A Stable and Theoretically Grounded Gromov-Wasserstein Distance for Reeb Graph Comparison using Persistence Images

TL;DR

This paper introduces a stable, theoretically grounded Gromov-Wasserstein distance tailored for Reeb graph comparison, leveraging persistence images to effectively incorporate topological features and ensure robustness against data perturbations.

Contribution

We propose a novel Reeb Gromov-Wasserstein distance with a symmetric Reeb radius and a probabilistic weighting scheme based on persistence images, providing stability and improved topological feature capture.

Findings

Proven stability of the proposed distance under data perturbations

Enhanced topological feature detection compared to existing metrics

Demonstrated effectiveness on multiple datasets

Abstract

Reeb graphs are a fundamental structure for analyzing the topological and geometric properties of scalar fields. Comparing Reeb graphs is crucial for advancing research in this domain, yet existing metrics are often computationally prohibitive or fail to capture essential topological features effectively. In this paper, we explore the application of the Gromov-Wasserstein distance, a versatile metric for comparing metric measure spaces, to Reeb graphs. We propose a framework integrating a symmetric variant of the Reeb radius for robust geometric comparison, and a novel probabilistic weighting scheme based on Persistence Images derived from extended persistence diagrams to effectively incorporate topological significance. A key contribution of this work is the rigorous theoretical proof of the stability of our proposed Reeb Gromov-Wasserstein distance with respect to perturbations in the underlying scalar fields. This ensures that small changes in the input data lead to small changes in the computed distance between Reeb graphs, a critical property for reliable analysis. We demonstrate the advantages of our approach, including its enhanced ability to capture topological features and its proven stability, through comparisons with other alternatives on several datasets, showcasing its practical utility and theoretical soundness.