Modeling of Persistent Homology

TL;DR

This paper advances statistical methods for analyzing persistence diagrams in Topological Data Analysis, enabling more effective inference from single large data sets by developing a broader class of models.

Contribution

It extends the parametric modeling approach for persistence diagrams, enhancing statistical inference capabilities in TDA for high-dimensional data.

Findings

Developed a wider class of models for persistence diagrams

Enhanced statistical inference methods for TDA

Demonstrated applicability on diverse data examples

Abstract

Topological Data Analysis (TDA) is a novel statistical technique, particularly powerful for the analysis of large and high dimensional data sets. Much of TDA is based on the tool of persistent homology, represented visually via persistence diagrams. In an earlier paper we proposed a parametric representation for the probability distributions of persistence diagrams, and based on it provided a method for their replication. Since the typical situation for big data is that only one persistence diagram is available, these replications allow for conventional statistical inference, which, by its very nature, requires some form of replication. In the current paper we continue this analysis, and further develop its practical statistical methodology, by investigating a wider class of examples than treated previously.