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

Toshiyuki Nakayama

TL;DR

The paper addresses the challenge of detecting structural changes in high-dimensional random point clouds by developing a statistically rigorous, nonparametric framework based on persistence statistics. It establishes moment bounds for total and maximum persistence under general distributions and Gaussian mixtures, and introduces a bounded, normalized statistic—PL+JS—constructed from persistence landscapes and the Jensen–Shannon divergence to enable stable, scale- and shift-invariant change detection via permutation tests. The authors prove Hölder continuity of the PL+JS-based change measure with respect to perturbations of input point clouds, enabling robust finite-sample inference, and demonstrate practical applicability with dynamic attribute data from decentralized governance. Overall, the work provides a theoretically grounded, computationally stable approach to identifying regime shifts in complex, high-dimensional data, with concrete guidance for parameter tuning and applicability to real-world dynamic systems.

Abstract

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.

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

TL;DR

The paper addresses the challenge of detecting structural changes in high-dimensional random point clouds by developing a statistically rigorous, nonparametric framework based on persistence statistics. It establishes moment bounds for total and maximum persistence under general distributions and Gaussian mixtures, and introduces a bounded, normalized statistic—PL+JS—constructed from persistence landscapes and the Jensen–Shannon divergence to enable stable, scale- and shift-invariant change detection via permutation tests. The authors prove Hölder continuity of the PL+JS-based change measure with respect to perturbations of input point clouds, enabling robust finite-sample inference, and demonstrate practical applicability with dynamic attribute data from decentralized governance. Overall, the work provides a theoretically grounded, computationally stable approach to identifying regime shifts in complex, high-dimensional data, with concrete guidance for parameter tuning and applicability to real-world dynamic systems.

Abstract

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.

Paper Structure

This paper contains 34 sections, 16 theorems, 167 equations, 3 figures, 7 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background and Related Work
  3. High-Dimensional and Dynamic Point Clouds
  4. Topological Data Analysis and Related Work
  5. Persistence Statistics and Moment Bounds
  6. Persistence Statistics
  7. Vietoris--Rips filtration.
  8. Continuity and Regularity of Persistence Statistics
  9. Moment Bounds for General Distributions
  10. Moment Bounds and Tightness under GMMs
  11. A Persistence-Based Statistic for Comparing Random Point Clouds
  12. Definition of the PL+JS Statistic
  13. Continuity of the Proposed Change-Point Detection Statistic
  14. Numerical Illustration: Dynamic Point Clouds
  15. Monte Carlo precision and empirical $p$-values.
  16. ...and 19 more sections

Key Result

Proposition 1

Under the setting of Section sec:general_preliminaries, the following follows.

Figures (3)

  • Figure 1: Comparison of cluster number selection by BIC and AIC (Real vs Synthetic)
  • Figure 2: Scale factor vs. GMM log-likelihood and $H_0$ persistence statistics
  • Figure 3: Scale factor vs. GMM log-likelihood and $H_1$ persistence statistics

Theorems & Definitions (61)

  • Remark 1
  • Example 1: Three-point configuration
  • Proposition 1: Basic Regularity
  • proof
  • Theorem 2: Moment Bounds of Persistence Statistics for General Distributions
  • proof
  • Remark 2
  • Remark 3: Gaussian distributions with possibly singular covariance matrices
  • Remark 4: Key Features of Our Model
  • Definition 1: Gaussian Moment Constant
  • ...and 51 more