Topological Optimal Transport for Geometric Cycle Matching

2024-03-28

arXiv.org

AI Summary

This work addresses the challenge of geometrically consistent matching of topological features—particularly persistent homology cycles—across disparate data systems. We propose Topological Optimal Transport (TpOT), the first framework that deeply integrates optimal transport with persistent homology. TpOT constructs a measure-topological network and defines a differentiable, geometry-aware topological-geometric joint distance within its non-negatively curved geodesic metric space, leveraging hypergraph optimal transport, measure theory, and Riemannian-geometric optimization of transport plans. On point cloud data, TpOT significantly reduces topological distortion while producing geometrically plausible and interpretable cycle-level correspondences. Theoretically, we prove that the proposed distance satisfies all metric axioms. TpOT establishes the first differentiable matching paradigm for topological data analysis that simultaneously ensures geometric fidelity and topological faithfulness.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Distributed Search

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Federated Web and WoT systems, including distributed, federated and edge-based data processing
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Data transparency and provenance

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.

Problem

• Match topological signals across systems
• Integrate geometric and topological information
• Minimise topological distortion in transport models

Innovation

• Optimal transport-based topological matching
• Integration of geometric and topological information
• Minimised topological distortion in point clouds