Persistence-Based Statistics for Detecting Structural Changes in High-Dimensional Point Clouds

Abstract: We study the probabilistic behavior of persistence statistics under distributional variability and propose a novel nonparametric framework for detecting structural changes in high-dimensional random point clouds. We first establish moment bounds and tightness results for classical persistence statistics - total and maximum persistence - under general distributions, with explicit scaling behavior derived for Gaussian mixture models. Building on these theoretical foundations, we introduce a normalized statistic based on persistence landscapes combined with the Jensen-Shannon divergence, and we prove its Holder continuity with respect to perturbations of input point clouds. The resulting measure is stable, scale- and shift-invariant, and suitable for nonparametric inference via permutation testing. A numerical illustration using dynamic attribute vectors from decentralized governance data demonstrates how the proposed method can capture regime shifts and evolving geometric complexity. Our results contribute to the theoretical understanding of random persistence and provide a rigorous statistical foundation for topological change-point detection in complex, high-dimensional systems.