# A Unified and Fast-Sampling Diffusion Bridge Framework via Stochastic Optimal Control

arXiv: https://arxiv.org/abs/2505.21528 · HTML: https://mlsanity.com/paper/2505.21528/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 65 of its 86 references that are in the map.
- First posted on arXiv 23 May 2025.

## Builds on (most cited first, 30 of 65)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [The Unreasonable Effectiveness of Deep Features as a Perceptual Metric](https://mlsanity.com/paper/1801.03924/) (2018)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [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)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Progressive Growing of GANs for Improved Quality, Stability, and Variation](https://mlsanity.com/paper/1710.10196/) (2017)
- [Diffusion Policy: Visuomotor Policy Learning via Action Diffusion](https://mlsanity.com/paper/2303.04137/) (2023)
- [Video Diffusion Models](https://mlsanity.com/paper/2204.03458/) (2022)
- [DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps](https://mlsanity.com/paper/2206.00927/) (2022)
- [Consistency Models](https://mlsanity.com/paper/2303.01469/) (2023)
- [RePaint: Inpainting using Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2201.09865/) (2022)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2201.11793/) (2022)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [DPM-Solver++: Fast Solver for Guided Sampling of Diffusion Probabilistic Models](https://mlsanity.com/paper/2211.01095/) (2022)
- [Adversarial Diffusion Distillation](https://mlsanity.com/paper/2311.17042/) (2023)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [Diffusion Policies as an Expressive Policy Class for Offline Reinforcement Learning](https://mlsanity.com/paper/2208.06193/) (2022)
- [Lumiere: A Space-Time Diffusion Model for Video Generation](https://mlsanity.com/paper/2401.12945/) (2024)
- [UniPC: A Unified Predictor-Corrector Framework for Fast Sampling of Diffusion Models](https://mlsanity.com/paper/2302.04867/) (2023)
- [Deep Joint Rain Detection and Removal from a Single Image](https://mlsanity.com/paper/1609.07769/) (2016)
- [VideoFusion: Decomposed Diffusion Models for High-Quality Video Generation](https://mlsanity.com/paper/2303.08320/) (2023)
- [DiffIR: Efficient Diffusion Model for Image Restoration](https://mlsanity.com/paper/2303.09472/) (2023)
- [Goal-Conditioned Imitation Learning using Score-based Diffusion Policies](https://mlsanity.com/paper/2304.02532/) (2023)
- [BBDM: Image-to-image Translation with Brownian Bridge Diffusion Models](https://mlsanity.com/paper/2205.07680/) (2022)
- [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)

## Built on by (most cited first, 2 of 2)

- [Diffusion Bridge or Flow Matching? A Unifying Framework and Comparative Analysis](https://mlsanity.com/paper/2509.24531/) (2025)
- [FASTER: Fast Adjoint Stochastic Transport for Endpoint Refinement in Reward-Guided Image Editing](https://mlsanity.com/paper/2610.04538/) (2026)

## Abstract (arXiv)

Recent advances in diffusion bridge models leverage Doob's h-transform to establish fixed endpoints between distributions, demonstrating promising results in image translation and restoration tasks. However, these approaches often produce blurred or excessively smoothed image details and lack a comprehensive theoretical foundation to explain these shortcomings. To address these limitations, we propose UniDB, a unified and fast-sampling framework for diffusion bridges based on Stochastic Optimal Control (SOC). We reformulate the problem through an SOC-based optimization, proving that existing diffusion bridges employing Doob's h-transform constitute a special case, emerging when the terminal penalty coefficient in the SOC cost function tends to infinity. By incorporating a tunable terminal penalty coefficient, UniDB achieves an optimal balance between control costs and terminal penalties, substantially improving detail preservation and output quality. To avoid computationally expensive costs of iterative Euler sampling methods in UniDB, we design a training-free accelerated algorithm by deriving exact closed-form solutions for UniDB's reverse-time SDE. It is further complemented by replacing conventional noise prediction with a more stable data prediction model, along with an SDE-Corrector mechanism that maintains perceptual quality for low-step regimes, effectively reducing error accumulation. Extensive experiments across diverse image restoration tasks validate the superiority and adaptability of the proposed framework, bridging the gap between theoretical generality and practical efficiency. Our code is available online https://github.com/2769433owo/UniDB-plusplus.