# Estimating Barycenters of Distributions with Neural Optimal Transport

arXiv: https://arxiv.org/abs/2402.03828 · HTML: https://mlsanity.com/paper/2402.03828/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 34 of its 59 references that are in the map.
- First posted on arXiv 6 Feb 2024.

## Builds on (most cited first, 30 of 34)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Model Fusion via Optimal Transport](https://mlsanity.com/paper/1910.05653/) (2019)
- [Fast Computation of Wasserstein Barycenters](https://mlsanity.com/paper/1310.4375/) (2013)
- [Optimal transport mapping via input convex neural networks](https://mlsanity.com/paper/1908.10962/) (2019)
- [Neural Optimal Transport](https://mlsanity.com/paper/2201.12220/) (2022)
- [Large-Scale Optimal Transport and Mapping Estimation](https://mlsanity.com/paper/1711.02283/) (2017)
- [Wasserstein-2 Generative Networks](https://mlsanity.com/paper/1909.13082/) (2019)
- [Do Neural Optimal Transport Solvers Work? A Continuous Wasserstein-2 Benchmark](https://mlsanity.com/paper/2106.01954/) (2021)
- [Smooth and Sparse Optimal Transport](https://mlsanity.com/paper/1710.06276/) (2017)
- [Generative Modeling with Optimal Transport Maps](https://mlsanity.com/paper/2110.02999/) (2021)
- [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)
- [Scalable Computations of Wasserstein Barycenter via Input Convex Neural Networks](https://mlsanity.com/paper/2007.04462/) (2020)
- [Entropic Neural Optimal Transport via Diffusion Processes](https://mlsanity.com/paper/2211.01156/) (2022)
- [Scalable Bayes via Barycenter in Wasserstein Space](https://mlsanity.com/paper/1508.05880/) (2015)
- [Generative Modeling through the Semi-dual Formulation of Unbalanced Optimal Transport](https://mlsanity.com/paper/2305.14777/) (2023)
- [Neural Monge Map estimation and its applications](https://mlsanity.com/paper/2106.03812/) (2021)
- [Neural Optimal Transport with General Cost Functionals](https://mlsanity.com/paper/2205.15403/) (2022)
- [Continuous Regularized Wasserstein Barycenters](https://mlsanity.com/paper/2008.12534/) (2020)
- [Kernel Neural Optimal Transport](https://mlsanity.com/paper/2205.15269/) (2022)
- [Energy-guided Entropic Neural Optimal Transport](https://mlsanity.com/paper/2304.06094/) (2023)
- [Wasserstein Iterative Networks for Barycenter Estimation](https://mlsanity.com/paper/2201.12245/) (2022)
- [Existence, Duality, and Cyclical monotonicity for weak transport costs](https://mlsanity.com/paper/1809.05893/) (2018)
- [Wasserstein Style Transfer](https://mlsanity.com/paper/1905.12828/) (2019)
- [Discrete Wasserstein Barycenters: Optimal Transport for Discrete Data](https://mlsanity.com/paper/1507.07218/) (2015)
- [Tree-Based Diffusion Schr"odinger Bridge with Applications to Wasserstein Barycenters](https://mlsanity.com/paper/2305.16557/) (2023)
- [An Optimal Transport Perspective on Unpaired Image Super-Resolution](https://mlsanity.com/paper/2202.01116/) (2022)
- [Energy-Guided Continuous Entropic Barycenter Estimation for General Costs](https://mlsanity.com/paper/2310.01105/) (2023)
- [(Martingale) Optimal Transport And Anomaly Detection With Neural Networks: A Primal-dual Algorithm](https://mlsanity.com/paper/1904.04546/) (2019)

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

- [Optimal Flow Matching: Learning Straight Trajectories in Just One Step](https://mlsanity.com/paper/2403.13117/) (2024)
- [Learning Continuous Wasserstein Barycenter Space for Generalized All-in-One Image Restoration](https://mlsanity.com/paper/2602.23169/) (2026)
- [Robust Barycenter Estimation using Semi-Unbalanced Neural Optimal Transport](https://mlsanity.com/paper/2410.03974/) (2024)
- [Learning of Population Dynamics: Inverse Optimization Meets JKO Scheme](https://mlsanity.com/paper/2506.01502/) (2025)
- [Improving Neural Optimal Transport via Displacement Interpolation](https://mlsanity.com/paper/2410.03783/) (2024)
- [Low-dimensional approximations of the conditional law of Volterra processes: a non-positive curvature approach](https://mlsanity.com/paper/2405.20094/) (2024)
- [A Statistical Learning Perspective on Semi-dual Adversarial Neural Optimal Transport Solvers](https://mlsanity.com/paper/2502.01310/) (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)
- [Wasserstein Gradient Flows for Scalable and Regularized Barycenter Computation](https://mlsanity.com/paper/2510.04602/) (2025)
- [Binding Multiple Modalities via Multimodal Wasserstein Barycenter](https://mlsanity.com/paper/2609.33800/) (2026)
- [Liouville PDE-based sliced-Wasserstein flow](https://mlsanity.com/paper/2505.17204/) (2025)
- [Simultaneous Neural Optimal Transport](https://mlsanity.com/paper/2609.37424/) (2026)
- [Towards Universal Wasserstein Barycenters through Flow Matching](https://mlsanity.com/paper/2609.38547/) (2026)

## Abstract (arXiv)

Given a collection of probability measures, a practitioner sometimes needs to find an "average" distribution which adequately aggregates reference distributions. A theoretically appealing notion of such an average is the Wasserstein barycenter, which is the primal focus of our work. By building upon the dual formulation of Optimal Transport (OT), we propose a new scalable approach for solving the Wasserstein barycenter problem. Our methodology is based on the recent Neural OT solver: it has bi-level adversarial learning objective and works for general cost functions. These are key advantages of our method since the typical adversarial algorithms leveraging barycenter tasks utilize tri-level optimization and focus mostly on quadratic cost. We also establish theoretical error bounds for our proposed approach and showcase its applicability and effectiveness in illustrative scenarios and image data setups. Our source code is available at https://github.com/justkolesov/NOTBarycenters.