# Continuous Regularized Wasserstein Barycenters

arXiv: https://arxiv.org/abs/2008.12534 · HTML: https://mlsanity.com/paper/2008.12534/

## Key facts

- Cited by 32 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Aug 2020.

## Built on by (most cited first, 30 of 32)

- [Score-based Generative Neural Networks for Large-Scale Optimal Transport](https://mlsanity.com/paper/2110.03237/) (2021)
- [Continuous Wasserstein-2 Barycenter Estimation without Minimax Optimization](https://mlsanity.com/paper/2102.01752/) (2021)
- [Averaging on the Bures-Wasserstein manifold: dimension-free convergence of gradient descent](https://mlsanity.com/paper/2106.08502/) (2021)
- [On amortizing convex conjugates for optimal transport](https://mlsanity.com/paper/2210.12153/) (2022)
- [Meta Optimal Transport](https://mlsanity.com/paper/2206.05262/) (2022)
- [Curriculum Reinforcement Learning using Optimal Transport via Gradual Domain Adaptation](https://mlsanity.com/paper/2210.10195/) (2022)
- [Scalable Optimal Transport Methods in Machine Learning: A Contemporary Survey](https://mlsanity.com/paper/2305.05080/) (2023)
- [Wasserstein Iterative Networks for Barycenter Estimation](https://mlsanity.com/paper/2201.12245/) (2022)
- [On Transportation of Mini-batches: A Hierarchical Approach](https://mlsanity.com/paper/2102.05912/) (2021)
- [Tree-Based Diffusion Schr"odinger Bridge with Applications to Wasserstein Barycenters](https://mlsanity.com/paper/2305.16557/) (2023)
- [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)
- [Quantitative Uniform Stability of the Iterative Proportional Fitting Procedure](https://mlsanity.com/paper/2108.08129/) (2021)
- [Doubly Regularized Entropic Wasserstein Barycenters](https://mlsanity.com/paper/2303.11844/) (2023)
- [Robust Barycenter Estimation using Semi-Unbalanced Neural Optimal Transport](https://mlsanity.com/paper/2410.03974/) (2024)
- [Variational Wasserstein Barycenters with c-Cyclical Monotonicity](https://mlsanity.com/paper/2110.11707/) (2021)
- [Learning to Generate Wasserstein Barycenters](https://mlsanity.com/paper/2102.12178/) (2021)
- [An Accelerated Stochastic Variance-Reduced Algorithm for Entropic Wasserstein Barycenters](https://mlsanity.com/paper/2203.00813/) (2022)
- [On the Benefit of Optimal Transport for Curriculum Reinforcement Learning](https://mlsanity.com/paper/2309.14091/) (2023)
- [A novel notion of barycenter for probability distributions based on optimal weak mass transport](https://mlsanity.com/paper/2102.13380/) (2021)
- [Fast Topological Clustering with Wasserstein Distance](https://mlsanity.com/paper/2112.00101/) (2021)
- [Iterative Alignment Flows](https://mlsanity.com/paper/2104.07232/) (2021)
- [Analysis of Kernel Mirror Prox for Measure Optimization](https://mlsanity.com/paper/2403.00147/) (2024)
- [Wasserstein Distributionally Robust Optimization with Wasserstein Barycenters](https://mlsanity.com/paper/2203.12136/) (2022)
- [BaryIR: Learning Multi-Source Unified Representation in Continuous Barycenter Space for Generalizable All-in-One Image Restoration](https://mlsanity.com/paper/2505.21637/) (2025)
- [Optimal Transport Barycenter via Nonconvex-Concave Minimax Optimization](https://mlsanity.com/paper/2501.14635/) (2025)
- [Sobolev Gradient Ascent for Optimal Transport: Barycenter Optimization and Convergence Analysis](https://mlsanity.com/paper/2505.13660/) (2025)
- [Schr"odinger Bridge Matching for Tree-Structured Costs and Entropic Wasserstein Barycentres](https://mlsanity.com/paper/2506.17197/) (2025)
- [A Spectral-Grassmann Wasserstein metric for operator representations of dynamical systems](https://mlsanity.com/paper/2509.24920/) (2025)

## Abstract (arXiv)

Wasserstein barycenters provide a geometrically meaningful way to aggregate probability distributions, built on the theory of optimal transport. They are difficult to compute in practice, however, leading previous work to restrict their supports to finite sets of points. Leveraging a new dual formulation for the regularized Wasserstein barycenter problem, we introduce a stochastic algorithm that constructs a continuous approximation of the barycenter. We establish strong duality and use the corresponding primal-dual relationship to parametrize the barycenter implicitly using the dual potentials of regularized transport problems. The resulting problem can be solved with stochastic gradient descent, which yields an efficient online algorithm to approximate the barycenter of continuous distributions given sample access. We demonstrate the effectiveness of our approach and compare against previous work on synthetic examples and real-world applications.