Schrödinger Bridge Matching for Tree-Structured Costs and Entropic Wasserstein Barycentres

• URL: http://arxiv.org/abs/2506.17197v1
• Date: Fri, 20 Jun 2025 17:47:47 GMT
• Title: Schrödinger Bridge Matching for Tree-Structured Costs and Entropic Wasserstein Barycentres
• Authors: Samuel Howard, Peter Potaptchik, George Deligiannidis,
• Abstract summary: Flow-based generative modelling has provided scalable methods for computing the Schr"odinger Bridge (SB) between distributions.<n>The IMF procedure solves the SB problem via sequential bridge-matching steps.<n>We extend the IMF procedure to solve for the tree-structured SB problem.
• Score: 5.397565689903148
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Recent advances in flow-based generative modelling have provided scalable methods for computing the Schr\"odinger Bridge (SB) between distributions, a dynamic form of entropy-regularised Optimal Transport (OT) for the quadratic cost. The successful Iterative Markovian Fitting (IMF) procedure solves the SB problem via sequential bridge-matching steps, presenting an elegant and practical approach with many favourable properties over the more traditional Iterative Proportional Fitting (IPF) procedure. Beyond the standard setting, optimal transport can be generalised to the multi-marginal case in which the objective is to minimise a cost defined over several marginal distributions. Of particular importance are costs defined over a tree structure, from which Wasserstein barycentres can be recovered as a special case. In this work, we extend the IMF procedure to solve for the tree-structured SB problem. Our resulting algorithm inherits the many advantages of IMF over IPF approaches in the tree-based setting. In the specific case of Wasserstein barycentres, our approach can be viewed as extending fixed-point approaches for barycentre computation to the case of flow-based entropic OT solvers.

Related papers

• Computing Optimal Transport Maps and Wasserstein Barycenters Using Conditional Normalizing Flows [1.8109081066789847]
  We present a novel method for efficiently computing optimal transport maps and Wasserstein barycenters in high-dimensional spaces.<n>Our approach uses conditional normalizing flows to approximate the input distributions as invertible pushforward transformations from a common latent space.<n>We show how this approach can be extended to compute Wasserstein barycenters by solving a conditional variance minimization problem.
  arXiv  Detail & Related papers  (2025-05-28T13:46:07Z)
• Diffusion & Adversarial Schrödinger Bridges via Iterative Proportional Markovian Fitting [87.37278888311839]
  Iterative Markovian Fitting (IMF) procedure successfully solves the Schr"odinger Bridge (SB) problem.<n>We show a close connection between IMF and the Iterative Proportional Fitting (IPF) procedure.<n>We refer to this combined approach as the Iterative Proportional Markovian Fitting (IPMF) procedure.
  arXiv  Detail & Related papers  (2024-10-03T15:43:17Z)
• Adversarial Schrödinger Bridge Matching [66.39774923893103]
  Iterative Markovian Fitting (IMF) procedure alternates between Markovian and reciprocal projections of continuous-time processes. We propose a novel Discrete-time IMF (D-IMF) procedure in which learning of processes is replaced by learning just a few transition probabilities in discrete time. We show that our D-IMF procedure can provide the same quality of unpaired domain translation as the IMF, using only several generation steps instead of hundreds.
  arXiv  Detail & Related papers  (2024-05-23T11:29:33Z)
• 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)
• Geometry-Aware Normalizing Wasserstein Flows for Optimal Causal Inference [0.0]
  This paper presents a groundbreaking approach to causal inference by integrating continuous normalizing flows with parametric submodels. We leverage optimal transport and Wasserstein gradient flows to develop causal inference methodologies with minimal variance in finite-sample settings. Preliminary experiments showcase our method's superiority, yielding lower mean-squared errors compared to standard flows.
  arXiv  Detail & Related papers  (2023-11-30T18:59:05Z)
• 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)
• Tree-Based Diffusion Schr\"odinger Bridge with Applications to Wasserstein Barycenters [44.75675404104031]
  We develop Tree-based Diffusion Schr"odinger Bridge (TreeDSB), an extension of the Diffusion Schr"odinger Bridge (DSB) algorithm. A notable use case of our methodology is to compute Wasserstein barycenters which can be recast as the solution of a mOT problem on a star-shaped tree.
  arXiv  Detail & Related papers  (2023-05-26T00:50:47Z)
• Markovian Sliced Wasserstein Distances: Beyond Independent Projections [51.80527230603978]
  We introduce a new family of SW distances, named Markovian sliced Wasserstein (MSW) distance, which imposes a first-order Markov structure on projecting directions. We compare distances with previous SW variants in various applications such as flows, color transfer, and deep generative modeling to demonstrate the favorable performance of MSW.
  arXiv  Detail & Related papers  (2023-01-10T01:58:15Z)
• 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)

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.