Multi-Marginal Gromov-Wasserstein Transport and Barycenters

TL;DR

This paper introduces multi-marginal Gromov-Wasserstein transport and barycenters, extending GW distances to multiple spaces with new regularized and fused variants, and proposes efficient numerical algorithms for their computation.

Contribution

It presents the novel concept of multi-marginal GW transport, its regularized and unbalanced versions, and develops a bi-convex relaxation with Sinkhorn-based algorithms for practical computation.

Findings

Relations to GW barycenters are established.

Numerical experiments demonstrate the method's potential.

Relaxation is tight in the balanced case with certain cost functions.

Abstract

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.