Gromov-Wasserstein Barycenters: The Analysis Problem

TL;DR

This paper addresses the analysis problem of estimating a matrix representing pairwise distances in a metric space using Gromov-Wasserstein barycenters, introducing two novel computational methods and demonstrating their applications.

Contribution

It formulates the GW barycenter estimation as an analysis problem and proposes fixed-point and differentiation-based methods for its solution.

Findings

Proposed fixed-point iteration method for GW barycenter computation.

Developed a differentiation-based approach utilizing a blow-up technique.

Validated the methods through numerical experiments and machine learning applications.

Abstract

This paper considers the problem of estimating a matrix that encodes pairwise distances in a finite metric space (or, more generally, the edge weight matrix of a network) under the barycentric coding model (BCM) with respect to the Gromov-Wasserstein (GW) distance function. We frame this task as estimating the unknown barycentric coordinates with respect to the GW distance, assuming that the target matrix (or kernel) belongs to the set of GW barycenters of a finite collection of known templates. In the language of harmonic analysis, if computing GW barycenters can be viewed as a synthesis problem, this paper aims to solve the corresponding analysis problem. We propose two methods: one utilizing fixed-point iteration for computing GW barycenters, and another employing a differentiation-based approach to the GW structure using a blow-up technique. Finally, we demonstrate the application of the proposed GW analysis approach in a series of numerical experiments and applications to machine learning.