# Tree-Based Diffusion Schr"odinger Bridge with Applications to Wasserstein Barycenters

arXiv: https://arxiv.org/abs/2305.16557 · HTML: https://mlsanity.com/paper/2305.16557/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 35 of its 82 references that are in the map.
- First posted on arXiv 26 May 2023.

## Builds on (most cited first, 30 of 35)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [End-to-End Object Detection with Transformers](https://mlsanity.com/paper/2005.12872/) (2020)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [A survey of the Schr"odinger problem and some of its connections with optimal transport](https://mlsanity.com/paper/1308.0215/) (2013)
- [Likelihood Training of Schr"odinger Bridge using Forward-Backward SDEs Theory](https://mlsanity.com/paper/2110.11291/) (2021)
- [Fast Computation of Wasserstein Barycenters](https://mlsanity.com/paper/1310.4375/) (2013)
- [Optimal transport mapping via input convex neural networks](https://mlsanity.com/paper/1908.10962/) (2019)
- [Diffusion Schr"odinger Bridge Matching](https://mlsanity.com/paper/2303.16852/) (2023)
- [Solving Schr"odinger Bridges via Maximum Likelihood](https://mlsanity.com/paper/2106.02081/) (2021)
- [Proximal Optimal Transport Modeling of Population Dynamics](https://mlsanity.com/paper/2106.06345/) (2021)
- [Diffusion Bridge Mixture Transports, Schr"odinger Bridge Problems and Generative Modeling](https://mlsanity.com/paper/2304.00917/) (2023)
- [Deep Shells: Unsupervised Shape Correspondence with Optimal Transport](https://mlsanity.com/paper/2010.15261/) (2020)
- [Continuous Wasserstein-2 Barycenter Estimation without Minimax Optimization](https://mlsanity.com/paper/2102.01752/) (2021)
- [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)
- [Deep Momentum Multi-Marginal Schr"odinger Bridge](https://mlsanity.com/paper/2303.01751/) (2023)
- [Optimal Transport for Diffeomorphic Registration](https://mlsanity.com/paper/1706.05218/) (2017)
- [An Optimal Transport approach for the Schr"odinger bridge problem and convergence of Sinkhorn algorithm](https://mlsanity.com/paper/1911.06850/) (2019)
- [Wasserstein Dictionary Learning: Optimal Transport-based unsupervised non-linear dictionary learning](https://mlsanity.com/paper/1708.01955/) (2017)
- [Continuous Regularized Wasserstein Barycenters](https://mlsanity.com/paper/2008.12534/) (2020)
- [Stochastic Wasserstein Barycenters](https://mlsanity.com/paper/1802.05757/) (2018)
- [Wasserstein Iterative Networks for Barycenter Estimation](https://mlsanity.com/paper/2201.12245/) (2022)
- [Estimating Barycenters of Measures in High Dimensions](https://mlsanity.com/paper/2007.07105/) (2020)
- [Measure-valued spline curves: an optimal transport viewpoint](https://mlsanity.com/paper/1801.03186/) (2018)

## Built on by (most cited first, 16 of 16)

- [Multimarginal generative modeling with stochastic interpolants](https://mlsanity.com/paper/2310.03695/) (2023)
- [Estimating Barycenters of Distributions with Neural Optimal Transport](https://mlsanity.com/paper/2402.03828/) (2024)
- [Energy-Guided Continuous Entropic Barycenter Estimation for General Costs](https://mlsanity.com/paper/2310.01105/) (2023)
- [Distribution Prototype Diffusion Learning for Open-set Supervised Anomaly Detection](https://mlsanity.com/paper/2502.20981/) (2025)
- [Robust Barycenter Estimation using Semi-Unbalanced Neural Optimal Transport](https://mlsanity.com/paper/2410.03974/) (2024)
- [DPBridge: Latent Diffusion Bridge for Dense Prediction](https://mlsanity.com/paper/2412.20506/) (2024)
- [Branched Schr"odinger Bridge Matching](https://mlsanity.com/paper/2506.09007/) (2025)
- [Stochastic Learning of Computational Resource Usage as Graph Structured Multimarginal Schr"odinger Bridge](https://mlsanity.com/paper/2405.12463/) (2024)
- [Sobolev Gradient Ascent for Optimal Transport: Barycenter Optimization and Convergence Analysis](https://mlsanity.com/paper/2505.13660/) (2025)
- [Neural Estimation for Scaling Entropic Multimarginal Optimal Transport](https://mlsanity.com/paper/2506.00573/) (2025)
- [Schr"odinger Bridge Matching for Tree-Structured Costs and Entropic Wasserstein Barycentres](https://mlsanity.com/paper/2506.17197/) (2025)
- [Beyond Continuity: Simulation-free Reconstruction of Discrete Branching Dynamics from Single-cell Snapshots](https://mlsanity.com/paper/2605.00545/) (2026)
- [Optimal Multimarginal Schr"odinger Bridge: Minimum Spanning Tree over Measure-valued Vertices](https://mlsanity.com/paper/2509.10626/) (2025)
- [Twisted Schr"odinger Bridge Matching](https://mlsanity.com/paper/2607.16987/) (2026)
- [Generative Profiling for Soft Real-Time Systems and its Applications to Resource Allocation](https://mlsanity.com/paper/2604.01441/) (2026)
- [Unsupervised Domain Adaptation for Enhanced Radiometer Image Precipitation Estimation using Conditional Flow Matching](https://mlsanity.com/paper/2610.01890/) (2026)

## Abstract (arXiv)

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.