Topological Optimal Transport for Geometric Cycle Matching

TL;DR

TpOT introduces a principled framework that fuses persistent homology with optimal transport to jointly match geometry and topology across datasets. By formulating measure topological networks and a distance $d_{\mathrm{TpOT},p}$, the approach yields geodesic, non-negatively curved metric spaces and provides entropic-regularised algorithms for practical computation. Theoretical results establish geodesics and curvature, while numerical experiments demonstrate robust joint matching of geometric cycles and topological generators, outperforming topology-only comparisons in preserving spatial arrangement. This topology-aware transport framework enables robust shape analysis, pattern tracking, and potential applications in protein folding, with clear guidelines on generating cycles and computational trade-offs.

Abstract

Topological data analysis is a powerful tool for describing topological signatures in real world data. An important challenge in topological data analysis is matching significant topological signals across distinct systems. In geometry and probability theory, optimal transport formalises notions of distance and matchings between distributions and structured objects. We propose to combine these approaches, constructing a mathematical framework for optimal transport-based matchings of topological features. Building upon recent advances in the domains of persistent homology and optimal transport for hypergraphs, we develop a transport-based methodology for topological data processing. We define measure topological networks, which integrate both geometric and topological information about a system, introduce a distance on the space of these objects, and study its metric properties, showing that it induces a geodesic metric space of non-negative curvature. The resulting Topological Optimal Transport (TpOT) framework provides a transport model on point clouds that minimises topological distortion while simultaneously yielding a geometrically informed matching between persistent homology cycles.

Figures (7)

• Figure 1: Topological Optimal Transport (TpOT). From left to right: (i) two input point clouds, (ii) their persistence diagrams, and (iii) corresponding PH-hypergraphs. The objective of the TpOT problem shares information across these three levels via a combination of distortion functionals: a Gromov-Wasserstein distortion on the point clouds (geometric information), (partial) Wasserstein matching of points in the persistence diagrams (topological information) and HyperCOT on the PH-hypergraphs (coupling of geometric and topological information).
• Figure 2: Computing PH and the PH-hypergraph(a) A point cloud and (b) a filtration of simplicial complexes built on it. (c) The $1$-dimensional persistence diagram of the filtration in (b). (d) The PH-hypergraph constructed after computing a representative cycle (orange and purple cycles in (b)) for each class in (c).
• Figure 3: (a) Source point cloud $X$, (b) (i) Pairwise affinity matrix $C$ encoding geometric information; (ii) PH-hypergraph incidence matrix $\omega$ encoding the membership of points to persistence features; (iii) 1-dimensional persistence diagram $D$, points coloured by lifespan; (iv) columns of $\omega$ for the top 4 persistence features. Node-level features in $\omega$ are computed using TOPF grande2024node, keeping the top 10 features. (c) Target point cloud $X'$. (d) Same as (b). (e) TpOT ($\alpha = 0.5, \beta = 1$) and GW matching. Inset: matching $\pi^e$ of the persistence diagrams $D, D'$ as found by TpOT. (f) A geodesic between the point clouds.
• Figure 4: (a) Source point cloud $X$ and its persistence diagram $D$. (b) Target point cloud $X'$ and its persistent diagram $D'$. (c) TpOT ($\alpha = 0.5, \beta = 1$) and GW matching. (d) Matching $\pi^e$ of the persistence diagrams $D, D'$ found by TpOT. (e) The variation of TpOT matchings as the parameters $\alpha$ and $\beta$ vary.
• Figure 5: (a) Source point cloud: a noisy mug in $\mathbb{R}^3$. (b) Target point cloud, a noisy solid torus in $\mathbb{R}^3$, with matching induced by TpOT and GW distance, respectively. (c) The variation of TpOT matching as the parameters $\alpha$ and $\beta$ vary.

Theorems & Definitions (29)

• Definition 1: Measure Topological Network
• Example 1: Topological Network from a Metric Measure Space
• Remark 1
• Example 2: Topological Network from a Curvature Set
• Remark 2: Finite and Infinite Diagrams
• Definition 2: Admissible Couplings
• Definition 3: Topological Optimal Transport (TpOT)
• Remark 3
• Proposition 1
• proof : Proof of Proposition \ref{['prop:realised']}