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Persistence stability for geometric complexes. (English) Zbl 1320.55003
The authors of the article define a persistence module \({\mathbb V}\) to be a direct system, indexed by the real numbers, in the category of vector spaces; that is, a collection of vector spaces \(\{V_a\mid a\in{\mathbb R}\}\) and linear maps \(v_a^b:V_a\rightarrow V_b\) for \(a\leq b\), such that \(v_b^c\circ v_a^b=v_a^c\) and \(v_a^a\) is the identity. A homomorphism \(\Phi\in\text{Hom}^\epsilon({\mathbb U},{\mathbb V})\) of degree \(\epsilon\) between persistence modules is defined to be a collection of linear maps \(\phi_a:U_a\rightarrow V_{a+\epsilon}\) such that \(v_{a+\epsilon}^{b+\epsilon}\circ\phi_a=\phi_b\circ u_a^b\), whenever \(a\leq b\). In particular the shift map \(1_{\mathbb V}^\epsilon\in\text{Hom}^\epsilon({\mathbb V},{\mathbb V})\), which denotes the collection of maps \(v_a^{a+\epsilon}\), is a homomorphism of degree \(\epsilon\). Persistence modules \({\mathbb U}\) and \({\mathbb V}\) are said to be \(\epsilon\)-interleaved if there are \(\Phi\in\text{Hom}^\epsilon({\mathbb U},{\mathbb V})\) and \(\Psi\in\text{Hom}^\epsilon({\mathbb V},{\mathbb U})\) such that \(\Psi\circ\Phi=1_{\mathbb U}^{2\epsilon}\) and \(\Phi\circ\Psi=1_{\mathbb V}^{2\epsilon}\).
Of primary concern in the article is the Vietoris-Rips complex of a metric space \(X\). Recall that this is the simplicial complex \(\text{Rips}(X,a)\) with vertex set \(X\), and whose simplices are finite sets of vertices such that the distance between any two pairs of vertices is at most \(a\). For a fixed homological dimension, the homology groups \(\{H(\text{Rips}(X,a))\mid a\in{\mathbb R}\}\) form a persistence module \(H(\text{Rips}(X))\). It is shown that for any two metric spaces \(X\) and \(Y\), the persistence modules \(H(\text{Rips}(X))\) and \(H(\text{Rips}(Y))\) are \(\epsilon\)-interleaved whenever the Gromov-Hausdorff distance between \(X\) and \(Y\) is less than \(\epsilon/2\). A similar result is shown to hold for the Čech complex of \(X\), as well as for other related simplicial complexes.
Using the techniques developed in the article, the authors deduce other properties of the Vietoris-Rips and Čech persistence modules, such as tameness, stability of persistence diagrams, and properties of their first and second homology groups.

55N05 Čech types
57Q05 General topology of complexes
Full Text: DOI
[1] Attali, D., Lieutier, A., Salinas, D.: Vietoris-Rips complexes also provide topologically correct reconstructions of sampled shapes. In: Proceedings of the 27th Annual ACM Symposium on Computational geometry, SoCG ’11, pp. 491-500. ACM, New York, NY, USA (2011). doi:10.1145/1998196.1998276 · Zbl 1283.68341
[2] Bartholdi, L; Schick, T; Smale, N; Smale, S; Baker, AW, Hodge theory on metric spaces, Found. Comput. Math., 12, 1-48, (2012) · Zbl 1366.58001
[3] Burago, D., Burago, Y., Ivanov, S.: A Course in Metric Geometry, Graduate Studies in Mathematics, vol. 33. American Mathematical Society, Providence, RI (2001) · Zbl 0981.51016
[4] Chazal, F., Cohen-Steiner, D., Glisse, M., Guibas, L., Oudot, S.: Proximity of persistence modules and their diagrams. In: SCG, pp. 237-246 (2009). doi:10.1145/1542362.1542407 · Zbl 1380.68387
[5] Chazal, F., Cohen-Steiner, D., Guibas, L.J., Mémoli, F., Oudot, S.Y.: Gromov-Hausdorff stable signatures for shapes using persistence. Computer Graphics Forum (Proceedings of the SGP 2009) pp. 1393-1403 (2009) · Zbl 1069.55003
[6] Chazal, F., Oudot, S.Y.: Towards persistence-based reconstruction in euclidean spaces. In: Proceedings of the Twenty-Fourth Annual Symposium on Computational Geometry, SCG ’08, pp. 232-241. ACM, New York, NY, USA (2008). doi:10.1145/1377676.1377719 · Zbl 1271.57058
[7] Chazal, F., de Silva, V., Glisse, M., Oudot, S.: The structure and stability of persistence modules (2012). ArXiv:1207.3674 [math.AT] · Zbl 1362.55002
[8] Dowker, CH, Homology groups of relations, Ann. Math., 56, 84-95, (1952) · Zbl 0046.40402
[9] Droz, J.M.: A subset of Euclidean space with large Vietoris-Rips homology (2012). ArXiv:1210.4097 [math.GT] · Zbl 0046.40402
[10] Edelsbrunner, H., Harer, J.: Computational Topology: An Introduction. American Mathematical Society, Providence, RI (2010) · Zbl 1193.55001
[11] Edelsbrunner, H; Letscher, D; Zomorodian, A, Topological persistence and simplification, Discret. Comput. Geom., 28, 511-533, (2002) · Zbl 1011.68152
[12] Ghys, E., de la Harpe, P.: Sur les groupes hyperboliques d’après Mikhael Gromov, vol. 83. Birkhäuser, Basel (1990) · Zbl 0731.20025
[13] Gromov, M.: Metric Structures for Riemannian and Non-Riemannian Spaces, 2nd edn. Birkhäuser, Basel (2007) · Zbl 1113.53001
[14] Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge, MA (2001). http://www.math.cornell.edu/ hatcher/ · Zbl 1044.55001
[15] Hausmann, JC, On the Vietoris-rips complexes and a cohomology theory for metric spaces, Ann. Math. Stud., 138, 175-188, (1995) · Zbl 0928.55003
[16] Latschev, J, Vietoris-rips complexes of metric spaces near a closed Riemannian manifold, Archiv der Mathematik, 77, 522-528, (2001) · Zbl 1001.53026
[17] Munkres, J.R.: Elements of Algebraic Topology. Westview Press, Boulder, CO (1984) · Zbl 0673.55001
[18] Zomorodian, A; Carlsson, G, Computing persistent homology, Discret. Comput. Geom., 33, 249-274, (2005) · Zbl 1069.55003
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