On the stability of persistent entropy and new summary functions for TDA

TL;DR

This paper establishes stability conditions for persistent entropy in topological data analysis, introduces two new stable summary functions combining entropy and Betti curves, and demonstrates their effectiveness in material classification tasks.

Contribution

It provides the first stability analysis for persistent entropy and proposes novel summary functions that enhance pattern recognition applications.

Findings

Persistent entropy is stable under small data perturbations.

Two new summary functions are introduced and shown to be stable.

The new functions improve material classification accuracy.

Abstract

Persistent homology and persistent entropy have recently become useful tools for patter recognition. In this paper, we find requirements under which persistent entropy is stable to small perturbations in the input data and scale invariant. In addition, we describe two new stable summary functions combining persistent entropy and the Betti curve. Finally, we use the previously defined summary functions in a material classification task to show their usefulness in machine learning and pattern recognition.