# BM2: Coupled Schr"odinger Bridge Matching

arXiv: https://arxiv.org/abs/2409.09376 · HTML: https://mlsanity.com/paper/2409.09376/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 29 references that are in the map.
- First posted on arXiv 14 Sep 2024.

## Builds on (most cited first, 16 of 16)

- [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)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [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)
- [I2SB: Image-to-Image Schr"odinger Bridge](https://mlsanity.com/paper/2302.05872/) (2023)
- [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)
- [Let us Build Bridges: Understanding and Extending Diffusion Generative Models](https://mlsanity.com/paper/2208.14699/) (2022)
- [Diffusion Bridge Mixture Transports, Schr"odinger Bridge Problems and Generative Modeling](https://mlsanity.com/paper/2304.00917/) (2023)
- [Light and Optimal Schr"odinger Bridge Matching](https://mlsanity.com/paper/2402.03207/) (2024)
- [Aligned Diffusion Schr"odinger Bridges](https://mlsanity.com/paper/2302.11419/) (2023)
- [Entropy-Regularized 2-Wasserstein Distance between Gaussian Measures](https://mlsanity.com/paper/2006.03416/) (2020)
- [Building the Bridge of Schr"odinger: A Continuous Entropic Optimal Transport Benchmark](https://mlsanity.com/paper/2306.10161/) (2023)
- [Light Schr"odinger Bridge](https://mlsanity.com/paper/2310.01174/) (2023)

## Built on by (most cited first, 4 of 4)

- [Branched Schr"odinger Bridge Matching](https://mlsanity.com/paper/2506.09007/) (2025)
- [Incorporating Pre-trained Diffusion Models in Solving the Schr"odinger Bridge Problem](https://mlsanity.com/paper/2508.18095/) (2025)
- [Multi-Marginal Schr"odinger Bridge Matching](https://mlsanity.com/paper/2510.16587/) (2025)
- [Statistical Analysis of the Sinkhorn Iterations for Two-Sample Schr"odinger Bridge Estimation](https://mlsanity.com/paper/2510.22560/) (2025)

## Abstract (arXiv)

A Schr"odinger bridge establishes a dynamic transport map between two target distributions via a reference process, simultaneously solving an associated entropic optimal transport problem. We consider the setting where samples from the target distributions are available, and the reference diffusion process admits tractable dynamics. We thus introduce Coupled Bridge Matching (BM2), a simple non-iterative approach for learning Schr"odinger bridges with neural networks. A preliminary theoretical analysis of the convergence properties of BM2 is carried out, supported by numerical experiments that demonstrate the effectiveness of our proposal.