# Reflected Schr"odinger Bridge Matching

arXiv: https://arxiv.org/abs/2607.03626 · HTML: https://mlsanity.com/paper/2607.03626/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 33 of its 49 references that are in the map.
- First posted on arXiv 3 Jul 2026.

## Placed in

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

## Builds on (most cited first, 30 of 33)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [The Unreasonable Effectiveness of Deep Features as a Perceptual Metric](https://mlsanity.com/paper/1801.03924/) (2018)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [FiLM: Visual Reasoning with a General Conditioning Layer](https://mlsanity.com/paper/1709.07871/) (2017)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [StarGAN v2: Diverse Image Synthesis for Multiple Domains](https://mlsanity.com/paper/1912.01865/) (2019)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [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)
- [Rectified Flow: A Marginal Preserving Approach to Optimal Transport](https://mlsanity.com/paper/2209.14577/) (2022)
- [Likelihood Training of Schr"odinger Bridge using Forward-Backward SDEs Theory](https://mlsanity.com/paper/2110.11291/) (2021)
- [Diffusion Schr"odinger Bridge Matching](https://mlsanity.com/paper/2303.16852/) (2023)
- [Constrained Synthesis with Projected Diffusion Models](https://mlsanity.com/paper/2402.03559/) (2024)
- [Reflected Diffusion Models](https://mlsanity.com/paper/2304.04740/) (2023)
- [Diffusion Bridge Mixture Transports, Schr"odinger Bridge Problems and Generative Modeling](https://mlsanity.com/paper/2304.00917/) (2023)
- [Diffusion Models for Constrained Domains](https://mlsanity.com/paper/2304.05364/) (2023)
- [Simulating Diffusion Bridges with Score Matching](https://mlsanity.com/paper/2111.07243/) (2021)
- [Schr"odinger Bridge Flow for Unpaired Data Translation](https://mlsanity.com/paper/2409.09347/) (2024)
- [Mirror Diffusion Models for Constrained and Watermarked Generation](https://mlsanity.com/paper/2310.01236/) (2023)
- [Metropolis Sampling for Constrained Diffusion Models](https://mlsanity.com/paper/2307.05439/) (2023)
- [Reflected Schr"odinger Bridge: Density Control with Path Constraints](https://mlsanity.com/paper/2003.13895/) (2020)

## Abstract (arXiv)

Recent advances in generative modeling have enabled the efficient computation of Schr"odinger bridges (SB) in high-dimensional settings by leveraging partially simulation-free training methods inspired by flow matching. However, these have not covered SBs with reflecting dynamics, a useful model choice with built-in guarantees that generated samples stay in the data domain. Existing alternatives for reflected SBs instead rely on more complex training based on forward--backward SDE theory, requiring expensive higher-order derivatives and sampling entire paths during training. In this article, we introduce a partially simulation-free framework that allows reflected SBs to be trained similarly to flow matching, using a new sampling method and regression target. We demonstrate our results by coupling pairs of well-known high-dimensional image datasets. Using reflected dynamics incurs negligible additional wall-clock time during both training and inference while maintaining or slightly improving generative performance.