We propose a new approach for unsupervised alignment of heterogeneous datasets, which maps data from two different domains without any known correspondences to a common metric space. Our method is based on an unbalanced optimal transport problem with Gromov-Wasserstein marginal penalization. It can be seen as a counterpart to the recently introduced joint multidimensional scaling method. We prove that there exists a minimizer of our functional and that for penalization parameters going to infinity, the corresponding sequence of minimizers converges to a minimizer of the so-called embedded Wasserstein distance. Our model can be reformulated as a quadratic, multi-marginal, unbalanced optimal transport problem, for which a bi-convex relaxation admits a numerical solver via block-coordinate descent. We provide numerical examples for joint embeddings in Euclidean as well as non-Euclidean spaces.

Abstract. The Gromov–Wasserstein (GW) transport problem is a generalization 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 multimarginal GW transport and GW barycenters, which are Fréchet means with respect to the GW distance. However, because the computation of GW itself already poses a quadratic and nonconvex 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 Fréchet 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 multimarginal 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 multigraph matching tasks.

Gromov-Wasserstein distances are generalization of Wasserstein distances, which are invariant under distance preserving transformations. Although a simplified version of optimal transport in Wasserstein spaces, called linear optimal transport (LOT), was successfully used in practice, there does not exist a notion of linear Gromov-Wasserstein distances so far. In this paper, we propose a definition of linear Gromov-Wasserstein distances. We motivate our approach by a generalized LOT model, which is based on barycentric projection maps of transport plans. Numerical examples illustrate that the linear Gromov-Wasserstein distances, similarly as LOT, can replace the expensive computation of pairwise Gromov-Wasserstein distances in applications like shape classification.

We present the integration of a neurosurgical microscope as an interface to a medical training simulator. The simulator is based on Virtual Reality and combines the use of real-time simulation algorithms with a native interface consisting of a real surgical microscope and original instruments. The position and pose of the head of the microscope are tracked with an optical tracking system that is mounted on the microscope. The camera tracks 3 infrared light-emitting diodes (LEDs) that are integrated in a phantom of the patient's head. The status of the microscope such as zoom and focus are read out via controller-area network (CAN). The oculars have been replaced by a stereoscopic display.