Linear and Multi-marginal Gromov–Wasserstein

Abstract:
Gromov—Wasserstein distances are a generalization of the classical optimal transport problem and allow for the matching and comparison of two arbitrary metric measure spaces. Due to its invariance under measure- and distance-preserving transformations, the metric has many applications in graph and shape analysis. Unfortunately, the computation of the Gromov—Wasserstein distance
is numerically expensive, limiting its application in machine learning like in classification tasks.
To overcome this issue, we propose a linear version of the Gromov-Wasserstein metric, which is based on the geometric structure of the Gromov—Wasserstein space. Furthermore, we introduce the concept of multi-marginal Gromov—Wasserstein transport between a set of metric measure spaces as well as its regularized and unbalanced versions. The multi-marginal Gromov-Wasserstein transport has a close relation to (unbalanced) Gromov—Wasserstein barycenter.
We present numerical examples to illustrate the proposed formulations.