Computing Wasserstein Barycenters through Gradient Flows

• URL: http://arxiv.org/abs/2510.04602v1
• Date: Mon, 06 Oct 2025 09:07:12 GMT
• Title: Computing Wasserstein Barycenters through Gradient Flows
• Authors: Eduardo Fernandes Montesuma, Yassir Bendou, Mike Gartrell,
• Abstract summary: We recast the original barycenter problem as a gradient flow in the Wasserstein space.<n>First, we achieve scalability by sampling mini-batches from the input measures.<n>Second, we incorporate functionals over probability measures, which regularize the barycenter problem through internal, potential, and interaction energies.
• Score: 10.299618586418761
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Wasserstein barycenters provide a powerful tool for aggregating probability measures, while leveraging the geometry of their ambient space. Existing discrete methods suffer from poor scalability, as they require access to the complete set of samples from input measures. We address this issue by recasting the original barycenter problem as a gradient flow in the Wasserstein space. Our approach offers two advantages. First, we achieve scalability by sampling mini-batches from the input measures. Second, we incorporate functionals over probability measures, which regularize the barycenter problem through internal, potential, and interaction energies. We present two algorithms for empirical and Gaussian mixture measures, providing convergence guarantees under the Polyak-{\L}ojasiewicz inequality. Experimental validation on toy datasets and domain adaptation benchmarks show that our methods outperform previous discrete and neural net-based methods for computing Wasserstein barycenters.

Related papers

• A Particle-Flow Algorithm for Free-Support Wasserstein Barycenters [0.31727619150610836]
  We develop a free-support algorithm for computing Wasserstein barycenters that avoids entropic regularization.<n>In our approach, barycenter atoms evolve as particles advected by averaged optimal-transport displacements.<n>This yields a geometry-aware particle-flow update that preserves sharp features of the Wasserstein barycenter while remaining computationally tractable.
  arXiv  Detail & Related papers  (2025-09-14T21:05:04Z)
• On Barycenter Computation: Semi-Unbalanced Optimal Transport-based Method on Gaussians [24.473522267391072]
  We develop algorithms on Bures-Wasserstein manifold, named the Exact Geodesic Gradient Descent and Hybrid Gradient Descent algorithms. We establish theoretical convergence guarantees for both methods and demonstrate that the Exact Geodesic Gradient Descent algorithm attains a dimension-free convergence rate.
  arXiv  Detail & Related papers  (2024-10-10T17:01:57Z)
• Robust Barycenter Estimation using Semi-Unbalanced Neural Optimal Transport [84.51977664336056]
  We propose a novel scalable approach for estimating the robust continuous barycenter.<n>Our method is framed as a min-max optimization problem and is adaptable to general cost functions.
  arXiv  Detail & Related papers  (2024-10-04T23:27:33Z)
• Estimating Barycenters of Distributions with Neural Optimal Transport [93.28746685008093]
  We propose a new scalable approach for solving the Wasserstein barycenter problem. Our methodology is based on the recent Neural OT solver. We also establish theoretical error bounds for our proposed approach.
  arXiv  Detail & Related papers  (2024-02-06T09:17:07Z)
• Energy-Guided Continuous Entropic Barycenter Estimation for General Costs [95.33926437521046]
  We propose a novel algorithm for approximating the continuous Entropic OT (EOT) barycenter for arbitrary OT cost functions.<n>Our approach is built upon the dual reformulation of the EOT problem based on weak OT.
  arXiv  Detail & Related papers  (2023-10-02T11:24:36Z)
• Wasserstein Iterative Networks for Barycenter Estimation [80.23810439485078]
  We present an algorithm to approximate the Wasserstein-2 barycenters of continuous measures via a generative model. Based on the celebrity faces dataset, we construct Ave, celeba! dataset which can be used for quantitative evaluation of barycenter algorithms.
  arXiv  Detail & Related papers  (2022-01-28T16:59:47Z)
• Continuous Wasserstein-2 Barycenter Estimation without Minimax Optimization [94.18714844247766]
  Wasserstein barycenters provide a geometric notion of the weighted average of probability measures based on optimal transport. We present a scalable algorithm to compute Wasserstein-2 barycenters given sample access to the input measures.
  arXiv  Detail & Related papers  (2021-02-02T21:01:13Z)
• Continuous Regularized Wasserstein Barycenters [51.620781112674024]
  We introduce a new dual formulation for the regularized Wasserstein barycenter problem. We establish strong duality and use the corresponding primal-dual relationship to parametrize the barycenter implicitly using the dual potentials of regularized transport problems.
  arXiv  Detail & Related papers  (2020-08-28T08:28:06Z)
• Scalable Computations of Wasserstein Barycenter via Input Convex Neural Networks [15.171726731041055]
  Wasserstein Barycenter is a principled approach to represent the weighted mean of a given set of probability distributions. We present a novel scalable algorithm to approximate the Wasserstein Barycenters aiming at high-dimensional applications in machine learning.
  arXiv  Detail & Related papers  (2020-07-08T22:41:18Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.