Beyond Distance: Quantifying Point Cloud Dynamics with Persistent Homology and Dynamic Optimal Transport
Authors: Yixin Wang, Ting Gao, Jinqiao Duan
Organizations: School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, China · Center for Mathematical Science, Huazhong University of Science and Technology, Wuhan, China · Steklov-Wuhan Institute for Mathematical Exploration, Huazhong University of Science and Technology, China · Department of Mathematics and Department of Physics, Great Bay University, Dongguan, China · Guangdong Provincial Key Laboratory of Mathematical and Neural Dynamical Systems, Dongguan, China.
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.
Figures & tables
Figure 1: Schematic overview of the hypergraph reconstruction and dynamic distortion framework. Row 1 (Initial TpOT): The optimal spatial coupling π⋆v is computed between the source X0 and target X1 point clouds via solving optimal TpOT problem. Row 2 (Geometric Interpolation): Metric interpolation and Multidimensional Scaling (MDS) generate intermediate spatial configurations Xt along the geodesic t∈[0,1] . Row 3 (Topological Structure Reconstruction): Persistent homology is computed based on the interpolated geometric point cloud to extract authentic intermediate features. Row 4 (Hypergraph Reconstruction: Measure topological networks Pt are assembled, where nodes represent regions and colored hulls represent topological hyperedges. Row 5 (Dynamic Distortion as Early Warning Indicators) The dynamic distortions ( Lt ) are rigorously computed by solving the TpOT distances between the global reference source P0 and each intermediate state Pt .
Figure 2: Ground truth evolution of the stochastic RVP oscillator. (Top) Scatter plots of the state space showing the transition from a limit cycle to a monostable point. Four sparse snapshots are chosen as our training density samples for topological optimal transport task (illustrated in red boxes ). (Bottom) The corresponding persistence diagrams tracking the birth and death of the 1-dimensional homological feature.
Figure 3: Baseline(ground truth) sequential evaluation directly computed between the reference state ( h=−1 ) and subsequent empirical snapshots. (a) The topological distortion peaks and flattens exactly as the limit cycle collapses. (b) Both persistence entropy and the proposed symmetric hypergraph entropy exhibit a discontinuous jump at the critical point h=0 . (c) The jump in symmetric entropy is primarily driven by the hyperedge-perspective component ( HEE ).To facilitate a direct visual comparison, both perspective entropies are normalized by their respective theoretical upper bounds.
Figure 4: Experimental validation via TpOT geodesic interpolation. We subsampled the empirical dataset into merely four equidistant keyframes and reconstructed the continuous topological evolution parameterized by τ∈[0,1] . (a) The MDS-based isometric embedding interpolates the intermediate spatial geometries between the sparse keyframes, approximating the collapse of the limit cycle. (b) The dynamic evaluation along the reconstructed geodesic reproduces the hierarchical macro-meso-micro distortion sequence and the abrupt entropic jumps at the critical transition point, closely aligning with the unobserved ground truth dynamics.
Figure 5: Phenomenological bifurcation in the double-well potential model. (Top) Scatter plots of the state space showing the transition from a bistable regime(two distinct clusters) to a monostable regime(a single fused cluster). Four sparse snapshots are chosen as our training density samples for topological optimal transport task (illustrated in red boxes ). (Bottom) The corresponding persistence diagrams tracking the birth and death of the 1-dimensional homological feature.
Figure 6: Ground truth baseline dynamics over parameter h . Because the system only undergoes a 0-dimensional transition, the 1-dimensional structural metrics ( Ltopo , Lhyper , and Entropies) remain completely suppressed, while the geometric distortion Lgeom rises to capture the spatial convergence.
Figure 7: Dimension-specificity and negative control evaluation. (a)Dynamic reconstruction via geodesic interpolation from only four keyframes successfully reproduces this exact dimension-specific decoupling. (b)The MDS-based isometric embedding and corresponding 1-D persistence diagram.
Figure 8: Ground truth evolution of the D’Orsogna biological aggregation model. (Top) The 2-dimensional spatial projection (x,y) of the 4-dimensional phase space shows particles self-organizing from a disorganized state into a mill with a hollow core. Four sparse snapshots are chosen as our training density samples for topological optimal transport task (illustrated in red boxes ). (Bottom) The corresponding 1-dimensional persistence diagrams track the emergence of a robust Betti-1 feature representing the rotational vortex.
Figure 9: Baseline dynamic evaluation of the 4D D’Orsogna model. (a) The topological distortion Ltopo rises significantly to capture the formation of the homological loop. (b, c) Unlike previous bifurcation experiments, the structural entropies undergo a significant drop, rigorously quantifying the thermodynamic transition from a disorganized chaotic swarm into a highly self-organized, low-entropy vortex state.
Figure 10: Reconstructed spatial geometry and persistence diagrams from only four keyframes. While the 2D visual projection exhibits minor alignment artifacts due to the rigid 4-dimensional optimal transport coupling, the 1-dimensional persistent homology accurately reconstructs the birth of the vortex core.
Figure 11: Experimental validation of the reconstructed structural dynamics. The TpOT geodesic interpolation tracks the ground truth dynamics from Figure 9 , recovering both the the increase in topological distortion and the subsequent entropy drop associated with biological self-organization.
Figure 12: Visualization of the stroke patient’s fMRI data and dimensionality reduction. (a) An exemplary 3D spatial slice of the patient’s raw fMRI volume. (b) 2D UMAP embeddings of the voxel feature vectors at 3-month (left) and 1-year (right) post-stroke time points. Points are colored by their corresponding Yeo7 functional network assignments.
Figure 13: Dynamic entropy and distortion heatmaps across seven functional brain networks (Yeo7) along the TpOT geodesic. The mesoscopic hypergraph entropies (a-c) capture asynchronous local rewirings and region-specific variations. In contrast, the macroscopic Persistence Entropy (d) and Topological Distortion (e) characterize global structural shifts (e.g., in Network 2). The Geometric Distortion (f) exhibits uniform linear growth, tracking overall spatial displacement rather than topological phase transitions.
Figure 14: Vertex-level hypergraph entropy analysis of the Dorsal Attention network.
Figure 15: Distortion and entropy trajectories along the TpOT geodesic of the Dorsal Attention network.
Persistence-based topological optimization deforms a point cloud X⊂Rd by minimizing objectives of the form L(X)=ℓ(Dgm(X)), where Dgm(X) is a persistence diagram. In practice, optimization is limited by two coupled issues: persistent homology is typically computed on subsamples, and the resulting topological gradients are highly sparse, with only a few anchor points receiving nonzero updates. Motivated by diffeomorphic interpolation, which extends sparse gradients to smooth ambient vector fields via Reproducing Kernel Hilbert Space (RKHS) interpolation, we propose a more scalable pipeline that improves both subsampling and gradient extension. We introduce subsampling via random slicing, a lightweight scheme that promotes iteration-wise geometric coverage and mitigates density bias. We further replace the costly kernel solve with a fast Nadaraya-Watson (NW) Gaussian convolution, producing a globally defined smooth update field at a fraction of the computational cost, while being more suited for topological optimization tasks. We provide theoretical guarantees for NW smoothing, including anchor approximation bounds and global Lipschitz estimates. Experiments in 2D and 3D show that combining random slicing with NW smoothing yields consistent speedups and improved objective values over other baselines on common persistence losses.
Point-cloud persistent homology (PH) -- computing alpha or Rips complexes on spin-position point clouds -- has been widely applied to detect phase transitions in classical spin models since Donato et al. (2016), with subsequent studies attributing the detection to the topological content of the persistence diagram. We ask a simple question that has not been posed: what fraction of the PH signal is genuinely topological? We introduce f_topo, a quantitative decomposition that separates the density-driven and topological contributions to any PH statistic by comparing real spin configurations against density-matched shuffled null models. Across the 2D Ising model (system sizes L = 16-128, ten temperatures) and Potts models (q = 3, 5), we find that H_0 statistics -- total persistence, persistence entropy, feature count -- are 94-100% density-driven (f_topo < 0.07). The density-matched shuffled null detects T_c at the identical location and with comparable peak height as real configurations, showing that density alone is sufficient for phase transition detection. However, H_1 statistics are partially topological: the topological fraction grows with system size as delta(TP_{H_1}) ~ L^{0.53} and follows a finite-size scaling collapse delta(T, L) = L^{0.53} g(tL^{1/nu}) with collapse quality CV = 0.27. The longest persistence bar is strongly topological (f_topo > 1) and scales with the correlation length. A scale-resolved analysis reveals that the topological excess shifts from large-scale to small-scale features as L increases. We propose that the TDA-for-phase-transitions community adopt shuffled null models as standard practice, and that H_1 rather than H_0 statistics be used when genuine topological information is sought.
Gromov--Wasserstein optimal transport (GW-OT) extends classical optimal transport by introducing structure-aware transport cost. This is particularly relevant for spatial transcriptomics, where dynamical reconstruction should preserve tissue structure in addition to matching expression patterns. While static formulations have been widely used for such structure-aware alignment, a general dynamic formulation for reconstructing continuous trajectories is still missing. We introduce Travelling Pair Dynamical Alignment and Trajectory Estimation (TP-DATE), a theoretical and computational framework to generalize GW-OT dynamically in a simulation-free manner. We formulate a broad class of static and dynamic Quadratic-form OT (QOT) through path actions and prove the static dynamic equivalence. We further develop travelling-pair flow matching, which allows interacting conditional paths and marginalizes their interactions into a single vector field. On synthetic and real spatial transcriptomics data, TP-DATE better preserves spatial structure and improves continuous 3D dynamics reconstruction.
Junda Ying, Zhiwei Zeng, Peijie Zhou +1
Beijing International Center for Mathematical Research, Peking University · Center for Quantitative Biology, Peking University · Center for Machine Learning Research, Peking University +4