# Persistent Entropy for Separating Topological Features from Noise in Vietoris-Rips Complexes

## TL;DR

This paper introduces persistent entropy as a stable measure for distinguishing meaningful topological features from noise in Vietoris-Rips filtrations, enhancing the analysis of persistent homology.

## Contribution

It presents new properties of persistent entropy for Cech and Vietoris-Rips filtrations and proposes a simple method to separate topological noise from features.

## Key findings

- Persistent entropy is stable under filtration perturbations.
- A new method effectively separates noise from features in Vietoris-Rips complexes.
- Properties of persistent entropy improve topological data analysis robustness.

## 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 a given filtration. In this paper we present new important properties of persistent entropy of Cech and Vietoris-Rips filtrations. Among the properties, we put a focus on the stability theorem that allows to use persistent entropy for comparing persistence barcodes. Later, we derive a simple method for separating topological noise from features in Vietoris-Rips filtrations.