Static and Dynamic Approaches to Computing Barycenters of Probability Measures on Graphs

2026-03-27

AI Summary

This work addresses the challenge of defining weighted averages (barycenters) of probability measures on graph structures, where classical optimal transport suffers from geometric degeneracy. By leveraging the Riemannian geometry of the probability simplex induced by dynamic optimal transport, the authors propose an intrinsic gradient descent method to compute barycenters on graphs. This approach approximates the exponential map and its inverse through numerical geodesic approximation and discretization of the continuity equation. In the analysis phase, barycentric coordinates are obtained by solving a quadratic program based on geodesic distances. Experimental results demonstrate that the proposed method significantly outperforms static optimal transport approaches relying on graph distances or entropic regularization, thereby validating the effectiveness and superiority of the introduced intrinsic geometric framework for measure synthesis and analysis on graphs.

Abstract

The optimal transportation problem defines a geometry of probability measures which leads to a definition for weighted averages (barycenters) of measures, finding application in the machine learning and computer vision communities as a signal processing tool. Here, we implement a barycentric coding model for measures which are supported on a graph, a context in which the classical optimal transport geometry becomes degenerate, by leveraging a Riemannian structure on the simplex induced by a dynamic formulation of the optimal transport problem. We approximate the exponential mapping associated to the Riemannian structure, as well as its inverse, by utilizing past approaches which compute action minimizing curves in order to numerically approximate transport distances for measures supported on discrete spaces. Intrinsic gradient descent is then used to synthesize barycenters, wherein gradients of a variance functional are computed by approximating geodesic curves between the current iterate and the reference measures; iterates are then pushed forward via a discretization of the continuity equation. Analysis of measures with respect to given dictionary of references is performed by solving a quadratic program formed by computing geodesics between target and reference measures. We compare our novel approach to one based on entropic regularization of the static formulation of the optimal transport problem where the graph structure is encoded via graph distance functions, we present numerical experiments validating our approach, and we conclude that intrinsic gradient descent on the probability simplex provides a coherent framework for the synthesis and analysis of measures supported on graphs.

Problem

• barycenters
• optimal transport
• graphs
• probability measures
• degenerate geometry

Innovation

• dynamic optimal transport
• Riemannian structure on simplex
• intrinsic gradient descent
• barycenter on graphs
• geodesic approximation