Persistent homology group explained

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,^[1] based largely on the persistence barcodes and the persistence algorithm, that were first described by Serguei Barannikov in the 1994 paper.^[2] Since then, the study of persistent homology groups has led to applications in data science,^[3] machine learning,^[4] materials science,^[5] biology,^{[6] [7]} and economics.^[8]

Definition

Let

K be a simplicial complex, and let f: K \to R be a real-valued monotonic function. Then for some values a _{0} < a _{1} < … < a _{n} \in R the sublevel-sets K(a) : = f ^{-1} ( -inf ty, a] yield a sequence of nested subcomplexes \emptyset = K _{0} \subseteq K _{1} \subseteq … \subseteq K _{n} = K known as a filtration of K .

Applying

p ^{th} homology to each complex yields a sequence of homology groups 0 = H _{p} (K _{0)} \to H _{p} (K _{1)} \to … \to H _{p} (K _{n)} = H _{p} (K) connected by homomorphisms induced by the inclusion maps of the underlying filtration. When homology is taken over a field, we get a sequence of vector spaces and linear maps known as a persistence module.

Let

f ^{i,j} _{p} : H _{p} (K _{i)} \to H _{p} (K _{j)} be the homomorphism induced by the inclusion K _{i} \hookrightarrow K _{j} . Then the p ^{th} persistent homology groups are defined as the images H ^{i,j} _{p} : = \operatorname{im} f ^{i,j} _{p} for all 1 \leq i \leq j \leq n . In particular, the persistent homology group H ^{i,i} _{p} = H _{p} (K _{i)} .

More precisely, the

p ^{th} persistent homology group can be defined as H ^{i,j} _{p} = Z _{p} (K _{i)} / \left( B _{p} (K _{j)} \cap Z _{p(K} _{i)} \right) , where Z _{p(K} _{\bullet)} and B _{p(K} _{\bullet)} are the standard p-cycle and p-boundary groups, respectively.^[9]

Birth and death of homology classes

Sometimes the elements of

H ^{i,j} _{p} are described as the homology classes that are "born" at or before K _{i} and that have not yet "died" entering K _{j} . These notions can be made precise as follows. A homology class \gamma \in H _{p} (K _{i)} is said to be born at K _{i} if it is not contained in the image of the previous persistent homology group, i.e., \gamma \notin H ^{i -1 , i} _{p} . Conversely, \gamma is said to die entering K _{j} if \gamma is subsumed (i.e., merges with) another older class as the sequence proceeds from K _{j-1} \to K _{j} . That is to say, f ^{i,j -1} _{p} (\gamma) \notin H ^{i -1 ,j -1} _{p} but f ^{i,j} _{p} (\gamma) \in H ^{i -1 ,j} _{p} . The determination that an older class persists if it merges with a younger class, instead of the other way around, is sometimes known as the Elder Rule.^{[10] [11]}

The indices

i,j at which a homology class \gamma is born and dies entering are known as the birth and death indices of \gamma . The difference j - i is known as the index persistence of \gamma , while the corresponding difference a _{j} - a _{i} in function values corresponding to those indices is known as the persistence of \gamma . If there exists no index at which \gamma dies, it is assigned an infinite death index. Thus, the persistence of each class can be represented as an interval in the extended real line R \cup \{\pm inf ty\} of either the form [a _{i,} a _{j)} or [a _{i',} inf ty) . Since, in the case of an infinite field, the infinite number of classes always have the same persistence, the collection over all classes of such intervals does not give meaningful multiplicities for a multiset of intervals. Instead, such multiplicities and a multiset of intervals in the extended real line are given by the structure theorem of persistence homology. This multiset is known as the persistence barcode.^[12]

Canonical form

Concretely, the structure theorem states that for any filtered complex over a field

F , there exists a linear transformation that preserves the filtration and converts the filtered complex into so called canonical form, a canonically defined direct sum of filtered complexes of two types: two-dimensional complexes with trivial homology d(e _{a _{j}} ) = e _{a _{i}} and one-dimensional complexes with trivial differential d(e _{a' _{i}} ) =0 .

Persistence diagram

Geometrically, a barcode can be plotted as a multiset of points (with possibly infinite coordinates)

(a _{i,} a _{j)} in the extended plane \left( R \cup \{\pm inf ty \} \right) ^{2} . By the above definitions, each point will lie above the diagonal, and the distance to the diagonal is exactly equal to the persistence of the corresponding class times (1)/(\sqrt 2) . This construction is known as the persistence diagram, and it provides a way of visualizing the structure of the persistence of homology classes in the sequence of persistent homology groups.

References

1. Edelsbrunner . Letscher . Zomorodian . 2002 . Topological Persistence and Simplification . Discrete & Computational Geometry . en . 28 . 4 . 511–533 . 10.1007/s00454-002-2885-2 . 0179-5376. free .
2. Framed Morse complex and its invariants . Advances in Soviet Mathematics . 1994. 93–115. 21. Sergey. Barannikov . ADVSOV . 10.1090/advsov/021/03. 9780821802373 . 125829976 .
3. Book: Chen, Li M. . Mathematical problems in data science : theoretical and practical methods . 2015 . Zhixun Su, Bo Jiang . 978-3-319-25127-1 . Cham . 120–124 . 932464024.
4. Book: 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 . 2017 . Andreas Holzinger, Peter Kieseberg, A. Min Tjoa, Edgar R. Weippl . 978-3-319-66808-6 . Cham . 23–24 . 1005114370.
5. Book: Hirata, Akihiko . Structural analysis of metallic glasses with computational homology . 2016 . Kaname Matsue, Mingwei Chen . 978-4-431-56056-2 . Japan . 63–65 . 946084762.
6. Book: Moraleda, Rodrigo Rojas . Computational topology for biomedical image and data analysis : theory and applications . 2020 . Nektarios A. Valous, Wei Xiong, Niels Halama . 978-0-429-81099-2 . Boca Raton, FL . 1108919429.
7. Book: Rabadán, Raúl . Topological data analysis for genomics and evolution : topology in biology . 2020 . Andrew J. Blumberg . 978-1-316-67166-5 . Cambridge, United Kingdom . 132–158 . 1129044889.
8. Yen . Peter Tsung-Wen . Cheong . Siew Ann . 2021 . Using Topological Data Analysis (TDA) and Persistent Homology to Analyze the Stock Markets in Singapore and Taiwan . Frontiers in Physics . 9 . 20 . 10.3389/fphy.2021.572216 . 2021FrP.....9...20Y . 2296-424X. free . 10356/155402 . free .
9. Book: Edelsbrunner, Herbert . Computational topology : an introduction . 2010 . American Mathematical Society . J. Harer . 978-0-8218-4925-5 . Providence, R.I. . 149–153 . 427757156.
10. Book: Progress in information geometry : theory and applications . 2021 . 978-3-030-65459-7 . Nielsen . Frank . Cham . 224 . 1243544872.
11. Book: Oudot, Steve Y. . Persistence theory : from quiver representations to data analysis . 2015 . 978-1-4704-2545-6 . Providence, Rhode Island . 2–3 . 918149730.
12. Ghrist . Robert . 2008 . Barcodes: The persistent topology of data . Bulletin of the American Mathematical Society . en . 45 . 1 . 61–75 . 10.1090/S0273-0979-07-01191-3 . 0273-0979. free .