Fused Gromov-Wasserstein distance for structured objects: theoretical foundations and mathematical properties

TL;DR

This paper introduces the Fused Gromov-Wasserstein distance, a new mathematical framework that combines feature and structural information for comparing structured objects, with proven properties and practical applications.

Contribution

It extends existing optimal transport distances by integrating feature and structure information into a unified metric, with theoretical foundations and convergence guarantees.

Findings

Proves the metric and interpolation properties of the Fused Gromov-Wasserstein distance.

Provides concentration results for finite sample convergence.

Demonstrates applications in structured object comparison.

Abstract

Optimal transport theory has recently found many applications in machine learning thanks to its capacity for comparing various machine learning objects considered as distributions. The Kantorovitch formulation, leading to the Wasserstein distance, focuses on the features of the elements of the objects but treat them independently, whereas the Gromov-Wasserstein distance focuses only on the relations between the elements, depicting the structure of the object, yet discarding its features. In this paper we propose to extend these distances in order to encode simultaneously both the feature and structure informations, resulting in the Fused Gromov-Wasserstein distance. We develop the mathematical framework for this novel distance, prove its metric and interpolation properties and provide a concentration result for the convergence of finite samples. We also illustrate and interpret its use in various contexts where structured objects are involved.