Multi-marginal Approximation of the Linear Gromov-Wasserstein Distance

TL;DR

This paper introduces an approximation framework that links the linear Gromov-Wasserstein distance to multi-marginal GW formulations, enhancing computational efficiency and enabling complex multi-input matching.

Contribution

It provides a theoretical approximation result showing the linear GW distance as a limit of multi-marginal GW, advancing the understanding of GW computational methods.

Findings

Linear GW can be approximated by multi-marginal GW formulations

The approximation characterizes the linear GW as a limit of multi-marginal approaches

Enhances computational efficiency for large-scale GW problems

Abstract

Recently, two concepts from optimal transport theory have successfully been brought to the Gromov--Wasserstein (GW) setting. This introduces a linear version of the GW distance and multi-marginal GW transport. The former can reduce the computational complexity when computing all GW distances of a large set of inputs. The latter allows for a simultaneous matching of more than two marginals, which can for example be used to compute GW barycenters. The aim of this paper is to show an approximation result which characterizes the linear version as a limit of a multi-marginal GW formulation.