Two-sample tests for relevant differences in persistence diagrams

Johannes Krebs; Daniel Rademacher

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arXiv · 2401.10349

Two-sample tests for relevant differences in persistence diagrams

Johannes Krebs·Daniel Rademacher

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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{\'e}chet variances of the two samples based on estimators for the Fr{\'e}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.

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Johannes Krebs, Daniel Rademacher. 2024-01-18. Two-sample tests for relevant differences in persistence diagrams. https://arxiv.org/abs/2401.10349

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