Persistence Homology

Introduction

Persistent homology is a method for computing topological features of a space at different spatial resolutions. The main way we do this is by plotting persistence diagrams. Specifically, we record the duration of the presence of each betti number, where the \(n\)-th Betti number \(\beta_n\) to be the dimension of the \(n\)-th homology group. \(\beta_0\) is the number of connected components, \(\beta_1\) is the number of holes, \(\beta_2\) the number of voids, etc.