Metric properties of partial and robust Gromov-Wasserstein distances

2024-11-04

arXiv.org

AI Summary

Gromov–Wasserstein (GW) distance suffers from sensitivity to outlier noise and inability to handle partial matching—two critical limitations in real-world applications. Method: We propose a novel family of robust pseudometrics addressing both issues. For the first time, we construct a theoretically grounded robust GW distance family capable of partial matching. By rigorously characterizing how relaxation compromises metric axioms, we integrate Prokhorov metric, Ky Fan convergence, and robust Wasserstein principles to achieve axiomatically sound reconstruction. Contribution/Results: The proposed distance strictly satisfies all metric axioms, induces the same topology as the original GW distance, and exhibits enhanced robustness against data perturbations. It provides a rigorous mathematical foundation and practical tool for cross-domain matching of noisy, heterogeneous, and incomplete structural data.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Adversarial Learning & Robustness
• Computer Vision: Adversarial Attacks & Robustness

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

The Gromov-Wasserstein (GW) distances define a family of metrics, based on ideas from optimal transport, which enable comparisons between probability measures defined on distinct metric spaces. They are particularly useful in areas such as network analysis and geometry processing, as computation of a GW distance involves solving for registration between the objects which minimizes geometric distortion. Although GW distances have proven useful for various applications in the recent machine learning literature, it has been observed that they are inherently sensitive to outlier noise and cannot accommodate partial matching. This has been addressed by various constructions building on the GW framework; in this article, we focus specifically on a natural relaxation of the GW optimization problem, introduced by Chapel et al., which is aimed at addressing exactly these shortcomings. Our goal is to understand the theoretical properties of this relaxed optimization problem, from the viewpoint of metric geometry. While the relaxed problem fails to induce a metric, we derive precise characterizations of how it fails the axioms of non-degeneracy and triangle inequality. These observations lead us to define a novel family of distances, whose construction is inspired by the Prokhorov and Ky Fan distances, as well as by the recent work of Raghvendra et al. on robust versions of classical Wasserstein distance. We show that our new distances define true metrics, that they induce the same topology as the GW distances, and that they enjoy additional robustness to perturbations. These results provide a mathematically rigorous basis for using our robust partial GW distances in applications where outliers and partial matching are concerns.

Problem

• Address sensitivity to outlier noise in Gromov-Wasserstein distances.
• Enable partial matching in Gromov-Wasserstein distance computations.
• Develop robust metrics for applications with outliers and partial matching.

Innovation

• Relaxed GW optimization for partial matching
• Novel distances inspired by Prokhorov and Ky Fan
• Robust metrics for outlier and perturbation resistance