Persistent Homology of Cover Refinements

Persistent homology is a key tool in Topological Data Analysis, used to study the shape of data. Its use in practice is often limited because standard constructions like the Vietoris–Rips filtration lead to an explosive growth in the number of simplices. This scaling problem restricts applications to small datasets and low-dimensional homology.

In this talk we´ll tackle this exponential increase in simplicies by focusing on underlying geometric structures that generate these complexes: covers of the data.

We present a framework to compute persistent homology starting from a sequence of cover refinements. The presence of cover refinements induces simplicial contractions that curb the growth of the complex, greatly reducing the number of simplices. Importantly, the method produces the same topological invariants as traditional techniques—without approximation.

By shifting the focus to covers, this perspective makes persistent homology more scalable and also clarifies connections between classical results, such as the relationship between Čech and Vietoris–Rips persistence.