The partial Gromov-Wasserstein (PGW) problem facilitates the comparison of measures with unequal masses residing in potentially distinct metric spaces, thereby enabling unbalanced and partial matching across these spaces. In this paper, we demonstrate that the PGW problem can be transformed into a variant of the Gromov-Wasserstein problem, akin to the conversion of the partial optimal transport problem into an optimal transport problem. This transformation leads to two new solvers, mathematically and computationally equivalent, based on the Frank-Wolfe algorithm, that provide efficient solutions to the PGW problem. We further establish that the PGW problem constitutes a metric for metric measure spaces. Finally, we validate the effectiveness of our proposed solvers in terms of computation time and performance on shape-matching and positive-unlabeled learning problems, comparing them against existing baselines.

翻译：部分Gromov-Wasserstein（PGW）问题有助于比较可能位于不同度量空间中、质量不等的测度，从而能够实现这些空间之间的非平衡与部分匹配。在本文中，我们证明PGW问题可转化为Gromov-Wasserstein问题的一个变体，类似于将部分最优运输问题转化为最优运输问题。这一转化基于Frank-Wolfe算法，催生了两种数学和计算上等价的求解器，为PGW问题提供了高效解法。我们进一步证明PGW问题构成度量测度空间上的一个度量。最后，我们通过形状匹配和正无标记学习问题，从计算时间和性能两方面验证了所提求解器的有效性，并与现有基线方法进行了比较。