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

arXiv: https://arxiv.org/abs/2603.15683 · HTML: https://mlsanity.com/paper/2603.15683/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 44 references that are in the map.
- First posted on arXiv 15 Mar 2026.

## Builds on (most cited first, 14 of 14)

- [Fused Gromov-Wasserstein distance for structured objects: theoretical foundations and mathematical properties](https://mlsanity.com/paper/1811.02834/) (2018)
- [CO-Optimal Transport](https://mlsanity.com/paper/2002.03731/) (2020)
- [Large Scale computation of Means and Clusters for Persistence Diagrams using Optimal Transport](https://mlsanity.com/paper/1805.08331/) (2018)
- [Topological Data Analysis](https://mlsanity.com/paper/1609.08227/) (2016)
- [Scalar Field Comparison with Topological Descriptors: Properties and Applications for Scientific Visualization](https://mlsanity.com/paper/2106.00157/) (2021)
- [Hypergraph Co-Optimal Transport: Metric and Categorical Properties](https://mlsanity.com/paper/2112.03904/) (2021)
- [Gromov-Wasserstein Averaging in a Riemannian Framework](https://mlsanity.com/paper/1910.04308/) (2019)
- [Tensor Entropy for Uniform Hypergraphs](https://mlsanity.com/paper/1912.09624/) (2019)
- [Flexible and Probabilistic Topology Tracking with Partial Optimal Transport](https://mlsanity.com/paper/2302.02895/) (2023)
- [Early warning indicators via latent stochastic dynamical systems](https://mlsanity.com/paper/2309.03842/) (2023)
- [Early Warning Prediction with Automatic Labeling in Epilepsy Patients](https://mlsanity.com/paper/2310.06059/) (2023)
- [Hypergraphs for multiscale cycles in structured data](https://mlsanity.com/paper/2210.07545/) (2022)
- [Covariance alignment: from maximum likelihood estimation to Gromov-Wasserstein](https://mlsanity.com/paper/2311.13595/) (2023)
- [Topological Optimal Transport for Geometric Cycle Matching](https://mlsanity.com/paper/2403.19097/) (2024)

## Abstract (arXiv)

We introduce a framework for analyzing topological tipping in time-evolutionary point clouds by extending the recently proposed Topological Optimal Transport (TpOT) distance. While TpOT unifies geometric, homological, and higher-order relations into one metric, its global scalar distance can obscure transient, localized structural reorganizations during dynamic phase transitions. To overcome this limitation, we present a hierarchical dynamic evaluation framework driven by a novel topological and hypergraph reconstruction strategy. Instead of directly interpolating abstract network parameters, our method interpolates the underlying spatial geometry and rigorously recomputes the valid topological structures, ensuring physical fidelity. Along this geodesic, we introduce a set of multi-scale indicators: macroscopic metrics (Topological Distortion and Persistence Entropy) to capture global shifts, and a novel mesoscopic dual-perspective Hypergraph Entropy (node-perspective and edge-perspective) to detect highly sensitive, asynchronous local rewirings. We further propagate the cycle-level entropy change onto individual vertices to form a point-level topological field. Extensive evaluations on physical dynamical systems (Rayleigh-Van der Pol limit cycles, Double-Well cluster fusion), high-dimensional biological aggregation (D'Orsogna model), and longitudinal stroke fMRI data demonstrate the utility of combining transport-based alignment with multi-scale entropy diagnostics for dynamic topological analysis.