# Variational Wasserstein Barycenters with c-Cyclical Monotonicity

arXiv: https://arxiv.org/abs/2110.11707 · HTML: https://mlsanity.com/paper/2110.11707/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 44 references that are in the map.
- First posted on arXiv 22 Oct 2021.

## Builds on (most cited first, 24 of 24)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Fast Computation of Wasserstein Barycenters](https://mlsanity.com/paper/1310.4375/) (2013)
- [Do Neural Optimal Transport Solvers Work? A Continuous Wasserstein-2 Benchmark](https://mlsanity.com/paper/2106.01954/) (2021)
- [Continuous Wasserstein-2 Barycenter Estimation without Minimax Optimization](https://mlsanity.com/paper/2102.01752/) (2021)
- [Scalable Computations of Wasserstein Barycenter via Input Convex Neural Networks](https://mlsanity.com/paper/2007.04462/) (2020)
- [Debiased Sinkhorn barycenters](https://mlsanity.com/paper/2006.02575/) (2020)
- [Continuous Regularized Wasserstein Barycenters](https://mlsanity.com/paper/2008.12534/) (2020)
- [Fast Discrete Distribution Clustering Using Wasserstein Barycenter with Sparse Support](https://mlsanity.com/paper/1510.00012/) (2015)
- [A Smoothed Dual Approach for Variational Wasserstein Problems](https://mlsanity.com/paper/1503.02533/) (2015)
- [Fixed-Support Wasserstein Barycenters: Computational Hardness and Fast Algorithm](https://mlsanity.com/paper/2002.04783/) (2020)
- [On Robust Optimal Transport: Computational Complexity and Barycenter Computation](https://mlsanity.com/paper/2102.06857/) (2021)
- [Semi-dual Regularized Optimal Transport](https://mlsanity.com/paper/1811.05527/) (2018)
- [Stochastic Wasserstein Barycenters](https://mlsanity.com/paper/1802.05757/) (2018)
- [Decentralize and Randomize: Faster Algorithm for Wasserstein Barycenters](https://mlsanity.com/paper/1806.03915/) (2018)
- [Discrete Wasserstein Barycenters: Optimal Transport for Discrete Data](https://mlsanity.com/paper/1507.07218/) (2015)
- [Estimating Barycenters of Measures in High Dimensions](https://mlsanity.com/paper/2007.07105/) (2020)
- [Parallel Streaming Wasserstein Barycenters](https://mlsanity.com/paper/1705.07443/) (2017)
- [Dimensionality Reduction for Wasserstein Barycenter](https://mlsanity.com/paper/2110.08991/) (2021)
- [Wasserstein Barycenter Model Ensembling](https://mlsanity.com/paper/1902.04999/) (2019)
- [Interior-Point Methods Strike Back: Solving the Wasserstein Barycenter Problem](https://mlsanity.com/paper/1905.12895/) (2019)
- [A Fast Globally Linearly Convergent Algorithm for the Computation of Wasserstein Barycenters](https://mlsanity.com/paper/1809.04249/) (2018)
- [Probabilistic Multilevel Clustering via Composite Transportation Distance](https://mlsanity.com/paper/1810.11911/) (2018)
- [Sampling From the Wasserstein Barycenter](https://mlsanity.com/paper/2105.01706/) (2021)

## Built on by (most cited first, 7 of 7)

- [Wasserstein Iterative Networks for Barycenter Estimation](https://mlsanity.com/paper/2201.12245/) (2022)
- [Learning Continuous Wasserstein Barycenter Space for Generalized All-in-One Image Restoration](https://mlsanity.com/paper/2602.23169/) (2026)
- [Estimating Barycenters of Distributions with Neural Optimal Transport](https://mlsanity.com/paper/2402.03828/) (2024)
- [Energy-Guided Continuous Entropic Barycenter Estimation for General Costs](https://mlsanity.com/paper/2310.01105/) (2023)
- [Robust Barycenter Estimation using Semi-Unbalanced Neural Optimal Transport](https://mlsanity.com/paper/2410.03974/) (2024)
- [BaryIR: Learning Multi-Source Unified Representation in Continuous Barycenter Space for Generalizable All-in-One Image Restoration](https://mlsanity.com/paper/2505.21637/) (2025)
- [Sobolev Gradient Ascent for Optimal Transport: Barycenter Optimization and Convergence Analysis](https://mlsanity.com/paper/2505.13660/) (2025)

## Abstract (arXiv)

Wasserstein barycenter, built on the theory of optimal transport, provides a powerful framework to aggregate probability distributions, and it has increasingly attracted great attention within the machine learning community. However, it suffers from severe computational burden, especially for high dimensional and continuous settings. To this end, we develop a novel continuous approximation method for the Wasserstein barycenters problem given sample access to the input distributions. The basic idea is to introduce a variational distribution as the approximation of the true continuous barycenter, so as to frame the barycenters computation problem as an optimization problem, where parameters of the variational distribution adjust the proxy distribution to be similar to the barycenter. Leveraging the variational distribution, we construct a tractable dual formulation for the regularized Wasserstein barycenter problem with c-cyclical monotonicity, which can be efficiently solved by stochastic optimization. We provide theoretical analysis on convergence and demonstrate the practical effectiveness of our method on real applications of subset posterior aggregation and synthetic data.