Vorträge in der Woche 20.05.2024 bis 26.05.2024

Freitag, 24.05.2024: Tangential fixed-point iterations for Gromov-Wasserstein barycenters

Florian Beier (TU Berlin)

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 Fréchet means with respect to the GW distance. This talk aims at providing a thorough exploration of the GW transport problem, the associated distance and its multi-marginal version. A major focus are GW barycenters for which we give a one-to-one characterization with respect to solutions of the multi-marginal problem. In order to tackle the barycenter/multi-marginal problem numerically, we employ a known procedure for the computation of generalized Fréchet means in Riemannian manifolds. This requires us to study the geometry of the GW space. We provide strong numerical evidence of the potential of this method, including multi 3d-shape interpolations.