# Conditional Simulation Using Diffusion Schr"odinger Bridges

arXiv: https://arxiv.org/abs/2202.13460 · HTML: https://mlsanity.com/paper/2202.13460/

## Key facts

- Cited by 60 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 34 of its 60 references that are in the map.
- First posted on arXiv 27 Feb 2022.

## Placed in

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

## Builds on (most cited first, 30 of 34)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [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)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [SDEdit: Guided Image Synthesis and Editing with Stochastic Differential Equations](https://mlsanity.com/paper/2108.01073/) (2021)
- [Image Super-Resolution via Iterative Refinement](https://mlsanity.com/paper/2104.07636/) (2021)
- [Variational Diffusion Models](https://mlsanity.com/paper/2107.00630/) (2021)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2201.11793/) (2022)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [ILVR: Conditioning Method for Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2108.02938/) (2021)
- [WaveGrad: Estimating Gradients for Waveform Generation](https://mlsanity.com/paper/2009.00713/) (2020)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [CSDI: Conditional Score-based Diffusion Models for Probabilistic Time Series Imputation](https://mlsanity.com/paper/2107.03502/) (2021)
- [Tackling the Generative Learning Trilemma with Denoising Diffusion GANs](https://mlsanity.com/paper/2112.07804/) (2021)
- [SRDiff: Single Image Super-Resolution with Diffusion Probabilistic Models](https://mlsanity.com/paper/2104.14951/) (2021)
- [Knowledge Distillation in Iterative Generative Models for Improved Sampling Speed](https://mlsanity.com/paper/2101.02388/) (2021)
- [Come-Closer-Diffuse-Faster: Accelerating Conditional Diffusion Models for Inverse Problems through Stochastic Contraction](https://mlsanity.com/paper/2112.05146/) (2021)
- [A survey of the Schr"odinger problem and some of its connections with optimal transport](https://mlsanity.com/paper/1308.0215/) (2013)
- [Score-Based Generative Modeling with Critically-Damped Langevin Diffusion](https://mlsanity.com/paper/2112.07068/) (2021)
- [Gotta Go Fast When Generating Data with Score-Based Models](https://mlsanity.com/paper/2105.14080/) (2021)
- [Likelihood Training of Schr"odinger Bridge using Forward-Backward SDEs Theory](https://mlsanity.com/paper/2110.11291/) (2021)
- [SRFlow: Learning the Super-Resolution Space with Normalizing Flow](https://mlsanity.com/paper/2006.14200/) (2020)
- [Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1507.05021/) (2015)
- [Conditional Image Generation with Score-Based Diffusion Models](https://mlsanity.com/paper/2111.13606/) (2021)
- [SNIPS: Solving Noisy Inverse Problems Stochastically](https://mlsanity.com/paper/2105.14951/) (2021)
- [Learning Fast Samplers for Diffusion Models by Differentiating Through Sample Quality](https://mlsanity.com/paper/2202.05830/) (2022)
- [Solving Schr"odinger Bridges via Maximum Likelihood](https://mlsanity.com/paper/2106.02081/) (2021)
- [Conditional Diffusion Probabilistic Model for Speech Enhancement](https://mlsanity.com/paper/2202.05256/) (2022)

## Built on by (most cited first, 30 of 60)

- [Diffusion Models: A Comprehensive Survey of Methods and Applications](https://mlsanity.com/paper/2209.00796/) (2022)
- [Diffusion Models in Vision: A Survey](https://mlsanity.com/paper/2209.04747/) (2022)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Practical and Asymptotically Exact Conditional Sampling in Diffusion Models](https://mlsanity.com/paper/2306.17775/) (2023)
- [I2SB: Image-to-Image Schr"odinger Bridge](https://mlsanity.com/paper/2302.05872/) (2023)
- [Denoising Diffusion Bridge Models](https://mlsanity.com/paper/2309.16948/) (2023)
- [Diffusion Schr"odinger Bridge Matching](https://mlsanity.com/paper/2303.16852/) (2023)
- [Stochastic interpolants with data-dependent couplings](https://mlsanity.com/paper/2310.03725/) (2023)
- [Unpaired Image-to-Image Translation via Neural Schr"odinger Bridge](https://mlsanity.com/paper/2305.15086/) (2023)
- [Simulation-free Schr"odinger bridges via score and flow matching](https://mlsanity.com/paper/2307.03672/) (2023)
- [Image Denoising: The Deep Learning Revolution and Beyond -- A Survey Paper --](https://mlsanity.com/paper/2301.03362/) (2023)
- [Sequential Neural Score Estimation: Likelihood-Free Inference with Conditional Score Based Diffusion Models](https://mlsanity.com/paper/2210.04872/) (2022)
- [Compositional Score Modeling for Simulation-based Inference](https://mlsanity.com/paper/2209.14249/) (2022)
- [JPEG Artifact Correction using Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2209.11888/) (2022)
- [Deep Momentum Multi-Marginal Schr"odinger Bridge](https://mlsanity.com/paper/2303.01751/) (2023)
- [Provably Convergent Schr"odinger Bridge with Applications to Probabilistic Time Series Imputation](https://mlsanity.com/paper/2305.07247/) (2023)
- [Generative Modeling with Phase Stochastic Bridges](https://mlsanity.com/paper/2310.07805/) (2023)
- [Unifying Generative Models with GFlowNets and Beyond](https://mlsanity.com/paper/2209.02606/) (2022)
- [Improved Convergence of Score-Based Diffusion Models via Prediction-Correction](https://mlsanity.com/paper/2305.14164/) (2023)
- [Consistency Diffusion Bridge Models](https://mlsanity.com/paper/2410.22637/) (2024)
- [Distribution learning via neural differential equations: a nonparametric statistical perspective](https://mlsanity.com/paper/2309.01043/) (2023)
- [Augmented Bridge Matching](https://mlsanity.com/paper/2311.06978/) (2023)
- [Warped Diffusion: Solving Video Inverse Problems with Image Diffusion Models](https://mlsanity.com/paper/2410.16152/) (2024)
- [Conditional sampling within generative diffusion models](https://mlsanity.com/paper/2409.09650/) (2024)
- [FlowDAS: A Stochastic Interpolant-based Framework for Data Assimilation](https://mlsanity.com/paper/2501.16642/) (2025)
- [Score-based generative models are provably robust: an uncertainty quantification perspective](https://mlsanity.com/paper/2405.15754/) (2024)
- [Inverse Bridge Matching Distillation](https://mlsanity.com/paper/2502.01362/) (2025)
- [Diffusion-Model-Assisted Supervised Learning of Generative Models for Density Estimation](https://mlsanity.com/paper/2310.14458/) (2023)
- [An Optimal Transport Formulation of Bayes' Law for Nonlinear Filtering Algorithms](https://mlsanity.com/paper/2203.11869/) (2022)
- [Nonlinear Filtering with Brenier Optimal Transport Maps](https://mlsanity.com/paper/2310.13886/) (2023)

## Abstract (arXiv)

Denoising diffusion models have recently emerged as a powerful class of generative models. They provide state-of-the-art results, not only for unconditional simulation, but also when used to solve conditional simulation problems arising in a wide range of inverse problems. A limitation of these models is that they are computationally intensive at generation time as they require simulating a diffusion process over a long time horizon. When performing unconditional simulation, a Schr"odinger bridge formulation of generative modeling leads to a theoretically grounded algorithm shortening generation time which is complementary to other proposed acceleration techniques. We extend the Schr"odinger bridge framework to conditional simulation. We demonstrate this novel methodology on various applications including image super-resolution, optimal filtering for state-space models and the refinement of pre-trained networks. Our code can be found at https://github.com/vdeborto/cdsb.