# nLab persistent homology

Contents

### Context

#### Representation theory

representation theory

geometric representation theory

# Contents

## 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.

## 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)

## References

### General

Introduction and survey:

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

The concept of persistent homology originates around:

• Herbert Edelsbrunner, D. Letscher, and A. Zomorodian, Topological persistence and simplification, Discrete & Computational Geometry, 28 (2002) 511–533 $[$doi:10.1007/s00454-002-2885-2$]$

The generalization to “zigzag persistence” with relation to A-type quiver representation-theory is due to:

and specifically for level sets:

The stability result is due to:

Bar-codes were introduced under the name of canonical forms invariants of filtered complexes in

• Serguei Barannikov, Framed Morse complex and its invariants, pdf Advances in Soviet Mathematics 21 93–115 (1994)

• A. Zomorodian, Gunnar Carlsson, Computing persistent homology, Discrete Comput. Geom. 33, 249–274 (2005) (doi:10.1007/s00454-004-1146-y)

• Gunnar Carlsson, V. de Silva, Zigzag persistence (arXiv:0812.0197)

• Robert J. Adler, Omer Bobrowski, Matthew S. Borman, Eliran Subag, Shmuel Weinberger, Persistent homology for random fields and complexes Institute of Mathematical Statistics Collections 6:124–143, 2010 (arxiv/1003.1001)

• Paweł Dłotko, Hubert Wagner, Computing homology and persistent homology using iterated Morse decomposition $[$arxiv/1210.1429$]$

• Robert MacPherson, Benjamin Schweinhart, Measuring shape with topology, J. Math. Phys. 53, 073516 (2012) (doi:10.1063/1.4737391)

• Robert J. Adler, Omer Bobrowski, Shmuel Weinberger, Crackle: the persistent homology of noise, arxiv/1301.1466

• D. Le Peutrec, N. Nier, C. Viterbo, Precise Arrhenius Law for p-forms: The Witten Laplacian and Morse–Barannikov Complex, Annales Henri Poincaré 14 (3): 567–610 (2013) doi

• Ulrich Bauer, Michael Kerber, Jan Reininghaus, Clear and compress: computing persistent homology in chunks, arxiv/1303.0477

• Sara Kališnik, Persistent homology and duality, 2013 (pdf, pdf)

• Francisco Belchí Guillamón, Aniceto Murillo Mas, A-infinity persistence, arxiv/1403.2395

• João Pita Costa, Mikael Vejdemo Johansson, Primož Škraba, Variable sets over an algebra of lifetimes: a contribution of lattice theory to the study of computational topology, arxiv/1409.8613

• Genki Kusano, Kenji Fukumizu, Yasuaki Hiraoka, Persistence weighted Gaussian kernel for topological data analysis, arxiv/1601.01741

• Heather A. Harrington, Nina Otter, Hal Schenck, Ulrike Tillmann, Stratifying multiparameter persistent homology, arxiv/1708.07390

• Nicolas Berkouk, Grégory Ginot, Steve Oudot, Level-sets persistence and sheaf theory $[$arXiv:1907.09759$]$

The following paper uses persistent homology to single out features relevant for training neural networks:

Application to topological data analysis in cosmological structure formation:

Application of topological data analysis (persistent homology) to analysis of phase transitions:

### 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:

Review: