Persistent entropy for separating topological features from noise in vietoris-rips complexes

Abstract

Persistent homology studies the evolution of k-dimensional holes along a nested sequence of simplicial complexes (called a filtration). The set of bars (i.e. intervals) representing birth and death times of k-dimensional holes along such sequence is called the persistence barcode. k-Dimensional holes with short lifetimes are informally considered to be “topological noise”, and those with long lifetimes are considered to be “topological features” associated to the filtration. Persistent entropy is defined as the Shannon entropy of the persistence barcode of the filtration. In this paper we present new important properties of persistent entropy of Vietoris-Rips filtrations. Later, using these properties, we derive a simple method for separating topological noise from features in Vietoris-Rips filtrations.