# Doubly Regularized Entropic Wasserstein Barycenters

arXiv: https://arxiv.org/abs/2303.11844 · HTML: https://mlsanity.com/paper/2303.11844/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 42 of its 94 references that are in the map.
- First posted on arXiv 21 Mar 2023.

## Builds on (most cited first, 30 of 42)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [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)
- [A Mean Field View of the Landscape of Two-Layers Neural Networks](https://mlsanity.com/paper/1804.06561/) (2018)
- [Learning Generative Models with Sinkhorn Divergences](https://mlsanity.com/paper/1706.00292/) (2017)
- [Fast Computation of Wasserstein Barycenters](https://mlsanity.com/paper/1310.4375/) (2013)
- [Statistical Aspects of Wasserstein Distances](https://mlsanity.com/paper/1806.05500/) (2018)
- [On Wasserstein Two Sample Testing and Related Families of Nonparametric Tests](https://mlsanity.com/paper/1509.02237/) (2015)
- [On the relation between optimal transport and Schr"odinger bridges: A stochastic control viewpoint](https://mlsanity.com/paper/1412.4430/) (2014)
- [Faster Wasserstein Distance Estimation with the Sinkhorn Divergence](https://mlsanity.com/paper/2006.08172/) (2020)
- [Mean-Field Langevin Dynamics: Exponential Convergence and Annealing](https://mlsanity.com/paper/2202.01009/) (2022)
- [Convex Analysis of the Mean Field Langevin Dynamics](https://mlsanity.com/paper/2201.10469/) (2022)
- [Continuous Wasserstein-2 Barycenter Estimation without Minimax Optimization](https://mlsanity.com/paper/2102.01752/) (2021)
- [Mean-Field Langevin Dynamics and Energy Landscape of Neural Networks](https://mlsanity.com/paper/1905.07769/) (2019)
- [Scalable Computations of Wasserstein Barycenter via Input Convex Neural Networks](https://mlsanity.com/paper/2007.04462/) (2020)
- [Scalable Bayes via Barycenter in Wasserstein Space](https://mlsanity.com/paper/1508.05880/) (2015)
- [An Optimal Transport approach for the Schr"odinger bridge problem and convergence of Sinkhorn algorithm](https://mlsanity.com/paper/1911.06850/) (2019)
- [Debiased Sinkhorn barycenters](https://mlsanity.com/paper/2006.02575/) (2020)
- [Entropy-Regularized 2-Wasserstein Distance between Gaussian Measures](https://mlsanity.com/paper/2006.03416/) (2020)
- [Continuous Regularized Wasserstein Barycenters](https://mlsanity.com/paper/2008.12534/) (2020)
- [Wasserstein barycenters are NP-hard to compute](https://mlsanity.com/paper/2101.01100/) (2021)
- [Analysis of a Two-Layer Neural Network via Displacement Convexity](https://mlsanity.com/paper/1901.01375/) (2019)
- [Sinkhorn Barycenters with Free Support via Frank-Wolfe Algorithm](https://mlsanity.com/paper/1905.13194/) (2019)
- [Semi-dual Regularized Optimal Transport](https://mlsanity.com/paper/1811.05527/) (2018)
- [Stochastic Wasserstein Barycenters](https://mlsanity.com/paper/1802.05757/) (2018)
- [Wasserstein barycenters can be computed in polynomial time in fixed dimension](https://mlsanity.com/paper/2006.08012/) (2020)
- [Decentralize and Randomize: Faster Algorithm for Wasserstein Barycenters](https://mlsanity.com/paper/1806.03915/) (2018)
- [Particle Dual Averaging: Optimization of Mean Field Neural Networks with Global Convergence Rate Analysis](https://mlsanity.com/paper/2012.15477/) (2020)
- [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)

## Built on by (most cited first, 10 of 10)

- [Tree-Based Diffusion Schr"odinger Bridge with Applications to Wasserstein Barycenters](https://mlsanity.com/paper/2305.16557/) (2023)
- [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)
- [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)
- [Linearized Wasserstein Barycenters: Synthesis, Analysis, Representational Capacity, and Applications](https://mlsanity.com/paper/2410.23602/) (2024)
- [Wasserstein Gradient Flows for Scalable and Regularized Barycenter Computation](https://mlsanity.com/paper/2510.04602/) (2025)
- [Computational Guarantees for Doubly Entropic Wasserstein Barycenters via Damped Sinkhorn Iterations](https://mlsanity.com/paper/2307.13370/) (2023)
- [On Barycenter Computation: Semi-Unbalanced Optimal Transport-based Method on Gaussians](https://mlsanity.com/paper/2410.08117/) (2024)

## Abstract (arXiv)

We study a general formulation of regularized Wasserstein barycenters that enjoys favorable regularity, approximation, stability and (grid-free) optimization properties. This barycenter is defined as the unique probability measure that minimizes the sum of entropic optimal transport (EOT) costs with respect to a family of given probability measures, plus an entropy term. We denote it (λ,τ)-barycenter, where λ is the inner regularization strength and τ the outer one. This formulation recovers several previously proposed EOT barycenters for various choices of λ,τ ≥ 0 and generalizes them. First, in spite of -- and in fact owing to -- being doubly regularized, we show that our formulation is debiased for τ=λ/2: the suboptimality in the (unregularized) Wasserstein barycenter objective is, for smooth densities, of the order of the strength λ2 of entropic regularization, instead of maxλ,τ in general. We discuss this phenomenon for isotropic Gaussians where all (λ,τ)-barycenters have closed form. Second, we show that for λ,τ>0, this barycenter has a smooth density and is strongly stable under perturbation of the marginals. In particular, it can be estimated efficiently: given n samples from each of the probability measures, it converges in relative entropy to the population barycenter at a rate n-1/2. And finally, this formulation lends itself naturally to a grid-free optimization algorithm: we propose a simple noisy particle gradient descent which, in the mean-field limit, converges globally at an exponential rate to the barycenter.