Tree-Based Diffusion Schrödinger Bridge with Applications to Wasserstein Barycenters

Abstract

Multi-marginal Optimal Transport (mOT), a generalization of OT, aims at minimizing the integral of a cost function with respect to a distribution with some prescribed marginals. In this paper, we consider an entropic version of mOT with a tree-structured quadratic cost, i.e., a function that can be written as a sum of pairwise cost functions between the nodes of a tree. To address this problem, we develop Tree-based Diffusion Schr\"odinger Bridge (TreeDSB), an extension of the Diffusion Schr\"odinger Bridge (DSB) algorithm. TreeDSB corresponds to a dynamic and continuous state-space counterpart of the multimarginal Sinkhorn 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. We demonstrate that our methodology can be applied in high-dimensional settings such as image interpolation and Bayesian fusion.

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Cite

Text

Noble et al. "Tree-Based Diffusion Schrödinger Bridge with Applications to Wasserstein Barycenters." Neural Information Processing Systems, 2023.

Markdown

[Noble et al. "Tree-Based Diffusion Schrödinger Bridge with Applications to Wasserstein Barycenters." Neural Information Processing Systems, 2023.](https://mlanthology.org/neurips/2023/noble2023neurips-treebased/)

BibTeX

@inproceedings{noble2023neurips-treebased,
  title     = {{Tree-Based Diffusion Schrödinger Bridge with Applications to Wasserstein Barycenters}},
  author    = {Noble, Maxence and De Bortoli, Valentin and Doucet, Arnaud and Durmus, Alain},
  booktitle = {Neural Information Processing Systems},
  year      = {2023},
  url       = {https://mlanthology.org/neurips/2023/noble2023neurips-treebased/}
}