# Diffusion Models for Constrained Domains

arXiv: https://arxiv.org/abs/2304.05364 · HTML: https://mlsanity.com/paper/2304.05364/

## Key facts

- Cited by 63 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 24 of its 86 references that are in the map.
- First posted on arXiv 11 Apr 2023.

## Placed in

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

## Builds on (most cited first, 24 of 24)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (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)
- [Photorealistic Text-to-Image Diffusion Models with Deep Language Understanding](https://mlsanity.com/paper/2205.11487/) (2022)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [DiffDock: Diffusion Steps, Twists, and Turns for Molecular Docking](https://mlsanity.com/paper/2210.01776/) (2022)
- [Torsional Diffusion for Molecular Conformer Generation](https://mlsanity.com/paper/2206.01729/) (2022)
- [Diffusion probabilistic modeling of protein backbones in 3D for the motif-scaffolding problem](https://mlsanity.com/paper/2206.04119/) (2022)
- [Riemannian Score-Based Generative Modelling](https://mlsanity.com/paper/2202.02763/) (2022)
- [SE(3) diffusion model with application to protein backbone generation](https://mlsanity.com/paper/2302.02277/) (2023)
- [A Variational Perspective on Diffusion-Based Generative Models and Score Matching](https://mlsanity.com/paper/2106.02808/) (2021)
- [Riemannian Diffusion Models](https://mlsanity.com/paper/2208.07949/) (2022)
- [Protein structure generation via folding diffusion](https://mlsanity.com/paper/2209.15611/) (2022)
- [Reflected Diffusion Models](https://mlsanity.com/paper/2304.04740/) (2023)
- [Fast and accurate optimization on the orthogonal manifold without retraction](https://mlsanity.com/paper/2102.07432/) (2021)
- [Convergence Rate of Riemannian Hamiltonian Monte Carlo and Faster Polytope Volume Computation](https://mlsanity.com/paper/1710.06261/) (2017)
- [Sampling with Riemannian Hamiltonian Monte Carlo in a Constrained Space](https://mlsanity.com/paper/2202.01908/) (2022)
- [Convergence of the Riemannian Langevin Algorithm](https://mlsanity.com/paper/2204.10818/) (2022)
- [Estimating Density Models with Truncation Boundaries using Score Matching](https://mlsanity.com/paper/1910.03834/) (2019)
- [Geodesic Walks in Polytopes](https://mlsanity.com/paper/1606.04696/) (2016)
- [Geomstats: A Python Package for Riemannian Geometry in Machine Learning](https://mlsanity.com/paper/2004.04667/) (2020)

## Built on by (most cited first, 30 of 63)

- [Solving Inverse Problems with Latent Diffusion Models via Hard Data Consistency](https://mlsanity.com/paper/2307.08123/) (2023)
- [Constrained Synthesis with Projected Diffusion Models](https://mlsanity.com/paper/2402.03559/) (2024)
- [Score-based Diffusion Models via Stochastic Differential Equations -- a Technical Tutorial](https://mlsanity.com/paper/2402.07487/) (2024)
- [Mirror Diffusion Models for Constrained and Watermarked Generation](https://mlsanity.com/paper/2310.01236/) (2023)
- [Diffusion Models as Constrained Samplers for Optimization with Unknown Constraints](https://mlsanity.com/paper/2402.18012/) (2024)
- [Contractive Diffusion Probabilistic Models](https://mlsanity.com/paper/2401.13115/) (2024)
- [Metropolis Sampling for Constrained Diffusion Models](https://mlsanity.com/paper/2307.05439/) (2023)
- [Generative Modeling on Manifolds Through Mixture of Riemannian Diffusion Processes](https://mlsanity.com/paper/2310.07216/) (2023)
- [Constrained Diffusion Models via Dual Training](https://mlsanity.com/paper/2408.15094/) (2024)
- [Training-Free Constrained Generation With Stable Diffusion Models](https://mlsanity.com/paper/2502.05625/) (2025)
- [Generative Modelling of Structurally Constrained Graphs](https://mlsanity.com/paper/2406.17341/) (2024)
- [Beta Diffusion](https://mlsanity.com/paper/2309.07867/) (2023)
- [Reflected Flow Matching](https://mlsanity.com/paper/2405.16577/) (2024)
- [DDAT: Diffusion Policies Enforcing Dynamically Admissible Robot Trajectories](https://mlsanity.com/paper/2502.15043/) (2025)
- [Reflected Schr"odinger Bridge for Constrained Generative Modeling](https://mlsanity.com/paper/2401.03228/) (2024)
- [Context-Guided Diffusion for Out-of-Distribution Molecular and Protein Design](https://mlsanity.com/paper/2407.11942/) (2024)
- [Constrained Diffusers for Safe Planning and Control](https://mlsanity.com/paper/2506.12544/) (2025)
- [Unbalanced Diffusion Schr"odinger Bridge](https://mlsanity.com/paper/2306.09099/) (2023)
- [Neural Approximate Mirror Maps for Constrained Diffusion Models](https://mlsanity.com/paper/2406.12816/) (2024)
- [Quantum State Generation with Structure-Preserving Diffusion Model](https://mlsanity.com/paper/2404.06336/) (2024)
- [Mirror Flow Matching with Heavy-Tailed Priors for Generative Modeling on Convex Domains](https://mlsanity.com/paper/2510.08929/) (2025)
- [Constrained Posterior Sampling: Time Series Generation with Hard Constraints](https://mlsanity.com/paper/2410.12652/) (2024)
- [Projected Coupled Diffusion for Test-Time Constrained Joint Generation](https://mlsanity.com/paper/2508.10531/) (2025)
- [Don't be so negative! Score-based Generative Modeling with Oracle-assisted Guidance](https://mlsanity.com/paper/2307.16463/) (2023)
- [Statistical guarantees for denoising reflected diffusion models](https://mlsanity.com/paper/2411.01563/) (2024)
- [Strictly Constrained Generative Modeling via Split Augmented Langevin Sampling](https://mlsanity.com/paper/2505.18017/) (2025)
- [Geometric Neural Network based on Phase Space for BCI-EEG decoding](https://mlsanity.com/paper/2403.05645/) (2024)
- [Equality Constrained Diffusion for Direct Trajectory Optimization](https://mlsanity.com/paper/2410.01939/) (2024)
- [Towards diffusion models for large-scale sea-ice modelling](https://mlsanity.com/paper/2406.18417/) (2024)
- [Generative Predictive Control: Flow Matching Policies for Dynamic and Difficult-to-Demonstrate Tasks](https://mlsanity.com/paper/2502.13406/) (2025)

## Abstract (arXiv)

Denoising diffusion models are a novel class of generative algorithms that achieve state-of-the-art performance across a range of domains, including image generation and text-to-image tasks. Building on this success, diffusion models have recently been extended to the Riemannian manifold setting, broadening their applicability to a range of problems from the natural and engineering sciences. However, these Riemannian diffusion models are built on the assumption that their forward and backward processes are well-defined for all times, preventing them from being applied to an important set of tasks that consider manifolds defined via a set of inequality constraints. In this work, we introduce a principled framework to bridge this gap. We present two distinct noising processes based on (i) the logarithmic barrier metric and (ii) the reflected Brownian motion induced by the constraints. As existing diffusion model techniques cannot be applied in this setting, we derive new tools to define such models in our framework. We then demonstrate the practical utility of our methods on a number of synthetic and real-world tasks, including applications from robotics and protein design.