Persistent Betti number

In persistent homology, a persistent Betti number is a multiscale analog of a Betti number that tracks the number of topological features that persist over multiple scale parameters in a filtration. Whereas the classical Betti number equals the rank of the homology group, the persistent Betti number is the rank of the persistent homology group. The concept of a persistent Betti number was introduced by Herbert Edelsbrunner, David Letscher, and Afra Zomorodian in the 2002 paper Topological Persistence and Simplification, one of the seminal papers in the field of persistent homology and topological data analysis.[1][2] Applications of the persistent Betti number appear in a variety of fields including data analysis,[3] machine learning,[4] and physics.[5]

Definition

Let be a simplicial complex, and let be a monotonic, i.e., non-decreasing function. Requiring monotonicity guarantees that the sublevel set is a subcomplex of for all . Letting the parameter vary, we can arrange these subcomplexes into a nested sequence for some natural number . This sequences defines a filtration on the complex .

Persistent homology concerns itself with the evolution of topological features across a filtration. To that end, by taking the homology group of every complex in the filtration we obtain a sequence of homology groups that are connected by homomorphisms induced by the inclusion maps in the filtration. When applying homology over a field, we get a sequence of vector spaces and linear maps commonly known as a persistence module.

In order to track the evolution of homological features as opposed to the static topological information at each individual index, one needs to count only the number of nontrivial homology classes that persist in the filtration, i.e., that remain nontrivial across multiple scale parameters.

For each , let denote the induced homomorphism . Then the persistent homology groups are defined to be the images of each induced map. Namely, for all .

In parallel to the classical Betti number, the persistent Betti numbers are precisely the ranks of the persistent homology groups, given by the definition .[6]

References

  1. Perea, Jose A. (2018-10-01). "A Brief History of Persistence". arXiv:1809.03624 [math.AT].
  2. Edelsbrunner; Letscher; Zomorodian (2002). "Topological Persistence and Simplification". Discrete & Computational Geometry. 28 (4): 511–533. doi:10.1007/s00454-002-2885-2. ISSN 0179-5376.
  3. Yvinec, M., Chazal, F., Boissonnat, J. (2018). Geometric and Topological Inference. pp. 211. United States: Cambridge University Press.
  4. Machine Learning and Knowledge Extraction : First IFIP TC 5, WG 8.4, 8.9, 12.9 International Cross-Domain Conference, CD-MAKE 2017, Reggio, Italy, August 29 - September 1, 2017, Proceedings. Andreas Holzinger, Peter Kieseberg, A. Min Tjoa, Edgar R. Weippl. Cham. 2017. pp. 23–24. ISBN 978-3-319-66808-6. OCLC 1005114370.{{cite book}}: CS1 maint: location missing publisher (link) CS1 maint: others (link)
  5. Morphology of condensed matter : physics and geometry of spatially complex systems. Klaus R. Mecke, Dietrich Stoyan. Berlin: Springer. 2002. pp. 261–274. ISBN 978-3-540-45782-4. OCLC 266958114.{{cite book}}: CS1 maint: others (link)
  6. Edelsbrunner, Herbert (2010). Computational topology : an introduction. J. Harer. Providence, R.I.: American Mathematical Society. pp. 178–180. ISBN 978-1-4704-1208-1. OCLC 946298151.
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