Gaussian Persistence Curves

TL;DR

This paper studies Gaussian persistence curves, a smooth, one-dimensional summary of persistence diagrams in topological data analysis, focusing on their stability and injectivity for better integration with machine learning.

Contribution

It introduces and analyzes Gaussian persistence curves, providing insights into their stability and injectivity, advancing the use of TDA summaries in data analysis.

Findings

Gaussian persistence curves are stable under data perturbations.

They are injective, preserving topological information.

The study enhances the integration of TDA with machine learning.

Abstract

Topological data analysis (TDA) is a rising field in the intersection of mathematics, statistics, and computer science/data science. The cornerstone of TDA is persistent homology, which produces a summary of topological information called a persistence diagram. To utilize machine and deep learning methods on persistence diagrams, These diagrams are further summarized by transforming them into functions. In this paper we investigate the stability and injectivity of a class of smooth, one-dimensional functional summaries called Gaussian persistence curves.