Identifying Topological Differences in Two Populations of Random Geometric Objects

TL;DR

This paper introduces a statistical framework using topological data analysis to detect differences between two populations of random geometric objects by embedding their topological signatures into Euclidean space for effective two-sample testing.

Contribution

It develops a novel embedding of persistence barcodes into Euclidean space via tropical geometry, enabling consistent two-sample testing of topological differences.

Findings

The embedding is a sufficient statistic for persistence barcodes.

The proposed test is shown to be consistent.

The framework effectively distinguishes topological differences in random geometric objects.

Abstract

We propose a statistical framework to identify topological differences in two populations of random geometric objects. The proposed framework involves first associating a topological signature with random geometric objects and then performing a two-sample test using the observed topological signatures. We associate persistence barcodes, a topological signature from topological data analysis, with each observed random geometric object. This, in turn, yields a two-sample problem on the space of persistence barcodes. As the space of persistence barcodes is not suitable for standard statistical analysis, we translate the two-sample problem on a suitable subset of a Euclidean space. In the course of this study, we embed the topological signatures in an ordered convex cone in a Euclidean space using functions from tropical geometry. We show that the embedding is a sufficient statistic for the persistence barcodes. This fact leads to the proposal of a two-sample test based on this sufficient statistic, and its equivalence to the two-sample problem on the barcode space is established. Finally, the consistency of the proposed test is studied.