The Z-Gromov-Wasserstein Distance

TL;DR

This paper introduces a generalized framework for Gromov-Wasserstein distances called Z-Gromov-Wasserstein, applicable to complex structured data like graphs, establishing its mathematical properties and providing computational bounds.

Contribution

It defines Z-Gromov-Wasserstein distance for Z-networks, unifying and extending existing GW variants with proven metric properties and practical computable bounds.

Findings

Z-GW is a metric on Z-networks with key properties.

The framework unifies many existing GW variants.

Provides computable bounds for practical use.

Abstract

The Gromov-Wasserstein (GW) distance is a powerful tool for comparing metric measure spaces which has found broad applications in data science and machine learning. Driven by the need to analyze datasets whose objects have increasingly complex structure (such as node and edge-attributed graphs), several variants of GW distance have been introduced in the recent literature. With a view toward establishing a general framework for the theory of GW-like distances, this paper considers a vast generalization of the notion of a metric measure space: for an arbitrary metric space $Z$, we define a $Z$-network to be a measure space endowed with a kernel valued in $Z$. We introduce a method for comparing $Z$-networks by defining a generalization of GW distance, which we refer to as $Z$-Gromov-Wasserstein ($Z$-GW) distance. This construction subsumes many previously known metrics and offers a unified approach to understanding their shared properties. This paper demonstrates that the $Z$-GW distance defines a metric on the space of $Z$-networks which retains desirable properties of $Z$, such as separability, completeness, and geodesicity. Many of these properties were unknown for existing variants of GW distance that fall under our framework. Our focus is on foundational theory, but our results also include computable lower bounds and approximations of the distance which will be useful for practical applications.