mlsanityData to 14 Oct 2026

Metropolis Sampling for Constrained Diffusion Models

arXiv 2307.05439 ↗First posted 11 Jul 2023cs.LGcited by 28

On the map: Diffusion models → Diffusion foundations and sampling

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.

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.

Builds on

19 of 79 references.

Closest ancestors · 6

Same direction, newest first

  1. 2023Diffusion Models for Constrained Domains11 Apr
  2. 2023Reflected Diffusion Models10 Apr
  3. 2022Riemannian Diffusion Models16 Aug
  4. 2022Riemannian Score-Based Generative Modelling6 Feb
  5. 2020Score-Based Generative Modeling through Stochastic Differential Equations26 Nov
  6. 2020Denoising Diffusion Probabilistic Models19 Jun

Other references · 13

  1. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,058
  2. 2022Photorealistic Text-to-Image Diffusion Models with Deep Language Understandingcited by 6,355
  3. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,101
  4. 2022DiffDock: Diffusion Steps, Twists, and Turns for Molecular Dockingcited by 313
  5. 2022Torsional Diffusion for Molecular Conformer Generationcited by 263
  6. 2022Diffusion probabilistic modeling of protein backbones in 3D for the motif-scaffolding problemcited by 236
  7. 2023SE(3) diffusion model with application to protein backbone generationcited by 219
  8. 2017Convergence Rate of Riemannian Hamiltonian Monte Carlo and Faster Polytope Volume Computationcited by 32
  9. 2022Sampling with Riemannian Hamiltonian Monte Carlo in a Constrained Spacecited by 28
  10. 2022Convergence of the Riemannian Langevin Algorithmcited by 25
  11. 2019Estimating Density Models with Truncation Boundaries using Score Matchingcited by 20
  12. 2016Geodesic Walks in Polytopescited by 13
1 more
  1. 2021On Riemannian Stochastic Approximation Schemes with Fixed Step-Sizecited by 12

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2024: 10; 2025: 13; 2026: 5.

28 papers.

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  1. 2024Constrained Synthesis with Projected Diffusion Modelscited by 89
  2. 2024Constrained Diffusion Models via Dual Trainingcited by 20
  3. 2025Training-Free Constrained Generation With Stable Diffusion Modelscited by 20
  4. 2024Generative Modelling of Structurally Constrained Graphscited by 19
  5. 2024Context-Guided Diffusion for Out-of-Distribution Molecular and Protein Designcited by 11
  6. 2024Neural Approximate Mirror Maps for Constrained Diffusion Modelscited by 9
  7. 2024Quantum State Generation with Structure-Preserving Diffusion Modelcited by 8
  8. 2025Constrained Generative Modeling with Manually Bridged Diffusion Modelscited by 8
  9. 2025Mirror Flow Matching with Heavy-Tailed Priors for Generative Modeling on Convex Domainscited by 8
  10. 2024Constrained Posterior Sampling: Time Series Generation with Hard Constraintscited by 7
  11. 2025Projected Coupled Diffusion for Test-Time Constrained Joint Generationcited by 7
  12. 2024Statistical guarantees for denoising reflected diffusion modelscited by 6
16 more
  1. 2024Equality Constrained Diffusion for Direct Trajectory Optimizationcited by 5
  2. 2025Generative Predictive Control: Flow Matching Policies for Dynamic and Difficult-to-Demonstrate Taskscited by 4
  3. 2025How Well do Diffusion Policies Learn Kinematic Constraint Manifolds?cited by 3
  4. 2025Diffusion-based Planning with Learned Viability Filterscited by 2
  5. 2025Score-based constrained generative modeling via Langevin diffusions with boundary conditionscited by 2
  6. 2026Inference-Time Attribute Distribution Alignment for Unconditional Diffusioncited by 2
  7. 2026Constrained Diffusion Models with Primal-Dual Inferencecited by 2
  8. 2025Softly Constrained Denoisers for Diffusion Models Applied to Partial Differential Equationscited by 1
  9. 2024Generative modeling assisted simulation of measurement-altered quantum criticalitycited by 0
  10. 2025A collaborative constrained graph diffusion model for the generation of realistic synthetic moleculescited by 0
  11. 2025Constraint-Guided Prediction Refinement via Deterministic Diffusion Trajectoriescited by 0
  12. 2025Neural Stochastic Differential Equations on Compact State Spaces: Theory, Methods, and Application to Suicide Risk Modelingcited by 0
  13. 2025Model-free filtering in high dimensions via projection and score-based diffusionscited by 0
  14. 2026Efficient Diffusion Models under Nonconvex Equality and Inequality constraints via Landingcited by 0
  15. 2026Generative Modeling with Flux Matchingcited by 0
  16. 2026Reflected Schr"odinger Bridge Matchingcited by 0

Similar, not cited

Close in wording, no citation either way

  1. 2022Convergence of denoising diffusion models under the manifold hypothesis
  2. 2024Diffusion Bridge Implicit Models
  3. 2026Manifold-Constrained Noise Optimization for Diverse Diffusion Sampling
  4. 2023Parallel Sampling of Diffusion Models
  5. 2023Restoration-Degradation Beyond Linear Diffusions: A Non-Asymptotic Analysis For DDIM-Type Samplers
  6. 2025Temporal Score Rescaling for Temperature Sampling in Diffusion and Flow Models
  7. 2026Sampling from multi-modal distributions on Riemannian manifolds with training-free stochastic interpolants
  8. 2023Infinite-Dimensional Diffusion Models