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The persistent topology of optimal transport based metric thickenings

Henry Adams, Facundo Mémoli, Michael Moy and Qingsong Wang

Algebraic & Geometric Topology 24 (2024) 393–447
Bibliography
1 M Adamaszek, H Adams, The Vietoris–Rips complexes of a circle, Pacific J. Math. 290 (2017) 1 MR3673078
2 M Adamaszek, H Adams, F Frick, Metric reconstruction via optimal transport, SIAM J. Appl. Algebra Geom. 2 (2018) 597 MR3871057
3 M Adamaszek, H Adams, S Reddy, On Vietoris–Rips complexes of ellipses, J. Topol. Anal. 11 (2019) 661 MR3999516
4 H Adams, J Bush, F Frick, The topology of projective codes and the distribution of zeros of odd maps, preprint (2021) arXiv:2106.14677
5 H Adams, M Heim, C Peterson, Metric thickenings and group actions, J. Topol. Anal. 14 (2022) 587 MR4493474
6 H Adams, J Mirth, Metric thickenings of Euclidean submanifolds, Topology Appl. 254 (2019) 69 MR3895304
7 H Barcelo, V Capraro, J A White, Discrete homology theory for metric spaces, Bull. Lond. Math. Soc. 46 (2014) 889 MR3262192
8 Y Baryshnikov, P Bubenik, M Kahle, Min-type Morse theory for configuration spaces of hard spheres, Int. Math. Res. Not. 2014 (2014) 2577 MR3207377
9 S Basu, Computing the top Betti numbers of semialgebraic sets defined by quadratic inequalities in polynomial time, Found. Comput. Math. 8 (2008) 45 MR2403530
10 S Basu, N Karisani, Persistent homology of semialgebraic sets, SIAM J. Appl. Algebra Geom. 7 (2023) 651 MR4646856
11 S Basu, R Pollack, M F Roy, Computing the first Betti number of a semi-algebraic set, Found. Comput. Math. 8 (2008) 97 MR2403532
12 U Bauer, Ripser : efficient computation of Vietoris–Rips persistence barcodes, J. Appl. Comput. Topol. 5 (2021) 391 MR4298669
13 V Berestovskii, C Plaut, Uniform universal covers of uniform spaces, Topology Appl. 154 (2007) 1748 MR2317077
14 V I Bogachev, Measure theory, II, Springer (2007) MR2267655
15 V I Bogachev, Weak convergence of measures, 234, Amer. Math. Soc. (2018) MR3837546
16 R Bott, Nondegenerate critical manifolds, Ann. of Math. 60 (1954) 248 MR0064399
17 R Bott, Lectures on Morse theory, old and new, Bull. Amer. Math. Soc. 7 (1982) 331 MR0663786
18 R Bott, Morse theory indomitable, Inst. Hautes Études Sci. Publ. Math. 68 (1988) 99 MR1001450
19 J Brazas, P Fabel, Thick Spanier groups and the first shape group, Rocky Mountain J. Math. 44 (2014) 1415 MR3295636
20 N Brodskiy, J Dydak, B Labuz, A Mitra, Rips complexes and covers in the uniform category, Houston J. Math. 39 (2013) 667 MR3080460
21 L N Bryzgalova, Maximum functions of a family of functions that depend on parameters, Funktsional. Anal. i Prilozhen. 12 (1978) 66 MR0487233
22 P S Bullen, D S Mitrinović, P M Vasić, Means and their inequalities, 31, D Reidel (1988) MR0947142
23 D Burago, Y Burago, S Ivanov, A course in metric geometry, 33, Amer. Math. Soc. (2001) MR1835418
24 P Bürgisser, F Cucker, P Lairez, Computing the homology of basic semialgebraic sets in weak exponential time, J. ACM 66 (2019) 5 MR3892564
25 G Carlsson, Topology and data, Bull. Amer. Math. Soc. 46 (2009) 255 MR2476414
26 M Cencelj, J Dydak, A Vavpetič, Ž Virk, A combinatorial approach to coarse geometry, Topology Appl. 159 (2012) 646 MR2868863
27 F Chazal, D Cohen-Steiner, L J Guibas, F Mémoli, S Y Oudot, Gromov–Hausdorff stable signatures for shapes using persistence, Computer Graphics Forum 28 (2009) 1393
28 F Chazal, S Y Oudot, Towards persistence-based reconstruction in Euclidean spaces, from: "Computational geometry (SCG’08)", ACM (2008) 232 MR2504289
29 F Chazal, V de Silva, M Glisse, S Oudot, The structure and stability of persistence modules, Springer (2016) MR3524869
30 F Chazal, V de Silva, S Oudot, Persistence stability for geometric complexes, Geom. Dedicata 173 (2014) 193 MR3275299
31 D Cohen-Steiner, H Edelsbrunner, J Harer, Stability of persistence diagrams, Discrete Comput. Geom. 37 (2007) 103 MR2279866
32 J Conant, V Curnutte, C Jones, C Plaut, K Pueschel, M Lusby, J Wilkins, Discrete homotopy theory and critical values of metric space, Fund. Math. 227 (2014) 97 MR3257953
33 A Dranishnikov, Cohomological approach to asymptotic dimension, Geom. Dedicata 141 (2009) 59 MR2520063
34 H Edelsbrunner, J L Harer, Computational topology: an introduction, Amer. Math. Soc. (2010) MR2572029
35 H Edelsbrunner, D Letscher, A Zomorodian, Topological persistence and simplification, from: "41st Annual Symposium on Foundations of Computer Science", IEEE Comput. Soc. (2000) 454 MR1931842
36 H Federer, Curvature measures, Trans. Amer. Math. Soc. 93 (1959) 418 MR0110078
37 P Frosini, C Landi, F Mémoli, The persistent homotopy type distance, Homology Homotopy Appl. 21 (2019) 231 MR3923782
38 V Gershkovich, H Rubinstein, Morse theory for Min-type functions, Asian J. Math. 1 (1997) 696 MR1621571
39 A L Gibbs, F E Su, On choosing and bounding probability metrics, Int. Stat. Rev. 70 (2002) 419
40 M Gromov, Filling Riemannian manifolds, J. Differential Geom. 18 (1983) 1 MR0697984
41 M Gromov, Asymptotic invariants of infinite groups, from: "Geometric group theory, II" (editors G A Niblo, M A Roller), London Math. Soc. Lecture Note Ser. 182, Cambridge Univ. Press (1993) 1 MR1253544
42 J C Hausmann, On the Vietoris–Rips complexes and a cohomology theory for metric spaces, from: "Prospects in topology" (editor F Quinn), Ann. of Math. Stud. 138, Princeton Univ. Press (1995) 175 MR1368659
43 N J Kalton, M I Ostrovskii, Distances between Banach spaces, Forum Math. 11 (1999) 17 MR1673915
44 H Karcher, Riemannian center of mass and mollifier smoothing, Comm. Pure Appl. Math. 30 (1977) 509 MR0442975
45 M Katz, The filling radius of two-point homogeneous spaces, J. Differential Geom. 18 (1983) 505 MR0723814
46 M Katz, Diameter-extremal subsets of spheres, Discrete Comput. Geom. 4 (1989) 117 MR0973541
47 M Katz, The filling radius of homogeneous manifolds, from: "Séminaire de théorie spectrale et géométrie", Univ. Grenoble I (1991) 103 MR1715933
48 M Katz, On neighborhoods of the Kuratowski imbedding beyond the first extremum of the diameter functional, Fund. Math. 137 (1991) 161 MR1110030
49 M Katz, The rational filling radius of complex projective space, Topology Appl. 42 (1991) 201 MR1137947
50 J Latschev, Vietoris–Rips complexes of metric spaces near a closed Riemannian manifold, Arch. Math. 77 (2001) 522 MR1879057
51 M Lesnick, The theory of the interleaving distance on multidimensional persistence modules, Found. Comput. Math. 15 (2015) 613 MR3348168
52 S Lim, F Mémoli, O Okutan, Vietoris–Rips persistent homology, injective metric spaces, and the filling radius,
53 S Lim, F Mémoli, Z Smith, The Gromov–Hausdorff distance between spheres, Geom. Topol. 27 (2023) 3733
54 V I Matov, Topological classification of the germs of functions of the maximum and minimax of families of functions in general position, Uspekhi Mat. Nauk 37 (1982) 167 MR0667989
55 F Mémoli, Gromov–Wasserstein distances and the metric approach to object matching, Found. Comput. Math. 11 (2011) 417 MR2811584
56 F Mémoli, Some properties of Gromov–Hausdorff distances, Discrete Comput. Geom. 48 (2012) 416 MR2946454
57 F Mémoli, A distance between filtered spaces via tripods, preprint (2017) arXiv:1704.03965
58 F Mémoli, O B Okutan, Quantitative simplification of filtered simplicial complexes, Discrete Comput. Geom. 65 (2021) 554 MR4212978
59 F Mémoli, L Zhou, Persistent homotopy groups of metric spaces, preprint (2019) arXiv:1912.12399
60 J Milnor, Morse theory, 51, Princeton Univ. Press (1963) MR0163331
61 J R Mirth, Vietoris–Rips metric thickenings and Wasserstein spaces, PhD thesis, Colorado State University (2020)
62 O B Okutan, Persistence, metric invariants, and simplification, PhD thesis, The Ohio State University (2019)
63 S Pavoine, S Ollier, D Pontier, Measuring diversity from dissimilarities with Rao’s quadratic entropy : are any dissimilarities suitable ?, Theoret. Population Biol. 67 (2005) 231
64 C Plaut, J Wilkins, Discrete homotopies and the fundamental group, Adv. Math. 232 (2013) 271 MR2989983
65 C Radhakrishna Rao, Diversity and dissimilarity coefficients: a unified approach, Theoret. Population Biol. 21 (1982) 24 MR0662520
66 A Rieser, Vietoris–Rips homology theory for semi-uniform spaces, preprint (2020) arXiv:2008.05739
67 J Roe, Coarse cohomology and index theory on complete Riemannian manifolds, 497, Amer. Math. Soc. (1993) MR1147350
68 K Sakai, Geometric aspects of general topology, Springer (2013) MR3099433
69 F Santambrogio, Euclidean, metric, and Wasserstein gradient flows : an overview, Bull. Math. Sci. 7 (2017) 87 MR3625852
70 C Thäle, 50 years sets with positive reach, Surv. Math. Appl. 3 (2008) 123 MR2443192
71 L Vietoris, Über den höheren Zusammenhang kompakter Räume und eine Klasse von zusammenhangstreuen Abbildungen, Math. Ann. 97 (1927) 454 MR1512371
72 C Villani, Optimal transport: old and new, 338, Springer (2009) MR2459454
73 Ž Virk, Rips complexes as nerves and a functorial Dowker-nerve diagram, Mediterr. J. Math. 18 (2021) 58 MR4218370
74 Ž Virk, Footprints of geodesics in persistent homology, Mediterr. J. Math. 19 (2022) 160 MR4443111
75 M C B Zaremsky, Bestvina–Brady discrete Morse theory and Vietoris–Rips complexes, Amer. J. Math. 144 (2022) 1177 MR4494179
76 A Zomorodian, G Carlsson, Computing persistent homology, Discrete Comput. Geom. 33 (2005) 249 MR2121296