The Gromov–Wasserstein problem is a non-convex optimization problem over the polytope of transportation plans between two probability measures supported on two spaces, each equipped with a cost function evaluating similarities between points. Akin to the standard optimal transportation problem, it is natural to ask for conditions guaranteeing some structure on the optimizers, for instance, if these are induced by a (Monge) map. We study this question in Euclidean spaces when the cost functions are either given by (i) inner products or (ii) squared distances, two standard choices in the literature. We establish the existence of an optimal map in case (i) and of an optimal 2-map (the union of the graphs of two maps) in case (ii), both under an absolute continuity condition on the source measure. Additionally, in case (ii) and in dimension one, we numerically design situations where optimizers of the Gromov–Wasserstein problem are 2-maps but are not maps. This suggests that our result cannot be improved in general for this cost. Still in dimension one, we additionally establish the optimality of monotone maps under some conditions on the measures, thereby giving insight into why such maps often appear to be optimal in numerical experiments.

Gromov-Wasserstein 问题是一个非凸优化问题，涉及两个空间上支持的两个概率度量之间的运输计划的多面体，每个概率度量都配备了评估点之间相似性的成本函数。类似于标准的最优运输问题，很自然地要求保证优化器上的某些结构的条件，例如，如果这些结构是由（Monge）图诱导的。当成本函数由（i）内积或（ii）平方距离（文献中的两个标准选择）给出时，我们在欧几里得空间中研究这个问题。我们在情况 (i) 中确定了最优映射的存在性，在情况 (ii) 中确定了最优 2-映射（两个映射的图的并集）的存在性，两者都在源测度的绝对连续性条件下存在。此外，在情况 (ii) 和维度一中，我们以数值方式设计了 Gromov-Wasserstein 问题的优化器是 2 映射但不是映射的情况。这表明我们的结果总体上无法以这种成本得到改善。仍然在第一维中，我们还建立了在某些测量条件下单调图的最优性，从而深入了解为什么此类图在数值实验中通常看起来是最优的。