# SDDBMs: Soft Denoising Diffusion Bridge Models

arXiv: https://arxiv.org/abs/2608.08594 · HTML: https://mlsanity.com/paper/2608.08594/

## 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 35 of its 48 references that are in the map.
- First posted on arXiv 9 Aug 2026.

## Placed in

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

## Builds on (most cited first, 30 of 35)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [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)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Scalable Diffusion Models with Transformers](https://mlsanity.com/paper/2212.09748/) (2022)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [SDXL: Improving Latent Diffusion Models for High-Resolution Image Synthesis](https://mlsanity.com/paper/2307.01952/) (2023)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [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)
- [Progressive Growing of GANs for Improved Quality, Stability, and Variation](https://mlsanity.com/paper/1710.10196/) (2017)
- [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)
- [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)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [Mean Flows for One-step Generative Modeling](https://mlsanity.com/paper/2505.13447/) (2025)
- [Fast Sampling of Diffusion Models with Exponential Integrator](https://mlsanity.com/paper/2204.13902/) (2022)
- [One Step Diffusion via Shortcut Models](https://mlsanity.com/paper/2410.12557/) (2024)
- [Deep Joint Rain Detection and Removal from a Single Image](https://mlsanity.com/paper/1609.07769/) (2016)
- [MAXIM: Multi-Axis MLP for Image Processing](https://mlsanity.com/paper/2201.02973/) (2022)
- [BBDM: Image-to-image Translation with Brownian Bridge Diffusion Models](https://mlsanity.com/paper/2205.07680/) (2022)
- [Image Restoration with Mean-Reverting Stochastic Differential Equations](https://mlsanity.com/paper/2301.11699/) (2023)
- [Denoising Diffusion Bridge Models](https://mlsanity.com/paper/2309.16948/) (2023)
- [An optimal control perspective on diffusion-based generative modeling](https://mlsanity.com/paper/2211.01364/) (2022)

## Abstract (arXiv)

Diffusion bridge models leverage Doob's (h)-transform to construct stochastic transports between arbitrary endpoint distributions, and have shown strong potential in image-to-image translation and restoration. However, most existing bridge models rely on hard endpoint conditioning, which forces the terminal state to match a prescribed target exactly. This hard constraint induces terminal-boundary singularities: the terminal law collapses to a Dirac measure, and the resulting drift coefficients become ill-conditioned near the endpoint. In this paper, we propose Soft Denoising Diffusion Bridge Models (SDDBMs), a generalized framework that regularizes diffusion bridges directly at the level of their terminal constraints. Instead of imposing an exact endpoint, SDDBMs prescribe a non-degenerate Gaussian terminal marginal under the transformed path measure, with a flexible terminal center and variance. Starting from this prescribed marginal, we develop a complete closed-form construction of the soft bridge, including the Gaussian terminal reweighting and soft (h)-function, the induced Gaussian forward marginals and (x0)-free dynamics. Theoretically, SDDBMs provide a unified probabilistic perspective that encompasses existing diffusion bridge models, including DDBMs, GOUB, and UniDB, as special cases under specific parameter choices. Extensive experiments on image restoration tasks demonstrate that SDDBMs achieve improved numerical stability and superior generation quality over existing bridge-based methods.