A Two-Sample Test for Random Persistence Diagrams Against Alternatives Characterized by Persistence Intensity Differences

Abstract

Persistence intensity functions provide informative first-order summaries of random persistence diagram distributions. We study the two-sample comparison of persistence diagram distributions through their persistence intensity functions. In particular, for persistence diagram distributions $P$ and $Q$ with respective intensities $p$ and $q$, we consider the two-sample testing problem: $ \mathrm{H}_0: P=Q \ \text{versus}\ \mathrm{H}_1 :p\neq q. $ We propose a kernel-based permutation test for this problem. The test has finite-sample validity under the distributional null, and its power is characterized in terms of the $L^2$ discrepancy between the weighted persistence intensity functions. To accommodate persistence diagrams with possibly unbounded cardinality, we introduce regularity conditions that control the effect of cardinality variation and yield a sharp variance bound for the test statistic. We further show that every probability density on $\{(x,y)\in\mathbb{R}^2:y>x > 0\}$ can be realized as the intensity function of a random diagram. Using these results, we establish that the proposed test attains minimax-optimal separation rates over anisotropic Sobolev balls. Since the optimal bandwidth is not directly accessible in practice, we adopt a bandwidth aggregation framework. Simulations demonstrate validity under the distributional null and high empirical power against intensity-separated alternatives.