Geom. 105/106 (2022), Paper No. 101879, 33. MR 4402576, DOI 10.1016/j.comgeo.2022.101879
Håvard Bakke Bjerkevik, On the stability of interval decomposable persistence modules, Discrete Comput. Geom. 66 (2021), no. 1, 92–121. MR 4270636, DOI 10.1007/s00454-021-00298-0
Magnus Bakke Botnan, Vadim Lebovici, and Steve Oudot, On rectangle-decomposable 2-parameter persistence modules, Discrete Comput. Geom. 68 (2022), no. 4, 1078–1101. MR 4517095, DOI 10.1007/s00454-022-00383-y
Peter Bubenik and Alex Elchesen, Universality of persistence diagrams and the bottleneck and Wasserstein distances, Comput. Geom. 105/106 (2022), Paper No. 101882, 18. MR 4414770, DOI 10.1016/j.comgeo.2022.101882
Peter Bubenik and Alex Elchesen, Virtual persistence diagrams, signed measures, Wasserstein distances, and Banach spaces, J. Appl. Comput. Topol. 6 (2022), no. 4, 429–474. MR 4496687, DOI 10.1007/s41468-022-00091-9
Peter Bubenik and Iryna Hartsock, Topological and metric properties of spaces of generalized persistence diagrams, arXiv:2205.08506 [math.AT], 2022.
Peter Bubenik and Tane Vergili, Topological spaces of persistence modules and their properties, J. Appl. Comput. Topol. 2 (2018), no. 3-4, 233–269. MR 3927353, DOI 10.1007/s41468-018-0022-4
Dmitri Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, Graduate Studies in Mathematics, vol. 33, American Mathematical Society, Providence, RI, 2001. MR 1835418, DOI 10.1090/gsm/033
Gunnar Carlsson and Mikael Vejdemo-Johansson, Topological data analysis with applications, Cambridge University Press, Cambridge, 2022. MR 4346385, DOI 10.1017/9781108975704
Mauricio Che, Fernando Galaz-García, Luis Guijarro, and Ingrid Membrillo Solis, Metric geometry of spaces of persistence diagrams, arXiv:2109.14697 [math.MG], 2021.
David Cohen-Steiner, Herbert Edelsbrunner, and John Harer, Stability of persistence diagrams, Discrete Comput. Geom. 37 (2007), no. 1, 103–120. MR 2279866, DOI 10.1007/s00454-006-1276-5
David Cohen-Steiner, Herbert Edelsbrunner, John Harer, and Yuriy Mileyko, Lipschitz functions have $L_p$-stable persistence, Found. Comput. Math. 10 (2010), no. 2, 127–139. MR 2594441, DOI 10.1007/s10208-010-9060-6
Vincent Divol and Théo Lacombe, Understanding the topology and the geometry of the space of persistence diagrams via optimal partial transport, J. Appl. Comput. Topol. 5 (2021), no. 1, 1–53. MR 4224153, DOI 10.1007/s41468-020-00061-z
Isaac Goldbring, Ultrafilters throughout mathematics, Graduate Studies in Mathematics, vol. 220, American Mathematical Society, Providence, RI, [2022] ©2022. MR 4454845, DOI 10.1090/gsm/220
M. Kapovich and B. Leeb, On asymptotic cones and quasi-isometry classes of fundamental groups of $3$-manifolds, Geom. Funct. Anal. 5 (1995), no. 3, 582–603. MR 1339818, DOI 10.1007/BF01895833
Yuriy Mileyko, Sayan Mukherjee, and John Harer, Probability measures on the space of persistence diagrams, Inverse Problems 27 (2011), no. 12, 124007, 22. MR 2854323, DOI 10.1088/0266-5611/27/12/124007
Steve Y. Oudot, Persistence theory: from quiver representations to data analysis, Mathematical Surveys and Monographs, vol. 209, American Mathematical Society, Providence, RI, 2015. MR 3408277, DOI 10.1090/surv/209
Jacek Skryzalin and Gunnar Carlsson, Numeric invariants from multidimensional persistence, J. Appl. Comput. Topol. 1 (2017), no. 1, 89–119. MR 3975550, DOI 10.1007/s41468-017-0003-z
Katharine Turner, Yuriy Mileyko, Sayan Mukherjee, and John Harer, Fréchet means for distributions of persistence diagrams, Discrete Comput. Geom. 52 (2014), no. 1, 44–70. MR 3231030, DOI 10.1007/s00454-014-9604-7
Afra Zomorodian and Gunnar Carlsson, Computing persistent homology, Discrete Comput. Geom. 33 (2005), no. 2, 249–274. MR 2121296, DOI 10.1007/s00454-004-1146-y