Gromov-Wasserstein (GW) distances are combinations of Gromov-Hausdorff and Wasserstein distances that allow the comparison of two different metric measure spaces (mm-spaces). Due to their invariance under measure- and distance-preserving transformations, they are well suited for many applications in graph and shape analysis. In this paper, we introduce the concept of multi-marginal GW transport between a set of mm-spaces as well as its regularized and unbalanced versions. As a special case, we discuss multi-marginal fused variants, which combine the structure information of an mm-space with label information from an additional label space. To tackle the new formulations numerically, we consider the bi-convex relaxation of the multi-marginal GW problem, which is tight in the balanced case if the cost function is conditionally negative definite. The relaxed model can be solved by an alternating minimization, where each step can be performed by a multi-marginal Sinkhorn scheme. We show relations of our multi-marginal GW problem to (unbalanced, fused) GW barycenters and present various numerical results, which indicate the potential of the concept.

翻译：Gromov-Wasserstein（GW）距离是Gromov-Hausdorff距离与Wasserstein距离的结合，能够比较两个不同的度量测度空间（mm-空间）。由于其在保测度和保距离变换下的不变性，该距离非常适合图分析与形状分析中的众多应用。本文提出了一组mm-空间之间的多边际GW输运概念，及其正则化与非平衡版本。作为特例，我们讨论了结合mm-空间结构信息与附加标签空间标签信息的多边际融合变体。为数值求解新公式，我们考虑了多边际GW问题的双凸松弛方法——当代价函数条件负定时，该松弛在平衡情形下是紧致的。松弛模型可通过交替最小化求解，其中每一步均可通过多边际Sinkhorn方案实现。我们展示了所提多边际GW问题与（非平衡、融合）GW重心之间的关系，并给出多项数值结果，表明该概念的潜力。