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

Persistence-based topological optimization deforms a point cloud $X \subset \mathbb{R}^d$ by minimizing objectives of the form $L(X) = \ell(\mathrm{Dgm}(X))$, where $\mathrm{Dgm}(X)$ 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 $2$D and $3$D 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.