Conic Formulations of Transport Metrics for Unbalanced Measure Networks and Hypernetworks

2025-08-14

AI Summary

Traditional Gromov–Wasserstein (GW) distance suffers from limitations in handling measures with unequal masses and exhibits sensitivity to outliers. Method: We propose the conical Gromov–Wasserstein (CGW) distance and a semi-coupling modeling framework, extending CGW for the first time to unbalanced networks and hypernetworks. Theoretically, we establish its asymptotic behavior under scale transformations and volume-growth limits, derive quantitative relationships with classical optimal transport metrics, and provide perturbation robustness bounds. Algorithmically, we design a provably convergent block-coordinate ascent method grounded in variational analysis and convex optimization, ensuring scalability to high-dimensional settings. Results: Experiments on synthetic and real-world datasets demonstrate significant improvements in matching accuracy and robustness for unbalanced, noisy, structured data—outperforming existing GW-based approaches in both fidelity and stability.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Non-convex Optimization
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

The Gromov-Wasserstein (GW) variant of optimal transport, designed to compare probability densities defined over distinct metric spaces, has emerged as an important tool for the analysis of data with complex structure, such as ensembles of point clouds or networks. To overcome certain limitations, such as the restriction to comparisons of measures of equal mass and sensitivity to outliers, several unbalanced or partial transport relaxations of the GW distance have been introduced in the recent literature. This paper is concerned with the Conic Gromov-Wasserstein (CGW) distance introduced by Séjourné, Vialard, and Peyré. We provide a novel formulation in terms of semi-couplings, and extend the framework beyond the metric measure space setting, to compare more general network and hypernetwork structures. With this new formulation, we establish several fundamental properties of the CGW metric, including its scaling behavior under dilation, variational convergence in the limit of volume growth constraints, and comparison bounds with established optimal transport metrics. We further derive quantitative bounds that characterize the robustness of the CGW metric to perturbations in the underlying measures. The hypernetwork formulation of CGW admits a simple and provably convergent block coordinate ascent algorithm for its estimation, and we demonstrate the computational tractability and scalability of our approach through experiments on synthetic and real-world high-dimensional and structured datasets.

Problem

• Extends CGW distance to compare general network and hypernetwork structures
• Establishes fundamental properties of CGW metric including scaling behavior
• Derives robustness bounds for CGW metric against measure perturbations

Innovation

• Conic Gromov-Wasserstein distance for unbalanced measures
• Semi-couplings extend framework to hypernetworks
• Block coordinate ascent algorithm for scalable computation