# Constrained Generative Modeling with Manually Bridged Diffusion Models

arXiv: https://arxiv.org/abs/2502.20371 · HTML: https://mlsanity.com/paper/2502.20371/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 26 of its 42 references that are in the map.
- First posted on arXiv 27 Feb 2025.

## Placed in

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

## Builds on (most cited first, 26 of 26)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Self-Attention with Relative Position Representations](https://mlsanity.com/paper/1803.02155/) (2018)
- [StarGAN v2: Diverse Image Synthesis for Multiple Domains](https://mlsanity.com/paper/1912.01865/) (2019)
- [Planning with Diffusion for Flexible Behavior Synthesis](https://mlsanity.com/paper/2205.09991/) (2022)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [Flexible Diffusion Modeling of Long Videos](https://mlsanity.com/paper/2205.11495/) (2022)
- [Likelihood Training of Schr"odinger Bridge using Forward-Backward SDEs Theory](https://mlsanity.com/paper/2110.11291/) (2021)
- [Rethinking and Improving Relative Position Encoding for Vision Transformer](https://mlsanity.com/paper/2107.14222/) (2021)
- [Guided Conditional Diffusion for Controllable Traffic Simulation](https://mlsanity.com/paper/2210.17366/) (2022)
- [Diffusion-based Molecule Generation with Informative Prior Bridges](https://mlsanity.com/paper/2209.00865/) (2022)
- [Deep Generative Learning via Schr"odinger Bridge](https://mlsanity.com/paper/2106.10410/) (2021)
- [Constrained Synthesis with Projected Diffusion Models](https://mlsanity.com/paper/2402.03559/) (2024)
- [SceneGen: Learning to Generate Realistic Traffic Scenes](https://mlsanity.com/paper/2101.06541/) (2021)
- [Simulating Diffusion Bridges with Score Matching](https://mlsanity.com/paper/2111.07243/) (2021)
- [SceneDiffuser: Efficient and Controllable Driving Simulation Initialization and Rollout](https://mlsanity.com/paper/2412.12129/) (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)
- [Data Redaction from Pre-trained GANs](https://mlsanity.com/paper/2206.14389/) (2022)
- [A Diffusion-Model of Joint Interactive Navigation](https://mlsanity.com/paper/2309.12508/) (2023)
- [Actively Avoiding Nonsense in Generative Models](https://mlsanity.com/paper/1802.07229/) (2018)
- [Don't be so negative! Score-based Generative Modeling with Oracle-assisted Guidance](https://mlsanity.com/paper/2307.16463/) (2023)

## Built on by (most cited first, 8 of 8)

- [Discrete-Guided Diffusion for Scalable and Safe Multi-Robot Motion Planning](https://mlsanity.com/paper/2508.20095/) (2025)
- [Chance-constrained Flow Matching for High-Fidelity Constraint-aware Generation](https://mlsanity.com/paper/2509.25157/) (2025)
- [Score-based constrained generative modeling via Langevin diffusions with boundary conditions](https://mlsanity.com/paper/2510.23985/) (2025)
- [Predict-Project-Renoise: Sampling Diffusion Models under Hard Constraints](https://mlsanity.com/paper/2601.21033/) (2026)
- [Improved Constrained Generation by Bridging Pretrained Generative Models](https://mlsanity.com/paper/2603.06742/) (2026)
- [Softly Constrained Denoisers for Diffusion Models Applied to Partial Differential Equations](https://mlsanity.com/paper/2512.14980/) (2025)
- [Neural Bridge Processes](https://mlsanity.com/paper/2508.07220/) (2025)
- [Integration Matters: Rollout-Based Training for Constrained Diffusion Models](https://mlsanity.com/paper/2607.14398/) (2026)

## Abstract (arXiv)

In this paper we describe a novel framework for diffusion-based generative modeling on constrained spaces. In particular, we introduce manual bridges, a framework that expands the kinds of constraints that can be practically used to form so-called diffusion bridges. We develop a mechanism for combining multiple such constraints so that the resulting multiply-constrained model remains a manual bridge that respects all constraints. We also develop a mechanism for training a diffusion model that respects such multiple constraints while also adapting it to match a data distribution. We develop and extend theory demonstrating the mathematical validity of our mechanisms. Additionally, we demonstrate our mechanism in constrained generative modeling tasks, highlighting a particular high-value application in modeling trajectory initializations for path planning and control in autonomous vehicles.