Decentralized Free-Support Wasserstein Barycenteredit

Decentralized Free-Support Wasserstein Barycenter is a manuscript by Xinbao Qiao, Bokai Hou, Peihua Mai, Wenqian Li, Wenjing Yan, and Ying-Jun Angela Zhang, currently under review. It studies how a network can build a shared distributional reference while allowing that reference to adapt to the geometry of locally held data.¹

Overviewedit

Many decentralized Wasserstein barycenter solvers place the barycenter on a grid chosen in advance and optimize the mass assigned to each location. A shared grid simplifies agreement between nodes, but it can limit geometric fidelity or require a large representation. This manuscript takes the complementary approach: it fixes equal atom masses and learns continuously movable locations. It does not jointly optimize atom masses and locations.

Each node computes a local optimal-transport plan and converts it into targets for the barycenter support. Neighbor gossip combines these support-sized updates within a majorization–minimization framework. Local measures and transport plans remain at their respective nodes; this data-access boundary does not by itself provide a formal privacy guarantee.

Key takeawaysedit

• Representation is part of the communication problem. A compact, movable support can preserve distributional geometry without committing the network to a dense common grid.
• Barycenter quality and network agreement have different time scales. The manuscript reports that a useful network-average barycenter can emerge with shallow gossip, while further communication mainly improves agreement between local copies.
• Communication budgets change the guarantee. Fixed-depth gossip controls disagreement and average motion; stronger stationarity conclusions require additional conditions and increasing communication.
• Shared geometric references can support decisions beyond averaging. The work evaluates barycenters as nominal distributions for cooperative distributionally robust optimization.

Evidence and boundariesedit

The manuscript reports experiments on synthetic measures, image distributions, and 3D point clouds, comparing against representative fixed-support decentralized solvers. It reports lower barycenter objectives, better geometric fidelity, and substantial computation and communication savings in the tested settings, together with a cooperative distributionally robust optimization application. These are manuscript-reported findings for a work under review.

For the quadratic-cost analysis with exact local transport plans, exact aggregation gives monotone descent and a best-iterate Clarke-stationarity guarantee. Fixed gossip depth instead gives topology-dependent bounds on disagreement and network-average motion, which do not establish stationarity. Under the stated compact-support and mixing assumptions, sufficiently increasing gossip depth with summable mixing errors yields vanishing disagreement and Clarke-stationary accumulation points. Stationarity does not establish global optimality. The regularized analysis assumes exact entropic subproblem solutions and does not justify a finite number of Sinkhorn iterations.

Placementedit

The paper connects AI and Networks, Distributed Wasserstein Barycenter, Wasserstein Geometry, and Data Centric Machine Learning. It extends Qiao's work on collaborative distributional references from using such references for evaluation to computing them under decentralized access and communication constraints.

Review statusedit

The manuscript is currently under review.

Footnotesedit

1. Author-provided manuscript, including the abstract, introduction, methodology, theory, experiments, and conclusion, with bibliographic metadata supplied by the author on 7 October 2026. ↩