Decentralized Free-Support Wasserstein Barycenter

We study the decentralized computation of free-support Wasserstein barycenters (WBs) problem over arbitrary connected undirected graphs. Existing decentralized WB methods typically prescribe a shared support and optimize only its associated weights. Although this formulation leads to a tractable convex consensus problem, it prevents the WB support from adapting to the geometry of the input measures. We instead propose a decentralized free-support WB solver that learns continuously movable locations for an equal-mass atomic barycenter within a majorization-minimization (MM) framework. We establish theoretical guarantees under two distinct aggregation schemes: under exact aggregation, we prove monotone MM descent and a best-iterate Clarke-stationarity guarantee; under gossip-based aggregation, we characterize two regimes: using a fixed gossip depth yields topology-dependent bounds on disagreement and the network-average update, whereas increasing gossip sufficiently over rounds ensures that disagreement vanishes and every accumulation point of the exact-transport iterates is Clarke stationary. To our knowledge, this is the first fully decentralized, peer-to-peer neighbor-gossip framework for WB computation with continuously movable support locations. Experiments on synthetic measures, image distributions, and 3D point clouds show that learning movable supports improves both barycenter quality and resource efficiency. Compared with representative fixed-support decentralized solvers, our method achieves lower WB objectives and better geometric fidelity while reducing computation and communication by up to three orders of magnitude, with reductions exceeding three orders of magnitude in selected settings. The resulting WBs also provide effective nominal distributions for cooperative data-driven distributionally robust optimization (DRO), achieving the lowest validation loss among the decentralized-DRO baselines. Code available at [**Anonymous GitHub**](https://anonymous.4open.science/r/Decentralized-Free-Support-Wasserstein-Barycenter-FED9/README.md).