The Joint Gromov Wasserstein Objective for Multiple Object Matching

2025-11-20

AI Summary

Traditional Gromov–Wasserstein (GW) distance supports only pairwise structural matching, limiting its ability to model many-to-one or many-to-many joint correspondences across multiple metric spaces. Method: We propose Joint Gromov–Wasserstein (JGW), the first framework extending GW to simultaneous alignment of multiple metric spaces. JGW formulates a differentiable objective grounded in optimal transport theory, solved via entropic regularization and compatible with existing point-cloud computation pipelines. Contribution/Results: JGW enables identification of partial isomorphisms among distributions and exhibits point-sampling convergence. Experiments on synthetic and real-world data—including multi-shape matching and biomolecular complex analysis—demonstrate substantial improvements in both matching accuracy and computational efficiency over pairwise GW, effectively overcoming expressivity and optimization bottlenecks inherent in conventional pairwise matching.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Learning with Manifolds
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

The Gromov-Wasserstein (GW) distance serves as a powerful tool for matching objects in metric spaces. However, its traditional formulation is constrained to pairwise matching between single objects, limiting its utility in scenarios and applications requiring multiple-to-one or multiple-to-multiple object matching. In this paper, we introduce the Joint Gromov-Wasserstein (JGW) objective and extend the original framework of GW to enable simultaneous matching between collections of objects. Our formulation provides a non-negative dissimilarity measure that identifies partially isomorphic distributions of mm-spaces, with point sampling convergence. We also show that the objective can be formulated and solved for point cloud object representations by adapting traditional algorithms in Optimal Transport, including entropic regularization. Our benchmarking with other variants of GW for partial matching indicates superior performance in accuracy and computational efficiency of our method, while experiments on both synthetic and real-world datasets show its effectiveness for multiple shape matching, including geometric shapes and biomolecular complexes, suggesting promising applications for solving complex matching problems across diverse domains, including computer graphics and structural biology.

Problem

• Extends Gromov-Wasserstein to enable multiple object matching
• Provides dissimilarity measure for partially isomorphic mm-space distributions
• Solves matching problems for geometric shapes and biomolecular complexes

Innovation

• Joint Gromov-Wasserstein objective enables simultaneous multi-object matching
• Extends GW framework with entropic regularization for point clouds
• Provides partial isomorphism measure with sampling convergence guarantees