# Parallel Streaming Wasserstein Barycenters

arXiv: https://arxiv.org/abs/1705.07443 · HTML: https://mlsanity.com/paper/1705.07443/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 May 2017.

## Built on by (most cited first, 14 of 14)

- [Continuous Wasserstein-2 Barycenter Estimation without Minimax Optimization](https://mlsanity.com/paper/2102.01752/) (2021)
- [Fair and Optimal Classification via Post-Processing](https://mlsanity.com/paper/2211.01528/) (2022)
- [Wasserstein barycenters are NP-hard to compute](https://mlsanity.com/paper/2101.01100/) (2021)
- [Tree-Based Diffusion Schr"odinger Bridge with Applications to Wasserstein Barycenters](https://mlsanity.com/paper/2305.16557/) (2023)
- [Dimensionality Reduction for Wasserstein Barycenter](https://mlsanity.com/paper/2110.08991/) (2021)
- [Doubly Regularized Entropic Wasserstein Barycenters](https://mlsanity.com/paper/2303.11844/) (2023)
- [Variational Wasserstein Barycenters with c-Cyclical Monotonicity](https://mlsanity.com/paper/2110.11707/) (2021)
- [A novel notion of barycenter for probability distributions based on optimal weak mass transport](https://mlsanity.com/paper/2102.13380/) (2021)
- [Continual Learning of Generative Models with Limited Data: From Wasserstein-1 Barycenter to Adaptive Coalescence](https://mlsanity.com/paper/2101.09225/) (2021)
- [Schr"odinger Bridge Matching for Tree-Structured Costs and Entropic Wasserstein Barycentres](https://mlsanity.com/paper/2506.17197/) (2025)
- [Decentralized Entropic Optimal Transport for Distributed Distribution Comparison](https://mlsanity.com/paper/2301.12065/) (2023)
- [ScoreFusion: Fusing Score-based Generative Models via Kullback-Leibler Barycenters](https://mlsanity.com/paper/2406.19619/) (2024)
- [An Asynchronous Decentralized Algorithm for Wasserstein Barycenter Problem](https://mlsanity.com/paper/2304.11653/) (2023)
- [Field Codes for Distributed Coupling Samplers and Certified Empirical Transport](https://mlsanity.com/paper/2607.27078/) (2026)

## Abstract (arXiv)

Efficiently aggregating data from different sources is a challenging problem, particularly when samples from each source are distributed differently. These differences can be inherent to the inference task or present for other reasons: sensors in a sensor network may be placed far apart, affecting their individual measurements. Conversely, it is computationally advantageous to split Bayesian inference tasks across subsets of data, but data need not be identically distributed across subsets. One principled way to fuse probability distributions is via the lens of optimal transport: the Wasserstein barycenter is a single distribution that summarizes a collection of input measures while respecting their geometry. However, computing the barycenter scales poorly and requires discretization of all input distributions and the barycenter itself. Improving on this situation, we present a scalable, communication-efficient, parallel algorithm for computing the Wasserstein barycenter of arbitrary distributions. Our algorithm can operate directly on continuous input distributions and is optimized for streaming data. Our method is even robust to nonstationary input distributions and produces a barycenter estimate that tracks the input measures over time. The algorithm is semi-discrete, needing to discretize only the barycenter estimate. To the best of our knowledge, we also provide the first bounds on the quality of the approximate barycenter as the discretization becomes finer. Finally, we demonstrate the practical effectiveness of our method, both in tracking moving distributions on a sphere, as well as in a large-scale Bayesian inference task.