nLab persistent homology

Contents

Context

Homological algebra

Representation theory

Idea
Properties
Software
Related concepts
References

General
Cohomotopy in topological data analysis
Further variants

Idea

Persistent homology is the study of sequences of linear maps between vector spaces

\cdots \xrightarrow{\;} V_i \xrightarrow{\phi_i} V_{i + 1} \xrightarrow{\phi_{i + 1}} V_{i + 2} \xrightarrow{\phi_{i + 2}} \cdots \,,

under the aspect of which elements (vectors) $v_i \in V_i$ “persist” (ie. remain non-zero) under which number of iterations of these linear maps. One then refers to these sequences as persistence modules (hence a concept with an attitude).

In the archetypical applications of interest, these vector spaces are ordinary homology groups $V_i \,=\, H_n(X_i)$ (over a given ground field) of topological spaces $X_i$ which themselves are stages

\cdots \hookrightarrow X_i \xhookrightarrow{\iota_i} X_{i+ 1} \xhookrightarrow{\iota_{i + 1}} X_{i + 2} \xhookrightarrow{} \cdots \xhookrightarrow{x} X

of a filtered topological space, and one is interested in deducing “relevant” properties of this filtration by finding those homology cycles which stand out as persisting over a large range of steps.

This question, in turn, has been motivated from problems in topological data analysis, where sequences of spaces $X_i$ arise as coarse-grained versions, at varying stages of resolution $i$ (say as measured in some ambient metric space) of a given discrete set of data points. In this example, the more the homology cycles persist as the resolution $i$ is varied, the more one is willing to assume that they signal relevant structure that is hidden in the data, while cycles that do not persist for long would be interpreted as irrelevant noise effects.

Of course, these simple basic ideas may be (and still are being) refined and generalized in a number of ways.

A particularly important generalization turns out to be that from directed 1-dimensional sequences of linear maps to arbitrary zigzags of these, then called zigzag persistence modules. Again, this is naturally motivated in the case of homology groups of a filtered topological space, where one may form the sequence of cospans

\cdots \leftarrow X_i \to X_i \cup X_{i + 1} \leftarrow X_{i + 1} \to X_{i + 1} \cup X_{i + 2} \leftarrow \cdots

of inclusions into unions of consecutive filter stages.

Seen in this generality, the notion of finite-length zigzag persistence modules happens to coincide with that of A-type quiver representations, a fruitful coincidence that serves to bring well-developed tools of quiver-representation theory to bear on persistent homology theory.

In particular, the founding result of quiver representation theory – namely Gabriel's theorem – serves, in hindsight, to establish the formal notion of persistence itself: The theorem implies (see there) that every (zigzag) persistence module is the direct sum of interval modules $I_{a,b}$ , namely those which are zero except over the interval $i \in [a,b]$ , where they are given by the identity on the ground field (the latter regarded as the 1-dimensional vector space over itself).

Hence the canonical linear basis-elements of these interval modules $I_{a,b}$ are exactly the persistent cycles which persist from $a$ to $b$ , so that Gabriel's theorem guarantees that any (zigzag) persistence module may be completely decomposed into (the linear span of) these elements. Numerous generalizations of this theorem exist (taking it out of the context of quiver-theory) and show that this state of affairs remains true notably when one allows the indices $i$ to be taken from sets with infinite cardinality.

In other words, the isomorphism class of any (zigzag) persistence module is equivalently encoded in a multiset of intervals. These multisets are known as barcodes (due to the evident graphical representation of a multiset of intervals) or as persistence diagrams (the latter usually when $[a,b]$ is regarded as a point $(a,b) \in \mathbb{R}^2$ in the plane). These persistence diagrams are meant to be the invariants of interest in persistent homology theory.

Besides the foundational theorem that guarantees the existence of persistence diagrams, the fundamental theorem of persistent homology has come to be the stability theorem: This says that persistence diagrams are indeed useful invariants, namely in that they remain “stable” under small variations (think: noise, measurement errors) of input data such as the above topological filter stages $X_i$ .

For example, as one changes a little the metric with which to produce the topological space $X_i$ of coarse grained data at resolution stage $i$ , the homology groups $H(X_i)$ may change dramatically, but the stability theorem ensures that effect on the persistence diagram remains small.

It is in this way that persistent homology may be used, in particular, to produce robust invariants of noisy data, which has made it the archetypical tool in topological data analysis.

Properties

stability of persistence diagrams

(Cohen-Steiner, Edelsbrunner & Harer 2007)
diamond principle

(Carlsson et al.)

Software

One can compute intervals for homological features algorithmically over field coefficients and software packages are available for this purpose. See for instance Perseus. The principal algorithm is based on bringing the filtered complex to its canonical form by upper-triangular matrices from (Barannikov1994, §2.1)

Related concepts

References

General

Introduction and survey:

Robert Ghrist, Barcodes: The Persistent Topology of Data, Bull. Amer. Math. Soc. 45 (2008), 61-75 (doi:10.1090/S0273-0979-07-01191-3, pdf)
Herbert Edelsbrunner, John Harer, Persistent homology – a survey, in: Surveys on Discrete and Computational Geometry: Twenty Years Later, Contemporary Mathematics 453 (2008) $[$ doi:10.1090/conm/453 $]$
Gunnar Carlsson, Topology and data, Bull. Amer. Math. Soc. 46 (2009), no. 2, 255-308 $[$ doi:10.1090/S0273-0979-09-01249-X $]$
Herbert Edelsbrunner, Dmitriy Morozov, Persistent homology: theory and practice, in: European Congress of Mathematics Kraków, 2–7 July, 2012 EMS $[$ doi:10.4171/120-1/3, pdf $]$
Donald Pinckney, Topological Data Analysis and Persistent Homology (2019)
Stefan Huber, Persistent Homology in Data Science In: Data Science – Analytics and Applications, Springer (2021) $[$ doi:10.1007/978-3-658-32182-6_13, pdf $]$

Review with emphasis of zigzag persistence with relation to quiver representation theory:

Steve Y. Oudot, Persistence Theory: From Quiver Representations to Data Analysis, Mathematical Surveys and Monographs 209 AMS (2015) $[$ pdf, ISBN:978-1-4704-3443-4 $]$
Gunnar Carlsson, Persistent Homology and Applied Homotopy Theory, in: Handbook of Homotopy Theory, CRC Press (2019) $[$ arXiv:2004.00738, doi:10.1201/9781351251624 $]$

Cohomotopy in topological data analysis

Introducing persistent cohomotopy as a tool in topological data analysis, improving on the use of well groups from persistent homology:

Peter Franek, Marek Krčál, On Computability and Triviality of Well Groups, Discrete Comput Geom 56 (2016) 126 (arXiv:1501.03641, doi:10.1007/s00454-016-9794-2)
Peter Franek, Marek Krčál, Persistence of Zero Sets, Homology, Homotopy and Applications, 19 2 (2017) (arXiv:1507.04310, doi:10.4310/HHA.2017.v19.n2.a16)
Peter Franek, Marek Krčál, Hubert Wagner, Solving equations and optimization problems with uncertainty, J Appl. and Comput. Topology 1 (2018) 297 (arxiv:1607.06344, doi:10.1007/s41468-017-0009-6)

Review:

Peter Franek, Marek Krčál, Cohomotopy groups capture robust Properties of Zero Sets via Homotopy Theory, talk at ACAT meeting 2015 $[$ pdf $]$
Urs Schreiber on joint work with Hisham Sati: New Foundations for TDA – Cohomotopy, (May 2022)