GEOMETRIC BOUNDS FOR PERSISTENCE

[en] In this paper, we offer a new perspective on persistent homology by integrating key concepts from metric geometry. For a given compact subset X of a Banach space Y, we analyze the topological features arising in the family of nested neighborhoods of X in Y and provide several geometric bounds on their persistence (lifespans). We begin by examining the lifespans of these homology classes in terms of their filling radii in Y, establishing connections between these lifespans and fundamental invariants in metric geometry, such as the Urysohn width. We then derive bounds on these lifespans by considering the l^\infinity-principal components of X, also known as Kolmogorov widths. Additionally, we introduce and investigate the concept of extinction time of a metric space X: the critical threshold beyond which no homological features persist in any degree. We propose methods for estimating the Čech and Vietoris-Rips extinction times of X by relating X to its convex hull and to its tight span, respectively.

Disciplines :

Mathematics

Author, co-author :

BALITSKIY, Alexey ; University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH)

Coskunuzer, Baris

Mémoli, Facundo

External co-authors :

yes

Language :

English

Title :

GEOMETRIC BOUNDS FOR PERSISTENCE

Publication date :

2025

Journal title :

Transactions of the American Mathematical Society

ISSN :

0002-9947

eISSN :

1088-6850

Publisher :

American Mathematical Society (AMS)

Peer reviewed :

Peer Reviewed verified by ORBi

Additional URL :

https://www.ams.org/tran/0000-000-00/S0002-9947-2025-09514-X/tran9514_AM.pdf
https://arxiv.org/abs/2403.13980

Funders :

National Science Foundation
H2020 Marie Skłodowska-Curie Actions

Available on ORBilu :

since 15 October 2025

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Bibliography

[AA17] MichaƗ Adamaszek and Henry Adams, The Vietoris-Rips complexes of a circle, Pacific J. Math. 290 (2017), no. 1, 1–40, DOI 10.2140/pjm.2017.290.1. MR3673078
[AAF19] Mehmet E. Aktas, Esra Akbas, and Ahmed El Fatmaoui, Persistence homology of networks: methods and applications, Appl. Network Sci. 4 (2019), no. 1, 1–28.
[AAR19] MichaƗ Adamaszek, Henry Adams, and Samadwara Reddy, On Vietoris-Rips complexes of ellipses, J. Topol. Anal. 11 (2019), no. 3, 661–690, DOI 10.1142/S1793525319500274. MR3999516
[ABC+22] Henry Adams, Johnathan Bush, Nate Clause, Florian Frick, Mario G´omez, Michael Harrison, R. Amzi Jeffs, Evgeniya Lagoda, Sunhyuk Lim, Facundo Mémoli, et al., Gromov–Hausdorff distances, Borsuk–Ulam theorems, and Vietoris–Rips complexes, Preprint, arXiv:2301.00246, 2022.
[AC22] Henry Adams and Baris Coskunuzer, Geometric approaches to persistent homology, SIAM J. Appl. Algebra Geom. 6 (2022), no. 4, 685–710, DOI 10.1137/21M1422914. MR4522864
[AFV23] Henry Adams, Florian Frick, and Žiga Virk, Vietoris thickenings and complexes have isomorphic homotopy groups, J. Appl. Comput. Topol. 7 (2023), no. 2, 221–241, DOI 10.1007/s41468-022-00106-5. MR4588139
[Ale26] Paul Alexandroff, Notes supplémentaires au “Mémoire sur les multiplicités Cantoriennes”, rédigées d’après les papiers posthumes de Paul Urysohn, Fund. Math. 1 (1926), no. 8, 352–359.
[Ale28] Paul Alexandroff, ¨ Uber den allgemeinen Dimensionsbegriff und seine Beziehungen zur elementaren geometrischen Anschauung (German), Math. Ann. 98 (1928), no. 1, 617–635, DOI 10.1007/BF01451612. MR1512423
[Ale33] Paul Alexandroff, ¨ Uber die Urysohnschen Konstanten, Fund. Math. 20 (1933), no. 1, 140–150.
[ALS13] Dominique Attali, André Lieutier, and David Salinas, Vietoris-Rips complexes also provide topologically correct reconstructions of sampled shapes, Comput. Geom. 46 (2013), no. 4, 448–465, DOI 10.1016/j.comgeo.2012.02.009. MR3003901
[AMMW24] Henry Adams, Facundo Mémoli, Michael Moy, and Qingsong Wang, The persistent topology of optimal transport based metric thickenings, Algebr. Geom. Topol. 24 (2024), no. 1, 393–447, DOI 10.2140/agt.2024.24.393. MR4721372
[AP56] N. Aronszajn and P. Panitchpakdi, Extension of uniformly continuous transformations and hyperconvex metric spaces, Pacific J. Math. 6 (1956), 405–439. MR84762
[AQO+20] Erik J. Amézquita, Michelle Y. Quigley, Tim Ophelders, Elizabeth Munch, and Daniel H. Chitwood, The shape of things to come: TDA and biology, from molecules to organisms, Dev. Dyn. 249 (2020), no. 7, 816–833.
[Bal21] Alexey Balitskiy, Bounds on Urysohn Width, ProQuest LLC, Ann Arbor, MI, 2021. Thesis (Ph.D.)–Massachusetts Institute of Technology. MR4464413
[Bar94] S. A. Barannikov, The framed Morse complex and its invariants, Singularities and bifurcations, Adv. Soviet Math., vol. 21, Amer. Math. Soc., Providence, RI, 1994, pp. 93–115. MR1310596
[Bau21] Ulrich Bauer, Ripser: efficient computation of Vietoris-Rips persistence barcodes, J. Appl. Comput. Topol. 5 (2021), no. 3, 391–423, DOI 10.1007/s41468-021-00071-5. MR4298669
[BBI01] Dmitri Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, Graduate Studies in Mathematics, vol. 33, American Mathematical Society, Providence, RI, 2001, DOI 10.1090/gsm/033. MR1835418
[BH10] Michael Brunnbauer and Bernhard Hanke, Large and small group homology, J. Topol. 3 (2010), no. 2, 463–486, DOI 10.1112/jtopol/jtq014. MR2651368
[BHK15] Imre Bárány, Andreas F. Holmsen, and Roman Karasev, Topology of geometric joins, Discrete Comput. Geom. 53 (2015), no. 2, 402–413, DOI 10.1007/s00454-015-9665-2. MR3316229
[BHPG+21] Michael Bleher, Lukas Hahn, Juan Ángel Patino-Galindo, Mathieu Carriere, Ulrich Bauer, Ral Rabadán, and Andreas Ott, Topology identifies emerging adaptive mutations in SARS-CoV-2, medRxiv, 2021, pp. 2021–06.
[BHPW20] Peter Bubenik, Michael Hull, Dhruv Patel, and Benjamin Whittle, Persistent homology detects curvature, Inverse Problems 36 (2020), no. 2, 025008, 23, DOI 10.1088/1361-6420/ab4ac0. MR4063201
[BKRR23] Ulrich Bauer, Michael Kerber, Fabian Roll, and Alexander Rolle, A unified view on the functorial nerve theorem and its variations, Expo.Math. 41 (2023), no. 4, Paper No. 125503, 52, DOI 10.1016/j.exmath.2023.04.005. MR4652781
[BKS17] Omer Bobrowski, Matthew Kahle, and Primoz Skraba, Maximally persistent cycles in random geometric complexes, Ann. Appl. Probab. 27 (2017), no. 4, 2032–2060, DOI 10.1214/16-AAP1232. MR3693519
[BM24] Peter Bubenik and Nikola Milićević, Homotopy, homology, and persistent homology using closure spaces, J. Appl. Comput. Topol. 8 (2024), no. 3, 579–641, DOI 10.1007/s41468-024-00183-8. MR4799024
[Bor32] Karol Borsuk, ¨ Uber eine Klasse von lokal zusammenhängenden Räumen, Fund. Math. 19 (1932), no. 1, 220–242.
[Bor48] Karol Borsuk, On the imbedding of systems of compacta in simplicial complexes, Fund. Math. 35 (1948), 217–234, DOI 10.4064/fm-35-1-217-234. MR28019
[Bor75] Karol Borsuk, Theory of shape, Monografie Matematyczne [Mathematical Monographs], Tom 59, PWN—Polish Scientific Publishers, Warsaw, 1975. MR418088
[BS83] Enrico Bombieri and Leon Simon, On the Gehring link problem, Seminar on minimal submanifolds, Ann. of Math. Stud., vol. 103, Princeton Univ. Press, Princeton, NJ, 1983, pp. 271–274. MR795242
[Cai94] Mingliang Cai, On Gromov’s large Riemannian manifolds, Geom. Dedicata 50 (1994), no. 1, 37–45, DOI 10.1007/BF01263649. MR1280793
[Car09] Gunnar Carlsson, Topology and data, Bull. Amer. Math. Soc. (N.S.) 46 (2009), no. 2, 255–308, DOI 10.1090/S0273-0979-09-01249-X. MR2476414
[CB15] William Crawley-Boevey, Decomposition of pointwise finite-dimensional persistence modules, J. Algebra Appl. 14 (2015), no. 5, 1550066, 8, DOI 10.1142/S0219498815500668. MR3323327
[CCI+20] Mathieu Carrière, Frédéric Chazal, Yuichi Ike, Théo Lacombe, Martin Royer, and Yuhei Umeda, Perslay: a neural network layer for persistence diagrams and new graph topological signatures, International Conference on Artificial Intelligence and Statistics, pp. 2786–2796, PMLR, 2020.
[CCR13] Joseph Minhow Chan, Gunnar Carlsson, and Raul Rabadan, Topology of viral evolution, Proc. Natl. Acad. Sci. USA 110 (2013), no. 46, 18566–18571, DOI 10.1073/pnas.1313480110. MR3153945
[CCSG+09] Frédéric Chazal, David Cohen-Steiner, Leonidas J. Guibas, Facundo Mémoli, and Steve Y. Oudot, Gromov–Hausdorff stable signatures for shapes using persistence, Computer Graphics Forum, vol. 28, pp. 1393–1403, Wiley Online Library, 2009.
[CDSGO16] Frédéric Chazal, Vin de Silva, Marc Glisse, and Steve Oudot, The structure and stability of persistence modules, Springer Briefs in Mathematics, Springer, [Cham], 2016, DOI 10.1007/978-3-319-42545-0. MR3524869
[CDSO14] Frédéric Chazal, Vin de Silva, and Steve Oudot, Persistence stability for geometric complexes, Geom. Dedicata 173 (2014), 193–214, DOI 10.1007/s10711-013-9937-z. MR3275299
[Che97] Victor Chepoi, A TX-approach to some results on cuts and metrics, Adv. in Appl. Math. 19 (1997), no. 4, 453–470, DOI 10.1006/aama.1997.0549. MR1479014
[CI08] Carina Curto and Vladimir Itskov, Cell groups reveal structure of stimulus space, PLoS Comput. Biol. 4 (2008), no. 10, e1000205, 13, DOI 10.1371/journal.pcbi.1000205. MR2457124
[CVJ21] Gunnar Carlsson and Mikael Vejdemo-Johansson, Topological data analysis with applications, Cambridge University Press, Cambridge, 2022, DOI 10.1017/9781108975704. MR4346385
[CW17] Zixuan Cang and Guo-Wei Wei, Topologynet: topology based deep convolutional and multi-task neural networks for biomolecular property predictions, PLoS Comput. Biol. 13 (2017), no. 7, e1005690.
[Deh55] René Deheuvels, Topologie d’une fonctionnelle (French), Ann. of Math. (2) 61 (1955), 13–72, DOI 10.2307/1969619. MR69488
[Del01] Pedro Delicado, Another look at principal curves and surfaces, J. Multivariate Anal. 77 (2001), no. 1, 84–116, DOI 10.1006/jmva.2000.1917. MR1838715
[DMFC12] Yuri Dabaghian, Facundo Mémoli, Loren Frank, and Gunnar Carlsson, A topological paradigm for hippocampal spatial map formation using persistent homology, PLoS Comput. Biol. (2012).
[Dre84] Andreas W. M. Dress, Trees, tight extensions of metric spaces, and the cohomological dimension of certain groups: a note on combinatorial properties of metric spaces, Adv. in Math. 53 (1984), no. 3, 321–402, DOI 10.1016/0001-8708(84)90029-X. MR753872
[EH10] Herbert Edelsbrunner and John L. Harer, Computational topology, American Mathematical Society, Providence, RI, 2010. An introduction, DOI 10.1090/mbk/069. MR2572029
[ELZ02] Herbert Edelsbrunner, David Letscher, and Afra Zomorodian, Topological persistence and simplification, Discrete Comput. Geom. 28 (2002), no. 4, 511–533, DOI 10.1007/s00454-002-2885-2. Discrete and computational geometry and graph drawing (Columbia, SC, 2001). MR1949898
[FF60] Herbert Federer and Wendell H. Fleming, Normal and integral currents, Ann. of Math. (2) 72 (1960), 458–520, DOI 10.2307/1970227. MR123260
[Fre97] D. H. Fremlin, Skeletons and central sets, Proc. London Math. Soc. (3) 74 (1997), no. 3, 701–720, DOI 10.1112/S0024611597000233. MR1434446
[Fro90] Patrizio Frosini, A distance for similarity classes of submanifolds of a Euclidean space, Bull. Austral. Math. Soc. 42 (1990), no. 3, 407–416, DOI 10.1017/S0004972700028574. MR1083277
[GHI+15] Marcio Gameiro, Yasuaki Hiraoka, Shunsuke Izumi, Miroslav Kramar, Konstantin Mischaikow, and Vidit Nanda, A topological measurement of protein compressibility, Jpn. J. Ind. Appl. Math. 32 (2015), no. 1, 1–17, DOI 10.1007/s13160-014-0153-5. MR3318898
[Ghr18] Robert Ghrist, Homological algebra and data, The mathematics of data, IAS/Park City Math. Ser., vol. 25, Amer. Math. Soc., Providence, RI, 2018, pp. 273–325. MR3839171
[Gil24] Patrick Gillespie, Vietoris thickenings and complexes are weakly homotopy equivalent, J. Appl. Comput. Topol. 8 (2024), no. 1, 35–53, DOI 10.1007/s41468-023-00135-8. MR4707456
[GK18] Marian Gidea and Yuri Katz, Topological data analysis of financial time series: landscapes of crashes, Phys. A 491 (2018), 820–834, DOI 10.1016/j.physa.2017.09.028. MR3721543
[Gro83] Mikhael Gromov, Filling Riemannian manifolds, J. Differential Geom. 18 (1983), no. 1, 1–147. MR697984
[Gro86] M. Gromov, Large Riemannian manifolds, Curvature and topology of Riemannian manifolds (Katata, 1985), Lecture Notes in Math., vol. 1201, Springer, Berlin, 1986, pp. 108–121, DOI 10.1007/BFb0075649. MR859578
[Gro88] M. Gromov, Width and related invariants of Riemannian manifolds (English, with French summary), Astérisque 163-164 (1988), 6, 93–109, 282 (1989). On the geometry of differentiable manifolds (Rome, 1986). MR999972
[Gro20] Misha Gromov, No metrics with positive scalar curvatures on aspherical 5-manifolds, Preprint, arXiv:2009.05332, 2020.
[Gut11] Larry Guth, Volumes of balls in large Riemannian manifolds, Ann. Of Math. (2) 173 (2011), no. 1, 51–76, DOI 10.4007/annals.2011.173.1.2. MR2753599
[Gut17] Larry Guth, Volumes of balls in Riemannian manifolds and Uryson width, J. Topol. Anal. 9 (2017), no. 2, 195–219, DOI 10.1142/S1793525317500029. MR3622232
[Has09] Trevor Hastie, The elements of statistical learning: data mining, inference, and prediction, 2009.
[Hau95] Jean-Claude Hausmann, On the Vietoris-Rips complexes and a cohomology theory for metric spaces, Prospects in topology (Princeton, NJ, 1994), Ann. of Math. Stud., vol. 138, Princeton Univ. Press, Princeton, NJ, 1995, pp. 175–188. MR1368659
[HG16] Gregory Henselman and Robert Ghrist, Matroid filtrations and computational persistent homology, Preprint, arXiv:1606.00199, 2016.
[HGR+20] Christoph Hofer, Florian Graf, Bastian Rieck, Marc Niethammer, and Roland Kwitt, Graph filtration learning, International Conference on Machine Learning, pp. 4314–4323, PMLR, 2020.
[HJ23] Michael Harrison and R. Amzi Jeffs, Quantitative upper bounds on the Gromov–Hausdorff distance between spheres, Preprint, arXiv:2309.11237, 2023.
[HKN19] Christoph D. Hofer, Roland Kwitt, and Marc Niethammer, Learning representations of persistence barcodes, J. Mach. Learn. Res. 20 (2019), Paper No. 126, 45. MR4002880
[HKNU17] Christoph Hofer, Roland Kwitt, Marc Niethammer, and Andreas Uhl, Deep learning with topological signatures, Proceedings of the 31st International Conference on Neural Information Processing Systems, 2017, pp. 1633–1643.
[HMMN14] Shaun Harker, Konstantin Mischaikow, Marian Mrozek, and Vidit Nanda, Discrete Morse theoretic algorithms for computing homology of complexes and maps, Found. Comput. Math. 14 (2014), no. 1, 151–184, DOI 10.1007/s10208-013-9145-0. MR3160710
[HNH+16] Yasuaki Hiraoka, Takenobu Nakamura, Akihiko Hirata, Emerson G. Escolar, Kaname Matsue, and Yasumasa Nishiura, Hierarchical structures of amorphous solids characterized by persistent homology, Proc. Natl. Acad. Sci. USA 113 (2016), no. 26, 7035–7040.
[HS89] Trevor Hastie and Werner Stuetzle, Principal curves, J. Amer. Statist. Assoc. 84 (1989), no. 406, 502–516. MR1010339
[Hu65] Sze-tsen Hu, Theory of retracts, Wayne State University Press, Detroit, MI, 1965. MR181977
[Isb64] J. R. Isbell, Six theorems about injective metric spaces, Comment. Math. Helv. 39 (1964), 65–76, DOI 10.1007/BF02566944. MR182949
[JJ23] Parvaneh Joharinad and Jürgen Jost, Mathematical principles of topological and geometric data analysis, Mathematics of Data, vol. 2, Springer, Cham, [2023] c 2023, DOI 10.1007/978-3-031-33440-5. MR4648028
[Jun01] Heinrich Jung, Ueber die kleinste Kugel, die eine räumliche Figur einschliesst (German), J. Reine Angew. Math. 123 (1901), 241–257, DOI 10.1515/crll.1901.123.241. MR1580570
[Kat83] Mikhail Katz, The filling radius of two-point homogeneous spaces, J. Differential Geom. 18 (1983), no. 3, 505–511. MR723814
[Kat89] Mikhail Katz, Diameter-extremal subsets of spheres, Discrete Comput. Geom. 4 (1989), no. 2, 117–137, DOI 10.1007/BF02187719. MR973541
[Kat90] Mikhail Katz, The filling radius of homogeneous manifolds, Séminaire de Théorie Spectrale et Géométrie, No. 9, Année 1990–1991, Sémin. Théor. Spectr. Géom., vol. 9, Univ. Grenoble I, Saint-Martin-d’Hères, 1991, pp. 103–109, DOI 10.5802/tsg.88. MR1715933
[Kat91] Mikhail Katz, On neighborhoods of the Kuratowski imbedding beyond the first extremum of the diameter functional, Fund. Math. 137 (1991), no. 3, 161–175, DOI 10.4064/fm-137-3-161-175. MR1110030
[KK16] Mehmet Kılı¸c and S¸ahin Ko¸cak, Tight span of subsets of the plane with the maximum metric, Adv. Math. 301 (2016), 693–710, DOI 10.1016/j.aim.2016.05.026. MR3539386
[Kol36] A. Kolmogoroff, ¨ Uber die beste Annäherung von Funktionen einer gegebenen Funktionenklasse (German), Ann. of Math. (2) 37 (1936), no. 1, 107–110, DOI 10.2307/1968691. MR1503273
[Lan13] Urs Lang, Injective hulls of certain discrete metric spaces and groups, J. Topol. Anal. 5 (2013), no. 3, 297–331, DOI 10.1142/S1793525313500118. MR3096307
[Lat01] Janko Latschev, Vietoris-Rips complexes of metric spaces near a closed Riemannian manifold, Arch. Math. (Basel) 77 (2001), no. 6, 522–528, DOI 10.1007/PL00000526. MR1879057
[Lax02] Peter D. Lax, Functional analysis, Pure and Applied Mathematics (New York), Wiley-Interscience [John Wiley & Sons], New York, 2002. MR1892228
[LBD+18] Yongjin Lee, Senja D. Barthel, PaweƗ DƗotko, Seyed Mohamad Moosavi, Kathryn Hess, and Berend Smit, High-throughput screening approach for nanoporous materials genome using topological data analysis: application to zeolites, J. Chem. Theory Comput. 14 (2018), no. 8, 4427–4437.
[LLNR22] Yevgeny Liokumovich, Boris Lishak, Alexander Nabutovsky, and Regina Rotman, Filling metric spaces, Duke Math. J. 171 (2022), no. 3, 595–632, DOI 10.1215/00127094-2021-0039. MR4382977
[LMO20] Sunhyuk Lim, Facundo Memoli, and Osman Berat Okutan, Vietoris–Rips persistent homology, injective metric spaces, and the filling radius. arXiv preprint arXiv:2001.07588, 2020.
[LMO24] Sunhyuk Lim, Facundo Mémoli, and Osman Berat Okutan, Vietoris-Rips persistent homology, injective metric spaces, and the filling radius, Algebr. Geom. Topol. 24 (2024), no. 2, 1019–1100, DOI 10.2140/agt.2024.24.1019. MR4735051
[LMS23] Sunhyuk Lim, Facundo Mémoli, and Zane Smith, The Gromov-Hausdorff distance between spheres, Geom. Topol. 27 (2023), no. 9, 3733–3800, DOI 10.2140/gt.2023.27.3733. MR4674839
[Meg12] Robert E. Megginson, An introduction to Banach space theory, Graduate Texts in Mathematics, vol. 183, Springer-Verlag, New York, 1998, DOI 10.1007/978-1-4612-0603-3. MR1650235
[MN13] Konstantin Mischaikow and Vidit Nanda, Morse theory for filtrations and efficient computation of persistent homology, Discrete Comput. Geom. 50 (2013), no. 2, 330–353, DOI 10.1007/s00454-013-9529-6. MR3090522
[Mor30] Marston Morse, The foundations of a theory of the calculus of variations in the large in m-space. II, Trans. Amer. Math. Soc. 32 (1930), no. 4, 599–631, DOI 10.2307/1989343. MR1501555
[MS73] J. H. Michael and L. M. Simon, Sobolev and mean-value inequalities on generalized submanifolds of Rn, Comm. Pure Appl. Math. 26 (1973), 361–379, DOI 10.1002/cpa.3160260305. MR344978
[MZ24] Facundo Mémoli and Ling Zhou, Ephemeral persistence features and the stability of filtered chain complexes, J. Comput. Geom. 15 (2024), no. 2, 258–328. MR4878036
[NLC11] Monica Nicolau, Arnold J. Levine, and Gunnar Carlsson, Topology based data analysis identifies a subgroup of breast cancers with a unique mutational profile and excellent survival, Proc. Natl. Acad. Sci. USA 108 (2011), no. 17, 7265–7270.
[NRS21] Alexander Nabutovsky, Regina Rotman, and Stéphane Sabourau, Sweepouts of closed Riemannian manifolds, Geom. Funct. Anal. 31 (2021), no. 3, 721–766, DOI 10.1007/s00039-021-00575-3. MR4311582
[OE11] Umut Ozertem and Deniz Erdogmus, Locally defined principal curves and surfaces, J. Mach. Learn. Res. 12 (2011), 1249–1286. MR2804600
[ONH22] Ippei Obayashi, Takenobu Nakamura, and Yasuaki Hiraoka, Persistent homology analysis for materials research and persistent homology software: HomCloud, J. Phys. Soc. Japan 91 (2022), no. 9, 091013.
[Pat18] Amit Patel, Generalized persistence diagrams, J. Appl. Comput. Topol. 1 (2018), no. 3-4, 397–419, DOI 10.1007/s41468-018-0012-6. MR3975559
[Pea01] Karl Pearson, LIII. On lines and planes of closest fit to systems of points in space, Lond. Edinb. Dublin Philos. Mag. J. Sci. 2 (1901), no. 11, 559-572.
[PRSZ20] Leonid Polterovich, Daniel Rosen, Karina Samvelyan, and Jun Zhang, Topological persistence in geometry and analysis, University Lecture Series, vol. 74, American Mathematical Society, Providence, RI, [2020] c 2020. MR4249570
[RB19] Ral Rabadán and Andrew J. Blumberg, Topological data analysis for genomics and evolution: topology in biology, Cambridge University Press, 2019.
[Rie20] Antonio Rieser, Vietoris–Rips homology theory for semi-uniform spaces, Preprint, arXiv:2008.05739, 2020.
[RM24] Sal Rodr´ıguez Mart´ın, Gromov-Hausdorff distances from simply connected geodesic spaces to the circle, Proc. Amer. Math. Soc. Ser. B 11 (2024), 624–637, DOI 10.1090/bproc/243. MR4840249
[RNS+17] Michael W. Reimann, Max Nolte, Martina Scolamiero, Katharine Turner, Rodrigo Perin, Giuseppe Chindemi, PaweƗ DƗotko, Ran Levi, Kathryn Hess, and Henry Markram, Cliques of neurons bound into cavities provide a missing link between structure and function, Front. Comput. Neurosci. 11 (2017), 266051.
[Rob99] V. Robins, Towards computing homology from finite approximations, Proceedings of the 14th Summer Conference on General Topology and its Applications (Brookville, NY, 1999), Topology Proc. 24 (1999), no. Summer, 503–532 (2001). MR1876386
[RQD23] Simon Rudkin, Wanling Qiu, and PaweƗ DƗotko, Uncertainty, volatility and the persistence norms of financial time series, Expert Syst. Appl. 223 (2023), 119894.
[RSDFS16] Vanessa Robins, Mohammad Saadatfar, Olaf Delgado-Friedrichs, and Adrian P. Sheppard, Percolating length scales from topological persistence analysis of micro-CT images of porous materials, Water Resour. Res. 52 (2016), no. 1, 315–329.
[Sab20] Stéphane Sabourau, One-cycle sweepout estimates of essential surfaces in closed Riemannian manifolds, Amer. J. Math. 142 (2020), no. 4, 1051–1082, DOI 10.1353/ajm.2020.0031. MR4124115
[SBdB16] Punam K. Saha, Gunilla Borgefors, and Gabriella Sanniti di Baja, A survey on skeletonization algorithms and their applications, Pattern Recognit. Lett. 76 (2016), 3–12.
[SDB+22] S⊘ren S. S⊘rensen, Tao Du, Christophe A. N. Biscio, Lisbeth Fajstrup, and Morten M. Smedskjaer, Persistent homology: a tool to understand medium-range order glass structure, J. Non-Crystalline Solids X 16 (2022), 100123.
[SL22] Yara Skaf and Reinhard Laubenbacher, Topological data analysis in biomedicine: a review, J. Biomed. Inform. (2022), 104082.
[SMI+08] Gurjeet Singh, Facundo Memoli, Tigran Ishkhanov, Guillermo Sapiro, Gunnar Carlsson, and Dario L. Ringach, Topological analysis of population activity in visual cortex, J. Vision 8 (2008), no. 8, 11–11.
[Tik76] V. M. Tikhomirov, Nekotorye voprosy teorii priblizheni˘i (Russian), Izdat. Moskov. Univ., Moscow, 1976. MR487161
[Tur19] Katharine Turner, Rips filtrations for quasimetric spaces and asymmetric functions with stability results, Algebr. Geom. Topol. 19 (2019), no. 3, 1135–1170, DOI 10.2140/agt.2019.19.1135. MR3954277
[UZ16] Michael Usher and Jun Zhang, Persistent homology and Floer-Novikov theory, Geom. Topol. 20 (2016), no. 6, 3333–3430, DOI 10.2140/gt.2016.20.3333. MR3590354
[Vie27] L. Vietoris, ¨ Uber den h¨oheren Zusammenhang kompakter Räume und eine Klasse von zusammenhangstreuen Abbildungen (German), Math. Ann. 97 (1927), no. 1, 454–472, DOI 10.1007/BF01447877. MR1512371
[Vir20] Žiga Virk, 1-dimensional intrinsic persistence of geodesic spaces, J. Topol. Anal. 12 (2020), no. 1, 169–207, DOI 10.1142/S1793525319500444. MR4080099
[Vir21] Žiga Virk, Rips complexes as nerves and a functorial Dowker-nerve diagram, Mediterr. J. Math. 18 (2021), no. 2, Paper No. 58, 24, DOI 10.1007/s00009-021-01699-4. MR4218370
[Vir22] Žiga Virk, Footprints of geodesics in persistent homology, Mediterr. J. Math. 19 (2022), no. 4, Paper No. 160, 29, DOI 10.1007/s00009-022-02089-0. MR4443111
[Wil92] Frederick H. Wilhelm Jr., On the filling radius of positively curved manifolds, Invent. Math. 107 (1992), no. 3, 653–668, DOI 10.1007/BF01231906. MR1150606
[Wol85] Franz-Erich Wolter, Cut loci in bordered and unbordered Riemannian manifolds, Ph.D. Thesis, Technische Universität Berlin, 1985.
[Zar22] Matthew C. B. Zaremsky, Bestvina-Brady discrete Morse theory and Vietoris-Rips complexes, Amer. J. Math. 144 (2022), no. 5, 1177–1200. MR4494179
[ZYCW20] Qi Zhao, Ze Ye, Chao Chen, and Yusu Wang, Persistence enhanced graph neural network, International Conference on Artificial Intelligence and Statistics, pp. 2896–2906, PMLR, 2020.