Beyond Distance: Quantifying Point Cloud Dynamics with Persistent Homology and Dynamic Optimal Transport

2026-03-19 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Data Science & Analytics · Depth: Expert, extended

Summary

Researchers from Huazhong University of Science and Technology and Great Bay University introduce a novel framework for analyzing topological tipping points in time-evolving point clouds, extending the Topological Optimal Transport (TpOT) distance. This framework addresses TpOT's limitation of obscuring transient structural reorganizations by employing a hierarchical dynamic evaluation driven by a topological and hypergraph reconstruction strategy. Instead of interpolating abstract network parameters, their method interpolates underlying spatial geometry and rigorously re-computes valid topological structures. It introduces multi-scale indicators: macroscopic metrics like Topological Distortion and Persistence Entropy for global shifts, and a novel mesoscopic dual-perspective Hypergraph Entropy (node-perspective and edge-perspective) to detect sensitive local rewirings. The framework also propagates cycle-level entropy changes to individual vertices, forming a point-level topological field. Extensive evaluations on physical dynamical systems (Rayleigh-Van der Pol, Double-Well), biological aggregation (D'Orsogna model), and longitudinal stroke fMRI data demonstrate its utility.

Key takeaway

Research Scientists working with dynamic point cloud data should consider integrating this TpOT-based framework to gain deeper insights into complex system transitions. By leveraging the dynamic distortion curves and the dual-perspective hypergraph entropy, you can precisely identify and localize critical topological tipping points and structural reorganizations that traditional metrics might miss, even from sparse temporal observations. This approach offers a robust, multi-scale diagnostic for understanding evolving system dynamics in fields from physics to neuroimaging.

Key insights

A new framework quantifies dynamic topological changes in point clouds using multi-scale entropy and a geometry-preserving reconstruction strategy.

Principles

• Interpolate geometry, then reconstruct topology for physical fidelity.
• Multi-scale metrics reveal distinct aspects of structural change.
• Hypergraph entropy detects asynchronous local rewirings.

Method

The method involves computing initial TpOT spatial coupling, performing geometric interpolation, rigorously reconstructing valid topological and hypergraph structures, and extracting multi-scale early warning indicators including dynamic distortion and dual-perspective hypergraph entropy.

In practice

• Use dynamic distortion curves to localize geometric, topological, and incidence deformations.
• Apply persistence entropy to detect birth/death of significant cycles.
• Employ hypergraph entropy for efficient detection of local structural reorganizations.

Topics

• Topological Optimal Transport
• Persistent Homology
• Hypergraph Entropy
• Dynamic Systems Analysis
• Medical Imaging

Best for: Research Scientist, AI Researcher, AI Scientist, Data Scientist

Related on AIssential

• Static and Dynamic Approaches to Computing Barycenters of Probability Measures on Graphs — A dynamic optimal transport framework enables robust barycenter computation and analysis for probability measures on graphs.
• Tracking Representation Dynamics in Large Language Models with Persistent Homology — Persistent homology uncovers internal representation dynamics in LLMs during fine-tuning, showing early topological reorganization and objective-specific trajectories.
• FreeBridge: Variational Schrödinger Bridges for Cellular Transition Dynamics — Geometric grounding is crucial for biologically interpretable single-cell perturbation dynamics modeling.
• A Closed-Form Persistence-Landmark Pipeline for Certified Point-Cloud and Graph Classification — PLACE offers a closed-form pipeline for point cloud and graph classification with certified guarantees from persistent homology.
• Geometric Dictionary Learning of Dynamical Systems with Optimal Transport — DOODL learns shared spectral manifolds of dynamical systems for robust, data-efficient operator estimation and system identification.
• Persistence Diagrams Estimation of Multivariate Piecewise H{\"o}lder-continuous Signals — Algebraic stability offers considerable gains over sup-norm stability for persistence diagram estimation from noisy signals.

Open in AIssential →

Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.