# Schr"odinger Bridge Flow for Unpaired Data Translation

arXiv: https://arxiv.org/abs/2409.09347 · HTML: https://mlsanity.com/paper/2409.09347/

## Key facts

- Cited by 47 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 18 of its 90 references that are in the map.
- First posted on arXiv 14 Sep 2024.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 18 of 18)

- [Training Diffusion Models with Reinforcement Learning](https://mlsanity.com/paper/2305.13301/) (2023)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Aligning Text-to-Image Models using Human Feedback](https://mlsanity.com/paper/2302.12192/) (2023)
- [Rectified Flow: A Marginal Preserving Approach to Optimal Transport](https://mlsanity.com/paper/2209.14577/) (2022)
- [Using Human Feedback to Fine-tune Diffusion Models without Any Reward Model](https://mlsanity.com/paper/2311.13231/) (2023)
- [EMNIST: an extension of MNIST to handwritten letters](https://mlsanity.com/paper/1702.05373/) (2017)
- [Continual Diffusion: Continual Customization of Text-to-Image Diffusion with C-LoRA](https://mlsanity.com/paper/2304.06027/) (2023)
- [SODA: Bottleneck Diffusion Models for Representation Learning](https://mlsanity.com/paper/2311.17901/) (2023)
- [Probabilistic Forecasting with Stochastic Interpolants and F"ollmer Processes](https://mlsanity.com/paper/2403.13724/) (2024)
- [Target Score Matching](https://mlsanity.com/paper/2402.08667/) (2024)
- [A Computational Framework for Solving Wasserstein Lagrangian Flows](https://mlsanity.com/paper/2310.10649/) (2023)
- [Light and Optimal Schr"odinger Bridge Matching](https://mlsanity.com/paper/2402.03207/) (2024)
- [Augmented Bridge Matching](https://mlsanity.com/paper/2311.06978/) (2023)
- [Continual Learning of Diffusion Models with Generative Distillation](https://mlsanity.com/paper/2311.14028/) (2023)
- [Exploring Continual Learning of Diffusion Models](https://mlsanity.com/paper/2303.15342/) (2023)
- [Riemannian Diffusion Schr"odinger Bridge](https://mlsanity.com/paper/2207.03024/) (2022)
- [Variational Schr"odinger Diffusion Models](https://mlsanity.com/paper/2405.04795/) (2024)
- [Neural Shr"odinger Bridge Matching for Pansharpening](https://mlsanity.com/paper/2404.11416/) (2024)

## Built on by (most cited first, 30 of 47)

- [Temporal Difference Flows](https://mlsanity.com/paper/2503.09817/) (2025)
- [On the Relation between Rectified Flows and Optimal Transport](https://mlsanity.com/paper/2505.19712/) (2025)
- [Diffusion & Adversarial Schr"odinger Bridges via Iterative Proportional Markovian Fitting](https://mlsanity.com/paper/2410.02601/) (2024)
- [Hierarchical Refinement: Optimal Transport to Infinity and Beyond](https://mlsanity.com/paper/2503.03025/) (2025)
- [Improving Neural Optimal Transport via Displacement Interpolation](https://mlsanity.com/paper/2410.03783/) (2024)
- [InfoBridge: Mutual Information estimation via Bridge Matching](https://mlsanity.com/paper/2502.01383/) (2025)
- [Rethinking Losses for Diffusion Bridge Samplers](https://mlsanity.com/paper/2506.10982/) (2025)
- [LADB: Latent Aligned Diffusion Bridges for Semi-Supervised Domain Translation](https://mlsanity.com/paper/2509.08628/) (2025)
- [Field Matching: an Electrostatic Paradigm to Generate and Transfer Data](https://mlsanity.com/paper/2502.02367/) (2025)
- [Overcoming Spurious Solutions in Semi-Dual Neural Optimal Transport: A Smoothing Approach for Learning the Optimal Transport Plan](https://mlsanity.com/paper/2502.04583/) (2025)
- [Gradient Variance Reveals Failure Modes in Flow-Based Generative Models](https://mlsanity.com/paper/2510.18118/) (2025)
- [Scalable Simulation-free Entropic Unbalanced Optimal Transport](https://mlsanity.com/paper/2410.02656/) (2024)
- [Stochastic Flow Matching for Resolving Small-Scale Physics](https://mlsanity.com/paper/2410.19814/) (2024)
- [Paired Wasserstein Autoencoders for Conditional Sampling](https://mlsanity.com/paper/2412.07586/) (2024)
- [Categorical Schr"odinger Bridge Matching](https://mlsanity.com/paper/2502.01416/) (2025)
- [Branched Schr"odinger Bridge Matching](https://mlsanity.com/paper/2506.09007/) (2025)
- [System-Embedded Diffusion Bridge Models](https://mlsanity.com/paper/2506.23726/) (2025)
- [SerpentFlow: Generative Unpaired Domain Alignment via Shared-Structure Decomposition](https://mlsanity.com/paper/2601.01979/) (2026)
- [Learning Visual Feature-Based World Models via Residual Latent Action](https://mlsanity.com/paper/2605.07079/) (2026)
- [Fractional Diffusion Bridge Models](https://mlsanity.com/paper/2511.01795/) (2025)
- [FlowEO: Generative Unsupervised Domain Adaptation for Earth Observation](https://mlsanity.com/paper/2512.05140/) (2025)
- [Generalized Wasserstein Flow Matching: Transport Plans, Everywhere, All at Once](https://mlsanity.com/paper/2605.08424/) (2026)
- [Grasp2Grasp: Vision-Based Dexterous Grasp Translation via Schr"odinger Bridges](https://mlsanity.com/paper/2506.02489/) (2025)
- [Schr"odinger Bridge Matching for Tree-Structured Costs and Entropic Wasserstein Barycentres](https://mlsanity.com/paper/2506.17197/) (2025)
- [Auditing Sybil: Explaining Deep Lung Cancer Risk Prediction Through Generative Interventional Attributions](https://mlsanity.com/paper/2602.02560/) (2026)
- [Beyond Continuity: Simulation-free Reconstruction of Discrete Branching Dynamics from Single-cell Snapshots](https://mlsanity.com/paper/2605.00545/) (2026)
- [SFBD Flow: A Continuous-Optimization Framework for Training Diffusion Models with Noisy Samples](https://mlsanity.com/paper/2506.02371/) (2025)
- [Variational Entropic Optimal Transport](https://mlsanity.com/paper/2602.02241/) (2026)
- [Twisted Schr"odinger Bridge Matching](https://mlsanity.com/paper/2607.16987/) (2026)
- [Entering the Era of Discrete Diffusion Models: A Benchmark for Schr"odinger Bridges and Entropic Optimal Transport](https://mlsanity.com/paper/2509.23348/) (2025)

## Abstract (arXiv)

Mass transport problems arise in many areas of machine learning whereby one wants to compute a map transporting one distribution to another. Generative modeling techniques like Generative Adversarial Networks (GANs) and Denoising Diffusion Models (DDMs) have been successfully adapted to solve such transport problems, resulting in CycleGAN and Bridge Matching respectively. However, these methods do not approximate Optimal Transport (OT) maps, which are known to have desirable properties. Existing techniques approximating OT maps for high-dimensional data-rich problems, such as DDM-based Rectified Flow and Schr"odinger Bridge procedures, require fully training a DDM-type model at each iteration, or use mini-batch techniques which can introduce significant errors. We propose a novel algorithm to compute the Schr"odinger Bridge, a dynamic entropy-regularised version of OT, that eliminates the need to train multiple DDM-like models. This algorithm corresponds to a discretisation of a flow of path measures, which we call the Schr"odinger Bridge Flow, whose only stationary point is the Schr"odinger Bridge. We demonstrate the performance of our algorithm on a variety of unpaired data translation tasks.