# nLab persistent homology

cohomology

### Theorems

#### Constructivism, Realizability, Computability

intuitionistic mathematics

# Persistent homology

## Idea

Persistent homology is a homology theory adapted to a computational context, for instance, in analysis of large data sets. It keeps track of homology classes which stay ‘persistent’ when the approximate image of a space gets refined to higher resolutions.

## More detail

We suppose given a ‘data cloud of samples’, $P\subset \mathbb{R}^m$, from some space $X$, yielding a simplicial complex $S_\rho(X)$ for each $\rho \gt 0$ via one of the family of simplicial complex approximation methods that are listed below (TO BE ADDED). For these, the important idea to retain is that if $\rho \lt \rho^\prime$, then

$S_\rho(X) \hookrightarrow S_{\rho^\prime}(X),$

so we get a ‘filtration structure’ on the complex.

The idea of persistent homology is to look for features that persist for some range of parameter values. Typically a feature, such as a hole, will initially not be observed, then will appear, and after a range of values of the parameter it will disappear again. A typical feature will be a Betti number of the complex, $S_\rho(X)$, which then will vary with the parameter $\rho$.

## Software

One can compute intervals for homological features algorithmically over field coefficients and software packages are available for this purpose. See for instance Perseus.

## References

A clear introduction to the use of persistent homology in data analysis is

Other references:

• Robert MacPherson, Benjamin Schweinhart, Measuring shape with topology, J. Math. Phys. 53, 073516 (2012); doi
• A. Zomorodian, G. Carlsson, Computing persistent homology, Discrete Comput. Geom. 33, 249–274 (2005)
• Ulrich Bauer, Michael Kerber, Jan Reininghaus, Clear and compress: computing persistent homology in chunks, arxiv/1303.0477
• 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
• Robert J. Adler, Omer Bobrowski, Shmuel Weinberger, Crackle: the persistent homology of noise, arxiv/1301.1466
• Paweł Dłotko, Hubert Wagner, Computing homology and persistent homology using iterated Morse decomposition, arxiv/1210.1429
• G. Carlsson, V. de Silva, Zigzag persistence, arXiv:0812.0197
• G. Carlsson, Topology and data, Bull. Amer. Math. Soc. 46 (2009), no. 2, 255-308.
• 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
• H. Edelsbrunner, D. Morozov, Persistent homology: theory and practice pdf

Last revised on August 25, 2017 at 16:24:38. See the history of this page for a list of all contributions to it.