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.

翻译：近年来，Gromov-Wasserstein (GW) 距离在机器学习领域引起了越来越多的关注，因为它允许在不同度量空间中对测度进行比较。为了克服经典GW问题中要求等质量约束所带来的局限性，研究者们已开始探索其在非平衡设置下的应用。然而，非平衡GW (UGW) 只能被视为两个度量测度空间（mm-空间）之间的一种差异，而非严格的度量/距离。本文提出了一种UGW问题的特例，称为部分Gromov-Wasserstein (PGW)。我们证明了PGW是mm-空间之间一个定义良好的度量，并讨论了其理论性质，包括PGW问题极小解的存在性、PGW与GW之间的关系等。随后，我们提出了两种用于求解PGW问题的Frank-Wolfe算法变体，并证明它们在数学和计算上是等价的。此外，基于我们的PGW度量，我们引入了mm-空间的重心类比概念。最后，我们在形状匹配、形状检索和形状插值等应用中验证了所提出的PGW度量及相关求解器的有效性，并与现有基线方法进行了比较。