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.

翻译：Gromov-Wasserstein (GW) 输运问题是经典最优输运的松弛形式，旨在寻找两个测度间的输运方案，同时保持其内部几何结构。由于契合这一理论基础，GW成为分析无自然嵌入或需独立于嵌入进行研究的对象的有效工具，主要应用于形状匹配、分类及插值任务。针对后者，一种具有理论合理性的方法是采用多边际GW输运和GW重心，即关于GW距离的Fréchet均值。然而，由于GW计算本身已是二次非凸优化问题，求解GW重心极具挑战，相关算法也较为稀缺。本文重新审视了黎曼流形中通过切空间逼近确定Fréchet均值的经典流程，并将其应用于GW场景。我们给出了GW切空间中重心的刻画，进而推导出基于多边际规划逼近GW重心的不动点迭代方法。我们提出了该不动点迭代的松弛形式，并证明其单调降低重心损失。在某些情况下，该方法能自然生成重心嵌入。最终算法可在合理时间内对含数千个支撑点的多个三维形状进行高质量形状插值。此外，我们还在形状分类与多图匹配任务上验证了方法的有效性。