Recent advances in flow-based generative modelling have provided scalable methods for computing the Schrödinger 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.
This paper presents the TreeDSBM algorithm, which extends the Iterative Markovian Fitting (IMF) procedure to solve the Schrödinger Bridge (SB) problem for tree-structured costs. The work builds on recent advancements in flow-based generative modeling to provide an efficient framework for computing entropic Wasserstein barycentres. The TreeDSBM algorithm is positioned as a robust alternative to traditional methods like the Iterative Proportional Fitting (IPF) by leveraging flow-based approaches to improve empirical performance when computing barycentres. Key theoretical insights and empirical validations demonstrate significant advantages of TreeDSBM in capturing the complexities of barycentres, particularly its scalability and enhanced performance over existing methods in continuous settings. The authors also address potential limitations and future work directions, especially in integrating advanced techniques from flow matching and improving architectures for various data modalities.
This paper employs the following methods:
- Iterative Markovian Fitting (IMF)
- Iterative Proportional Fitting (IPF)
The following datasets were used in this research:
- Sinkhorn divergence
- BW 2 2 −UVP
- L 2 −UVP
- TreeDSBM shows significant improvement in empirical performance over TreeDSB
- Competitive performance against other continuous Wasserstein-2 barycentre algorithms
- Number of GPUs: None specified
- GPU Type: None specified
- Compute Requirements: None specified