  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.
