Tropical Sufficient Statistics for Persistent Homology

TL;DR

This paper introduces a tropical geometric embedding that creates stable, sufficient statistics for barcodes in persistent homology, enabling classical statistical inference methods to be applied directly to topological summaries.

Contribution

It presents a novel tropical geometric approach that yields sufficient statistics for barcodes, facilitating parametric statistical inference in topological data analysis.

Findings

Stable tropical geometric embeddings for barcodes

Construction of likelihood functions for persistent homology

Application to HIV and influenza data analysis

Abstract

We show that an embedding in Euclidean space based on tropical geometry generates stable sufficient statistics for barcodes. In topological data analysis, barcodes are multiscale summaries of algebraic topological characteristics that capture the `shape' of data; however, in practice, they have complex structures that make them difficult to use in statistical settings. The sufficiency result presented in this work allows for classical probability distributions to be assumed on the tropical geometric representation of barcodes. This makes a variety of parametric statistical inference methods amenable to barcodes, all while maintaining their initial interpretations. More specifically, we show that exponential family distributions may be assumed, and that likelihood functions for persistent homology may be constructed. We conceptually demonstrate sufficiency and illustrate its utility in persistent homology dimensions 0 and 1 with concrete parametric applications to human immunodeficiency virus and avian influenza data.