Gromov-Wasserstein Barycenters: The Analysis Problem
This addresses a theoretical analysis problem in metric space estimation, but appears incremental as it builds on existing synthesis methods for GW barycenters.
The paper tackles the problem of estimating a matrix representing distances in a metric space under the barycentric coding model with respect to the Gromov-Wasserstein distance, proposing two methods and demonstrating their application in numerical experiments and machine learning.
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.