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.

通过将部分 Gromov-Wasserstein 问题转化为 Gromov-Wasserstein 问题的变体，本文提出了两个新的使用 Frank-Wolfe 算法的求解器，并通过对形状匹配和正无标记学习问题的比较验证了其计算时间和性能的有效性。