# Metropolis Sampling for Constrained Diffusion Models

arXiv: https://arxiv.org/abs/2307.05439 · HTML: https://mlsanity.com/paper/2307.05439/

## Key facts

- Cited by 28 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 19 of its 79 references that are in the map.
- First posted on arXiv 11 Jul 2023.

## Placed in

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

## Builds on (most cited first, 19 of 19)

- [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)
- [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)
- [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)
- [Riemannian Diffusion Models](https://mlsanity.com/paper/2208.07949/) (2022)
- [Reflected Diffusion Models](https://mlsanity.com/paper/2304.04740/) (2023)
- [Diffusion Models for Constrained Domains](https://mlsanity.com/paper/2304.05364/) (2023)
- [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)
- [On Riemannian Stochastic Approximation Schemes with Fixed Step-Size](https://mlsanity.com/paper/2102.07586/) (2021)

## Built on by (most cited first, 28 of 28)

- [Constrained Synthesis with Projected Diffusion Models](https://mlsanity.com/paper/2402.03559/) (2024)
- [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)
- [Context-Guided Diffusion for Out-of-Distribution Molecular and Protein Design](https://mlsanity.com/paper/2407.11942/) (2024)
- [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)
- [Constrained Generative Modeling with Manually Bridged Diffusion Models](https://mlsanity.com/paper/2502.20371/) (2025)
- [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)
- [Statistical guarantees for denoising reflected diffusion models](https://mlsanity.com/paper/2411.01563/) (2024)
- [Equality Constrained Diffusion for Direct Trajectory Optimization](https://mlsanity.com/paper/2410.01939/) (2024)
- [Generative Predictive Control: Flow Matching Policies for Dynamic and Difficult-to-Demonstrate Tasks](https://mlsanity.com/paper/2502.13406/) (2025)
- [How Well do Diffusion Policies Learn Kinematic Constraint Manifolds?](https://mlsanity.com/paper/2510.01404/) (2025)
- [Diffusion-based Planning with Learned Viability Filters](https://mlsanity.com/paper/2502.19564/) (2025)
- [Score-based constrained generative modeling via Langevin diffusions with boundary conditions](https://mlsanity.com/paper/2510.23985/) (2025)
- [Inference-Time Attribute Distribution Alignment for Unconditional Diffusion](https://mlsanity.com/paper/2605.07456/) (2026)
- [Constrained Diffusion Models with Primal-Dual Inference](https://mlsanity.com/paper/2606.17192/) (2026)
- [Softly Constrained Denoisers for Diffusion Models Applied to Partial Differential Equations](https://mlsanity.com/paper/2512.14980/) (2025)
- [Generative modeling assisted simulation of measurement-altered quantum criticality](https://mlsanity.com/paper/2412.01513/) (2024)
- [A collaborative constrained graph diffusion model for the generation of realistic synthetic molecules](https://mlsanity.com/paper/2505.16365/) (2025)
- [Constraint-Guided Prediction Refinement via Deterministic Diffusion Trajectories](https://mlsanity.com/paper/2506.12911/) (2025)
- [Neural Stochastic Differential Equations on Compact State Spaces: Theory, Methods, and Application to Suicide Risk Modeling](https://mlsanity.com/paper/2508.17090/) (2025)
- [Model-free filtering in high dimensions via projection and score-based diffusions](https://mlsanity.com/paper/2510.23197/) (2025)
- [Efficient Diffusion Models under Nonconvex Equality and Inequality constraints via Landing](https://mlsanity.com/paper/2604.17838/) (2026)
- [Generative Modeling with Flux Matching](https://mlsanity.com/paper/2605.07319/) (2026)
- [Reflected Schr"odinger Bridge Matching](https://mlsanity.com/paper/2607.03626/) (2026)

## Abstract (arXiv)

Denoising diffusion models have recently emerged as the predominant paradigm for generative modelling on image domains. In addition, their extension to Riemannian manifolds has facilitated a range of applications across the natural sciences. While many of these problems stand to benefit from the ability to specify arbitrary, domain-informed constraints, this setting is not covered by the existing (Riemannian) diffusion model methodology. Recent work has attempted to address this issue by constructing novel noising processes based on the reflected Brownian motion and logarithmic barrier methods. However, the associated samplers are either computationally burdensome or only apply to convex subsets of Euclidean space. In this paper, we introduce an alternative, simple noising scheme based on Metropolis sampling that affords substantial gains in computational efficiency and empirical performance compared to the earlier samplers. Of independent interest, we prove that this new process corresponds to a valid discretisation of the reflected Brownian motion. We demonstrate the scalability and flexibility of our approach on a range of problem settings with convex and non-convex constraints, including applications from geospatial modelling, robotics and protein design.