With Rips complexes, the calculation of persistent homology is simply linear algebra. However, it takes a lot of computational time since even a few hundred data points yield lots of simplexes in the filtration.
There is also the R package TDA which can calculate the persistent homology of a density estimator of the data by looking at the super-level sets of the density estimator. (also it is computationally fast)
Two sloppy definitions (for the above sentence):
1) For a data set $\mathcal D \in \mathbb R^2$, a density estimator is a function $z= f(x,y)$ which takes larger values at denser areas.
2) A super-level set of this function is a set $\{(x,y) : f(x,y) \geq c \}$ for some $c \in \mathbb R$.
My question is, how does this package calculate persistent homology of density estimators? Does it also use linear algebra?
I could not find any information about this in the documentation. And I suspect that they do not use a Rips filtration, because the calculations with density estimators is much faster that the ones with Rips complexes.
Best Answer
I don't know exactly what is done in R TDA with their density estimator. However, typically when you have a function $f\colon \mathbb{R}^d\to \mathbb{R}$, especially for $d=2$ or 3, one proceeds as follows. Instead of storing the function values $f$ on all of $\mathbb{R}^d$, one instead just stores the values of $f$ on an grid of square (or cubical) grid of points in $\mathbb{R}^d$. So for $d=2$ the values of $f$ on this grid are stored as a matrix, and for $d=3$ the values of $f$ on this grid are stored as a 3D matrix (a tensor). Then, the sublevelsets (or superlevelsets) of $f$ will be cubical complexes (indeed, cubical complexes which are "aligned" with the square lattice in $\mathbb{R}^2$, or the cubical lattice in $\mathbb{R}^3$). One can compute the persistent homology of an increasing sequence of cubical complexes in much the same way as one does for an increasing sequence of simplicial complexes. For more information on the persistent homology of cubical complexes, see https://www.springer.com/gp/book/9780387408538 by Kaczynski, Mischaikow, Mrozek for a book reference from 2004, and (for example) the paper http://www2.im.uj.edu.pl/mpd/publications/Wagner_persistence.pdf by Wagner, Chen, Vucini for a newer reference.