Persistent Betti number

Last updated October 29, 2023

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 $n^{th}$ Betti number equals the rank of the $n^{th}$ homology group, the $n^{th}$ persistent Betti number is the rank of the $n^{th}$ 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]^[5]^[6] and physics.^[7]^[8]^[9]

Definition

Let $K$ be a simplicial complex, and let $f:K\to \mathbb {R}$ be a monotonic, i.e., non-decreasing function. Requiring monotonicity guarantees that the sublevel set $K(a):=f^{-1}(-\infty ,a]$ is a subcomplex of $K$ for all $a\in \mathbb {R}$ . Letting the parameter $a$ vary, we can arrange these subcomplexes into a nested sequence $\emptyset =K_{0}\subseteq K_{1}\subseteq \cdots \subseteq K_{n}=K$ for some natural number $n$ . This sequences defines a filtration on the complex $K$ .

Persistent homology concerns itself with the evolution of topological features across a filtration. To that end, by taking the $p^{th}$ homology group of every complex in the filtration we obtain a sequence of homology groups $0=H_{p}(K_{0})\to H_{p}(K_{1})\to \cdots \to H_{p}(K_{n})=H_{p}(K)$ 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 $i\leq j$ , let $f_{p}^{i,j}$ denote the induced homomorphism $H_{p}(K_{i})\to H_{p}(K_{j})$ . Then the $p^{th}$ persistent homology groups are defined to be the images of each induced map. Namely, $H_{p}^{i,j}:=\operatorname {im} f_{p}^{i,j}$ for all $0\leq i\leq j\leq n$ .

In parallel to the classical Betti number, the $p^{th}$ persistent Betti numbers are precisely the ranks of the $p^{th}$ persistent homology groups, given by the definition $\beta _{p}^{i,j}:=\operatorname {rank} H_{p}^{i,j}$ .^[10]

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The degree-Rips bifiltration is a simplicial filtration used in topological data analysis for analyzing the shape of point cloud data. It is a multiparameter extension of the Vietoris–Rips filtration that possesses greater stability to data outliers than single-parameter filtrations, and which is more amenable to practical computation than other multiparameter constructions. Introduced in 2015 by Lesnick and Wright, the degree-Rips bifiltration is a parameter-free and density-sensitive vehicle for performing persistent homology computations on point cloud data.

The offset filtration is a growing sequence of metric balls used to detect the size and scale of topological features of a data set. The offset filtration commonly arises in persistent homology and the field of topological data analysis. Utilizing a union of balls to approximate the shape of geometric objects was first suggested by Frosini in 1992 in the context of submanifolds of Euclidean space. The construction was independently explored by Robins in 1998, and expanded to considering the collection of offsets indexed over a series of increasing scale parameters, in order to observe the stability of topological features with respect to attractors. Homological persistence as introduced in these papers by Frosini and Robins was subsequently formalized by Edelsbrunner et al. in their seminal 2002 paper Topological Persistence and Simplification. Since then, the offset filtration has become a primary example in the study of computational topology and data analysis.

The multicover bifiltration is a two-parameter sequence of nested topological spaces derived from the covering of a finite set in a metric space by growing metric balls. It is a multidimensional extension of the offset filtration that captures density information about the underlying data set by filtering the points of the offsets at each index according to how many balls cover each point. The multicover bifiltration has been an object of study within multidimensional persistent homology and topological data analysis.

A persistence module is a mathematical structure in persistent homology and topological data analysis that formally captures the persistence of topological features of an object across a range of scale parameters. A persistence module often consists of a collection of homology groups corresponding to a filtration of topological spaces, and a collection of linear maps induced by the inclusions of the filtration. The concept of a persistence module was first introduced in 2005 as an application of graded modules over polynomial rings, thus importing well-developed algebraic ideas from classical commutative algebra theory to the setting of persistent homology. Since then, persistence modules have been one of the primary algebraic structures studied in the field of applied topology.

In topological data analysis, a subdivision bifiltration is a collection of filtered simplicial complexes, typically built upon a set of data points in a metric space, that captures shape and density information about the underlying data set. The subdivision bifiltration relies on a natural filtration of the barycentric subdivision of a simplicial complex by flags of minimum dimension, which encodes density information about the metric space upon which the complex is built. The subdivision bifiltration was first introduced by Donald Sheehy in 2011 as part of his doctoral thesis as a discrete model of the multicover bifiltration, a continuous construction whose underlying framework dates back to the 1970s. In particular, Sheehy applied the construction to both the Vietoris-Rips and Čech filtrations, two common objects in the field of topological data analysis. Whereas single parameter filtrations are not robust with respect to outliers in the data, the subdivision-Rips and -Cech bifiltrations satisfy several desirable stability properties.

In topological data analysis, the Vietoris-Rips filtration is the collection of nested Vietoris-Rips complexes on a metric space created by taking the sequence of Vietoris-Rips complexes over an increasing scale parameter. Often, the Vietoris-Rips filtration is used to create a discrete, simplicial model on point cloud data embedded in an ambient metric space. The Vietoris-Rips filtration is a multiscale extension of the Vietoris-Rips complex that enables researchers to detect and track the persistence of topological features, over a range of parameters, by way of computing the persistent homology of the entire filtration.

In persistent homology, a persistent homology group is a multiscale analog of a homology group that captures information about the evolution of topological features across a filtration of spaces. While the ordinary homology group represents nontrivial homology classes of an individual topological space, the persistent homology group tracks only those classes that remain nontrivial across multiple parameters in the underlying filtration. Analogous to the ordinary Betti number, the ranks of the persistent homology groups are known as the persistent Betti numbers. Persistent homology groups were first introduced by Herbert Edelsbrunner, David Letscher, and Afra Zomorodian in a 2002 paper Topological Persistence and Simplification, one of the foundational papers in the fields of persistent homology and topological data analysis, based largely on the persistence barcodes and the persistence algorithm, that were first described by Serguei Barannikov in the 1994 paper. Since then, the study of persistent homology groups has led to applications in data science, machine learning, materials science, biology, and economics.

In topological data analysis, a persistence barcode, sometimes shortened to barcode, is an algebraic invariant of a persistence module that characterizes the stability of topological features throughout a growing family of spaces. Formally, a persistence barcode consists of a multiset of intervals in the extended real line, where the length of each interval corresponds to the lifetime of a topological feature in a filtration, usually built on a point cloud, a graph, a function, or, more generally, a simplicial complex or a chain complex. Generally, longer intervals in a barcode correspond to more robust features, whereas shorter intervals are more likely to be noise in the data. A persistence barcode is a complete invariant that captures all the topological information in a filtration. In algebraic topology, the persistence barcodes were first introduced by Sergey Barannikov in 1994 as the "canonical forms" invariants consisting of a multiset of line segments with ends on two parallel lines, and later, in geometry processing, by Gunnar Carlsson et al. in 2004.

References

↑ Perea, Jose A. (2018-10-01). "A Brief History of Persistence". arXiv: 1809.03624 [math.AT].
↑ 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.
↑ Yvinec, M., Chazal, F., Boissonnat, J. (2018). Geometric and Topological Inference. pp. 211. United States: Cambridge University Press.
↑ Conti, F., Moroni, D., & Pascali, M. A. (2022). A Topological Machine Learning Pipeline for Classification. Mathematics, 10(17), 3086. https://doi.org/10.3390/math10173086
↑ Krishnapriyan, A. S., Montoya, J., Haranczyk, M., Hummelshøj, J., & Morozov, D. (2021, March 31). Machine learning with persistent homology and chemical word embeddings improves prediction accuracy and interpretability in metal-organic frameworks. arXiv. http://arxiv.org/abs/2010.00532. Accessed 28 October 2023
↑ 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)
↑ 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)
↑ Makarenko, I., Bushby, P., Fletcher, A., Henderson, R., Makarenko, N., & Shukurov, A. (2018). Topological data analysis and diagnostics of compressible magnetohydrodynamic turbulence. Journal of Plasma Physics, 84(4), 735840403. https://doi.org/10.1017/S0022377818000752
↑ Pranav, P., Edelsbrunner, H., van de Weygaert, R., Vegter, G., Kerber, M., Jones, B. J. T., & Wintraecken, M. (2017). The topology of the cosmic web in terms of persistent Betti numbers. Monthly Notices of the Royal Astronomical Society, 465(4), 4281–4310. https://doi.org/10.1093/mnras/stw2862
↑ 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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[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] Conti, F., Moroni, D., & Pascali, M. A. (2022). A Topological Machine Learning Pipeline for Classification. Mathematics, 10(17), 3086. https://doi.org/10.3390/math10173086

[5] Krishnapriyan, A. S., Montoya, J., Haranczyk, M., Hummelshøj, J., & Morozov, D. (2021, March 31). Machine learning with persistent homology and chemical word embeddings improves prediction accuracy and interpretability in metal-organic frameworks. arXiv. http://arxiv.org/abs/2010.00532. Accessed 28 October 2023

[6] 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)

[7] 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)

[8] Makarenko, I., Bushby, P., Fletcher, A., Henderson, R., Makarenko, N., & Shukurov, A. (2018). Topological data analysis and diagnostics of compressible magnetohydrodynamic turbulence. Journal of Plasma Physics, 84(4), 735840403. https://doi.org/10.1017/S0022377818000752

[9] Pranav, P., Edelsbrunner, H., van de Weygaert, R., Vegter, G., Kerber, M., Jones, B. J. T., & Wintraecken, M. (2017). The topology of the cosmic web in terms of persistent Betti numbers. Monthly Notices of the Royal Astronomical Society, 465(4), 4281–4310. https://doi.org/10.1093/mnras/stw2862

[10] 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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