Persistence-Based Statistics for Detecting Structural Changes in High-Dimensional Point Clouds
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866917171271041024 |
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| author | Nakayama, Toshiyuki |
| author_facet | Nakayama, Toshiyuki |
| contents | We study the probabilistic behavior of persistence-based statistics and propose a novel nonparametric framework for detecting structural changes in high-dimensional random point clouds. We establish moment bounds and tightness results for classical persistence statistics-total and maximum persistence-under general distributions, with explicit variance-scaling behavior derived for Gaussian mixture models. Building on these results, we introduce a bounded and normalized statistic based on persistence landscapes combined with the Jensen-Shannon divergence, and we prove its Holder continuity with respect to perturbations of the input point clouds. The resulting measure is stable, scale- and shift-invariant, and well suited for finite-sample nonparametric inference via permutation testing. An illustrative numerical study using dynamic attribute vectors from decentralized governance data demonstrates the practical applicability of the proposed method. Overall, this work provides a statistically rigorous and computationally stable approach to change-point detection in complex, high-dimensional data. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_00938 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Persistence-Based Statistics for Detecting Structural Changes in High-Dimensional Point Clouds Nakayama, Toshiyuki Statistics Theory Algebraic Topology Probability 62R40, 62G10, 55N31 We study the probabilistic behavior of persistence-based statistics and propose a novel nonparametric framework for detecting structural changes in high-dimensional random point clouds. We establish moment bounds and tightness results for classical persistence statistics-total and maximum persistence-under general distributions, with explicit variance-scaling behavior derived for Gaussian mixture models. Building on these results, we introduce a bounded and normalized statistic based on persistence landscapes combined with the Jensen-Shannon divergence, and we prove its Holder continuity with respect to perturbations of the input point clouds. The resulting measure is stable, scale- and shift-invariant, and well suited for finite-sample nonparametric inference via permutation testing. An illustrative numerical study using dynamic attribute vectors from decentralized governance data demonstrates the practical applicability of the proposed method. Overall, this work provides a statistically rigorous and computationally stable approach to change-point detection in complex, high-dimensional data. |
| title | Persistence-Based Statistics for Detecting Structural Changes in High-Dimensional Point Clouds |
| topic | Statistics Theory Algebraic Topology Probability 62R40, 62G10, 55N31 |
| url | https://arxiv.org/abs/2511.00938 |