Metric properties of partial and robust Gromov-Wasserstein distances

TL;DR

This work extends Gromov-Wasserstein distances to robust, partial settings by introducing PGW_{ε,p} and sPGW_{ε,p}, which allow controlled mass redistribution to address outliers and partial matching. It establishes foundational results including existence of optimal relaxed couplings, right-continuity in the relaxation parameter, and convergence to GW as ε→0, while revealing that PGW and sPGW are not true metrics in general but admit meaningful approximate properties. The authors then present a parameter-free robust partial GW distance PGW_p^k, prove it is a true metric and topologically equivalent to GW on compact mm-spaces, and demonstrate robustness to perturbations both theoretically and numerically. They relate their framework to existing partial/relaxed GW formulations, provide a Gluing-type lemma to support relaxed-triangle bounds, and supply numerical implementations illustrating improved resilience to noise in registration tasks. Overall, the paper provides a rigorous mathematical basis for applying robust partial GW distances in applications where outliers and partial matching are prevalent, while connecting to related unbalanced and outlier-robust transport literature.

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.

Figures (3)

• Figure 1: Illustration of $\mathsf{mPGW}_p^k$ on synthetic data (see Example \ref{['ex:circles_and_ellipses']} for details). (a) Synthetic dataset of noisy circles and ellipses. (b) Plots of $\mathsf{mPGW}_{1-\delta,2}(X_0,X)$ between the template shape $X_0$ and shapes $X$ in the dataset, with $\delta \in [0,0.3]$, and the line $y = \delta$. The values at zero are equal to $\mathsf{GW}_2(X_0,X)$, which we emphasize with colored points along the $y$-axis. The $\delta$-values where the curves intersect the $y = \delta$ line give $\mathsf{mPGW}_2^1(X_0,X)$, as we emphasize with colored points along the $x$-axis. Observe that the points on the $x$-axis give a more intuitive clustering pattern than those along the $y$-axis. (c) Visualization of the matchings induced by Gromov-Wasserstein distance (top) and robust partial Gromov-Wasserstein distance (bottom).
• Figure 2: Quantitative evaluation of $\mathsf{mPGW}_p^k$ on synthetic data (see Example \ref{['ex:clustering_circles_ellipses']} for details). (a)$K$-Nearest Neighbor scores for Gromov-Wasserstein $\mathsf{GW}_2$ and partial robust Gromov-Wasserstein $\mathsf{mPGW}_2^1$ distances on a synthetic dataset of noisy circles and ellipses. (b) Dendrogram of the Gromov-Wasserstein distance matrix, with respect to the Ward linkage. (c) Dendrogram of the robust partial Gromov-Wasserstein distance matrix, with respect to the Ward linkage. Observe that $\mathsf{mPGW}_2^1$ gives a more consistent clustering pattern with respect to shape class than $\mathsf{GW}_2$.
• Figure 3: Visual comparisons of the matching results for the airplane, person, and chair, using GW and mPGW.

Theorems & Definitions (66)

• Remark 2.1
• Definition 2.2: Relaxed Couplings
• Remark 2.3
• Remark 2.4
• Theorem 2.5: memoli2007memoli2011-gw
• Remark 2.6
• Definition 2.7: Partial Gromov-Wasserstein Distances
• Remark 2.8
• Remark 2.9
• Remark 2.10