nLab
persistent homology

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Constructivism, Realizability, Computability

Contents

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 mP\subset \mathbb{R}^m, from some space XX, yielding a simplicial complex S ρ(X)S_\rho(X) for each ρ>0\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 ρ<ρ \rho \lt \rho^\prime, then

S ρ(X)S ρ(X),S_\rho(X) \hookrightarrow S_{\rho'}(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 ρ(X)S_\rho(X), which then will vary with the parameter ρ\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. 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

See also

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

See also

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

The suggestion to regard cobordism theory of iso-hypersurfaces, and thus Pontryagin's theorem in Cohomotopy, as a tool in (persistent) topological data analysis (improving on homologuical well groups):

Further variants

See also