Partial Gromov-Wasserstein Metric

TL;DR

The paper defines Partial Gromov-Wasserstein (PGW) as a metric between general metric measure spaces with possibly unequal masses, by using a total-variation penalty and restricting transport plans to a mass-constrained set. It establishes PGW as a true metric on mm-spaces (up to a mass-equivalence) and shows its relationship to GW, including a limiting connection as the penalty grows. The authors develop two practically equivalent Frank-Wolfe solvers for the discrete PGW problem and extend the framework to PGW barycenters for shape interpolation. Through toy outlier tests, shape retrieval, and barycenter experiments, PGW demonstrates robustness to outliers and noise while maintaining competitive accuracy, offering a scalable and robust alternative to existing GW variants for cross-domain shape analysis. Their open-source implementation further enables application to shape matching, retrieval, and interpolation tasks in diverse domains where mass is not necessarily conserved.

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. Our code is available at https://github.com/mint-vu/PGW_Metric.

Figures (13)

• Figure 1: The set of red points comprises the source point cloud. The union of the dark blue (outliers) and light blue points comprises the target point cloud. For UGW, MPGW, and PGW, we set the mass for each point to be the same. For GW, we normalize the mass for the balanced mass constraint setting.
• Figure 2: In each row, the first figure visualizes an example shape from each class, and the second figure visualizes the resulting pairwise distance matrices. The first row corresponds to Dataset I, and the second corresponds to Dataset II.
• Figure 3: In the first column, the first and second figures are the source and target point clouds in the first experiment ($\eta=5\%$); the third and fourth figures are the source and target point clouds in the second experiment ($\eta=10\%$).
• Figure 4: We visualize the dataset in point cloud interpolation. The first row is the original images in https://github.com/gpeyre/2016-ICML-gromov-wasserstein. The second row is the point clouds obtained by the k-mean method, where $k=1024$.
• Figure 5: We test interpolation tasks in 3 scenarios: source data is clean, target data is selected from three cases as described in section dataset and data processing. In each scenario, we test $\eta=5\%,10\%$ respectively. In the first column, we present the source and target point cloud visualization in each task. In columns 2-9, we present GW, PGW barycenter for $t=0/7,1/7,\ldots, 7/7$.

Theorems & Definitions (61)

• Proposition 2.1
• Remark 3.1
• Proposition 3.2
• Proposition 3.3
• Proposition 3.4
• Proposition 3.5
• proof : Proof of Remark \ref{['remark: rewrite pGW']}
• Lemma C.1
• proof
• Lemma C.2