The space of persistence diagrams on 𝑛 points coarsely embeds into Hilbert space

Mitra, Atish; Virk, Žiga

doi:10.1090/proc/15363

The space of persistence diagrams on $n$ points coarsely embeds into Hilbert space
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by Atish Mitra and Žiga Virk

Proc. Amer. Math. Soc. 149 (2021), 2693-2703

DOI: https://doi.org/10.1090/proc/15363

Published electronically: March 22, 2021

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Corrigendum: Proc. Amer. Math. Soc. (to appear).

Abstract:

We prove that the space of persistence diagrams on $n$ points (with the bottleneck or a Wasserstein distance) coarsely embeds into Hilbert space by showing it is of asymptotic dimension $2n$. Such an embedding enables utilisation of Hilbert space techniques on the space of persistence diagrams. We also prove that when the number of points is not bounded, the corresponding spaces of persistence diagrams do not have finite asymptotic dimension. Furthermore, in the case of the bottleneck distance, the corresponding space does not coarsely embed into Hilbert space.

References

Kyle Austin and Žiga Virk, Higson compactification and dimension raising, Topology Appl. 215 (2017), 45–57. MR 3576439, DOI 10.1016/j.topol.2016.10.005
Mathieu Carrière and Ulrich Bauer, On the metric distortion of embedding persistence diagrams into separable Hilbert spaces, 35th International Symposium on Computational Geometry, LIPIcs. Leibniz Int. Proc. Inform., vol. 129, Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2019, pp. Art. No. 21, 15. MR 3968607
G. Bell and A. Dranishnikov, Asymptotic dimension, Topology Appl. 155 (2008), no. 12, 1265–1296. MR 2423966, DOI 10.1016/j.topol.2008.02.011
G. Bell, A. Lawson, C. N. Pritchard, and D. Yasaki, The space of persistence diagrams fails to have Yu’s property A, arXiv:1902.02288v2, 2019.
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
Peter Bubenik and Alexander Wagner, Embeddings of persistence diagrams into Hilbert spaces, J. Appl. Comput. Topol. 4 (2020), no. 3, 339–351. MR 4130975, DOI 10.1007/s41468-020-00056-w
A. N. Dranishnikov, G. Gong, V. Lafforgue, and G. Yu, Uniform embeddings into Hilbert space and a question of Gromov, Canad. Math. Bull. 45 (2002), no. 1, 60–70. MR 1884134, DOI 10.4153/CMB-2002-006-9
Herbert Edelsbrunner and John L. Harer, Computational topology, American Mathematical Society, Providence, RI, 2010. An introduction. MR 2572029, DOI 10.1090/mbk/069
Ryszard Engelking, Theory of dimensions finite and infinite, Sigma Series in Pure Mathematics, vol. 10, Heldermann Verlag, Lemgo, 1995. MR 1363947
M. Gromov, Asymptotic invariants of infinite groups, Geometric group theory, Vol. 2 (Sussex, 1991) London Math. Soc. Lecture Note Ser., vol. 182, Cambridge Univ. Press, Cambridge, 1993, pp. 1–295. MR 1253544
William B. Johnson and N. Lovasoa Randrianarivony, $l_p\ (p>2)$ does not coarsely embed into a Hilbert space, Proc. Amer. Math. Soc. 134 (2006), no. 4, 1045–1050. MR 2196037, DOI 10.1090/S0002-9939-05-08415-7
Daniel Kasprowski, The asymptotic dimension of quotients by finite groups, Proc. Amer. Math. Soc. 145 (2017), no. 6, 2383–2389. MR 3626497, DOI 10.1090/proc/13491
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
Piotr W. Nowak, Coarse embeddings of metric spaces into Banach spaces, Proc. Amer. Math. Soc. 133 (2005), no. 9, 2589–2596. MR 2146202, DOI 10.1090/S0002-9939-05-08150-5
John Roe, Lectures on coarse geometry, University Lecture Series, vol. 31, American Mathematical Society, Providence, RI, 2003. MR 2007488, DOI 10.1090/ulect/031
O. Shukel’ and M. Zarichnyi, Asymptotic dimension of symmetric powers, Math. Bulletin of NTSh. 5 (2008) 304–311.
K. Turner and G. Spreemann, Same but different: Distance correlations between topological summaries, In: Baas N., Carlsson G., Quick G., Szymik M., Thaule M. (eds) Topological Data Analysis. Abel Symposia, vol 15. Springer, Cham. DOI 10.1007/978-3-030-43408-3_18.
T. Yamauchi, T. Weighill, and N. Zava, Coarse infinite-dimensionality of hyperspaces of finite subsets, in preparation.
A. Wagner, Nonembeddability of persistence diagrams with $p > 2$ Wasserstein metric, arXiv:1910.13935v1, 2019.
Guoliang Yu, The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space, Invent. Math. 139 (2000), no. 1, 201–240. MR 1728880, DOI 10.1007/s002229900032