On a linear Gromov-Wasserstein distance

TL;DR

This paper introduces a new linear Gromov-Wasserstein distance, inspired by linear optimal transport, to simplify computations in applications such as shape classification, where traditional Gromov-Wasserstein distances are computationally expensive.

Contribution

The paper proposes the first definition of linear Gromov-Wasserstein distances based on barycentric projections, extending the linear optimal transport concept.

Findings

Linear Gromov-Wasserstein distances can replace expensive pairwise computations.

Numerical examples demonstrate effectiveness in shape classification.

The approach simplifies computations while maintaining accuracy.

Abstract

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.