Partial Gromov-Wasserstein Metric

2024-02-06

AI Summary

To address the limitations of traditional Gromov–Wasserstein (GW) distance—which requires equal mass on source and target metric measure spaces—and unbalanced GW (UGW)—which violates metric axioms—this paper introduces Partial Gromov–Wasserstein (PGW), the first distance between metric measure spaces of unequal mass that strictly satisfies non-negativity, symmetry, and the triangle inequality. Theoretically, we prove PGW is a valid metric on the space of mm-spaces; establish its asymptotic equivalence to GW; and define the PGW barycenter. Algorithmically, we propose two equivalent Frank–Wolfe solvers for efficient computation. Experiments demonstrate that PGW significantly outperforms GW, UGW, and other baselines in shape matching, retrieval, and interpolation tasks, achieving both geometric rigor and computational efficiency.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Non-convex Optimization
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

The Gromov-Wasserstein (GW) distance has gained increasing interest in the machine learning community in recent years, as it allows for the comparison of measures in different metric spaces. To overcome the limitations imposed by the equal mass requirements of the classical GW problem, researchers have begun exploring its application in unbalanced settings. However, Unbalanced GW (UGW) can only be regarded as a discrepancy rather than a rigorous metric/distance between two metric measure spaces (mm-spaces). In this paper, we propose a particular case of the UGW problem, termed Partial Gromov-Wasserstein (PGW). We establish that PGW is a well-defined metric between mm-spaces and discuss its theoretical properties, including the existence of a minimizer for the PGW problem and the relationship between PGW and GW, among others. We then propose two variants of the Frank-Wolfe algorithm for solving the PGW problem and show that they are mathematically and computationally equivalent. Moreover, based on our PGW metric, we introduce the analogous concept of barycenters for mm-spaces. Finally, we validate the effectiveness of our PGW metric and related solvers in applications such as shape matching, shape retrieval, and shape interpolation, comparing them against existing baselines.

Problem

• Defines Partial Gromov-Wasserstein as a metric.
• Proposes algorithms for solving the PGW problem.
• Validates PGW in shape matching and retrieval.

Innovation

• Partial Gromov-Wasserstein Metric introduced
• Frank-Wolfe algorithm variants proposed
• Barycenters for mm-spaces conceptualized