Two-sample tests for relevant differences in persistence diagrams

Abstract

We study two-sample tests for relevant differences in persistence diagrams obtained from $L^p$-$m$-approximable data $({\mathcal{X}}_t{)}_t$ and $({\mathcal{Y}}_t{)}_t$. To this end, we compare variance estimates w.r.t.\ the Wasserstein metrics on the space of persistence diagrams. In detail, we consider two test procedures. The first compares the Fr{é}chet variances of the two samples based on estimators for the Fr{é}chet mean of the observed persistence diagrams $PD({\mathcal{X}}_i)$ ($1\le i\le m$), resp., $PD({\mathcal{Y}}_j)$ ($1\le j\le n$) of a given feature dimension. We use classical functional central limit theorems to establish consistency of the testing procedure. The second procedure relies on a comparison of the so-called independent copy variances of the respective samples. Technically, this leads to functional central limit theorems for U-statistics built on $L^p$-$m$-approximable sample data.