group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
constructive mathematics, realizability, computability
propositions as types, proofs as programs, computational trinitarianism
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.
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
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 $\rho$.
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)
A clear introduction to the use of persistent homology in data analysis is
Bar-codes were discovered under the name of canonical forms invariants of filtered complexes in
Other references:
The following paper uses persistent homology to single out features relevant for training neural networks,
Last revised on May 24, 2019 at 21:23:34. See the history of this page for a list of all contributions to it.