Aug 2026· SIAM Journal on Mathematics of Data Science· 0 citations· 24 references
TL;DR
Two methods are proposed: 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.
Abstract
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 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.
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Defining a weighted mean over probability measures under probability metrics is a central tool in probabilistic machine learning. Under the Wasserstein metric, these are called \emph{Wasserstein barycenters}. While most approaches compute barycenters for a fixed weight vector, approximating the whole family of barycent...
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