The structure and stability of persistence modules.

*(English)*Zbl 1362.55002
SpringerBriefs in Mathematics. Cham: Springer (ISBN 978-3-319-42543-6/pbk; 978-3-319-42545-0/ebook). x, 120 p. (2016).

Assuming no more than rudimentary knowledge of linear algebra and topology, this book offers an excellent introduction to anyone interested in understanding the fundamentals of persistent homology. The exposition is clear, concise and easy to read. The book contains subtle – but deep – remarks throughout, making connections to other fields of mathematics such as category theory and order theory. A fair overview of similar results appearing elsewhere is given, and an extensive list of suggested further reading is provided for the inspired reader.

Persistent homology is perhaps the most prominent tool in the newly emerging field of topological data analysis. Its pipeline is as follows: given a filtered topological space, apply homology (in any degree) with coefficients in a field to get a sequence of vector spaces connected by linear maps. Such a sequence of vector spaces and linear maps constitutes an example of a persistence module.

There are two natural finiteness conditions that can be satisfied: 1) the above filtration consists of a finite number of spaces, or 2) all the vector spaces are finite dimensional. If either of the two is satisfied, then the resultant persistence module is completely understood by a collection of (decorated) pairs of points, called the persistence diagram.

Persistence modules satisfying both of the above finiteness conditions have been extensively studied, and early stability results were restricted to such modules. In this book the authors introduce the framework of a ‘measure theory’ to allow for the study of more general persistence modules. Good reasons for doing so are presented in Section 1.1, including: “Real-world data sets are always finite, but they may be statistical samples from an underlying continuous object or process. Ideally the persistent homology of a sample will be an approximation of the persistent homology of the continuous model. Formulating this requires a theory of continuous-parameter persistence”.

In short, the book offers a self-contained introduction to topics such as persistence modules, persistence diagrams, interleavings, and the famous algebraic stability theorem. A novelty is that all of this is generalized to rectangle measures on the extended plane. This in turn generalizes results such as the algebraic stability theorem to arbitrary persistence modules.

Although the results are more general, it should be noted that the core ideas and proofs are essentially the same as those presented in [D. Cohen-Steiner et al., Discrete Comput. Geom. 37, No. 1, 103–120 (2007; Zbl 1117.54027) and F. Chazal et al., “Proximity of persistence modules and their diagrams”, in: Proceedings of the twenty-fifth annual symposium on computational geometry, SGG’09. New York: ACM. 237–246 (2009; doi:10.1145/1542362.1542407)].

Persistent homology is perhaps the most prominent tool in the newly emerging field of topological data analysis. Its pipeline is as follows: given a filtered topological space, apply homology (in any degree) with coefficients in a field to get a sequence of vector spaces connected by linear maps. Such a sequence of vector spaces and linear maps constitutes an example of a persistence module.

There are two natural finiteness conditions that can be satisfied: 1) the above filtration consists of a finite number of spaces, or 2) all the vector spaces are finite dimensional. If either of the two is satisfied, then the resultant persistence module is completely understood by a collection of (decorated) pairs of points, called the persistence diagram.

Persistence modules satisfying both of the above finiteness conditions have been extensively studied, and early stability results were restricted to such modules. In this book the authors introduce the framework of a ‘measure theory’ to allow for the study of more general persistence modules. Good reasons for doing so are presented in Section 1.1, including: “Real-world data sets are always finite, but they may be statistical samples from an underlying continuous object or process. Ideally the persistent homology of a sample will be an approximation of the persistent homology of the continuous model. Formulating this requires a theory of continuous-parameter persistence”.

In short, the book offers a self-contained introduction to topics such as persistence modules, persistence diagrams, interleavings, and the famous algebraic stability theorem. A novelty is that all of this is generalized to rectangle measures on the extended plane. This in turn generalizes results such as the algebraic stability theorem to arbitrary persistence modules.

Although the results are more general, it should be noted that the core ideas and proofs are essentially the same as those presented in [D. Cohen-Steiner et al., Discrete Comput. Geom. 37, No. 1, 103–120 (2007; Zbl 1117.54027) and F. Chazal et al., “Proximity of persistence modules and their diagrams”, in: Proceedings of the twenty-fifth annual symposium on computational geometry, SGG’09. New York: ACM. 237–246 (2009; doi:10.1145/1542362.1542407)].

Reviewer: Magnus Bakke Botnan (München)

##### MSC:

55-02 | Research exposition (monographs, survey articles) pertaining to algebraic topology |

55N99 | Homology and cohomology theories in algebraic topology |

16G20 | Representations of quivers and partially ordered sets |

62-02 | Research exposition (monographs, survey articles) pertaining to statistics |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

68-02 | Research exposition (monographs, survey articles) pertaining to computer science |