Tangential Fixpoint Iterations for Gromov-Wasserstein Barycenters

The Gromov-Wasserstein (GW) transport problem is a relaxation of classic optimal transport, which seeks a transport between two measures while preserving their internal geometry. Due to meeting this theoretical underpinning, it is a valuable tool for the analysis of objects that do not possess a natural embedding or should be studied independently of it. Prime applications can thus be found in e.g. shape matching, classification and interpolation tasks. To tackle the latter, one theoretically justified approach is the employment of multi-marginal GW transport and GW barycenters, which are Fr\'echet means with respect to the GW distance. However, because the computation of GW itself already poses a quadratic and non-convex optimization problem, the determination of GW barycenters is a hard task and algorithms for their computation are scarce. In this paper, we revisit a known procedure for the determination of Fr\'echet means in Riemannian manifolds via tangential approximations in the context of GW. We provide a characterization of barycenters in the GW tangent space, which ultimately gives rise to a fixpoint iteration for approximating GW barycenters using multi-marginal plans. We propose a relaxation of this fixpoint iteration and show that it monotonously decreases the barycenter loss. In certain cases our proposed method naturally provides us with barycentric embeddings. The resulting algorithm is capable of producing qualitative shape interpolations between multiple 3d shapes with support sizes of over thousands of points in reasonable time. In addition, we verify our method on shape classification and multi-graph matching tasks.