A Two-Sample Test on Weighted Persistence Intensity Functions in Topological Data Analysis

The intensity function, defined as the Lebesgue density of the expected measure of a persistence diagram, is a fundamental summary of the probability distribution of persistence diagrams in topological data analysis (TDA). Although several methods have been proposed for estimating intensity functions, statistical hypothesis testing for intensity functions remains largely unexplored. In particular, little is known about the power properties of hypothesis tests based on persistence diagrams.
We propose a kernel-based permutation test and analyze its power against alternatives characterized by differences in persistence intensity functions. We introduce assumptions that control the effect of the possibly unbounded cardinality of persistence diagrams and yield a sharp variance bound for the test statistic. We also show that our probability model is broad enough to include all probability densities on the subset of $\mathbb{R}^2$ where $y>x\geq 0$. Using these results, we establish minimax optimality of the proposed test. Along the way, we derive an explicit characterization of the persistence diagram of the Čech complex on the circle.
Since the optimal bandwidth is not directly accessible in practice, we adopt a bandwidth aggregation framework. Simulations and real-data applications demonstrate validity and high empirical power.