mlsanityData to 7 Oct 2026

Diffusion Models for Constrained Domains

arXiv 2304.05364 ↗First posted 11 Apr 2023cs.LG stat.MLcited by 60

On the map: Diffusion models → Diffusion foundations and sampling

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.

Key facts. Cited by 60 papers in the map (data through 7 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.

Builds on

24 of 86 references.

Closest ancestors · 7

Same direction, newest first

  1. 2023Reflected Diffusion Models10 Apr
  2. 2022Riemannian Diffusion Models16 Aug
  3. 2022Riemannian Score-Based Generative Modelling6 Feb
  4. 2021A Variational Perspective on Diffusion-Based Generative Models and Score Matching5 Jun
  5. 2021Diffusion Models Beat GANs on Image Synthesis11 May
  6. 2020Score-Based Generative Modeling through Stochastic Differential Equations26 Nov
  7. 2020Denoising Diffusion Probabilistic Models19 Jun

Other references · 17

  1. 2014Adam: A Method for Stochastic Optimizationcited by 36,960
  2. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,005
  3. 2022Photorealistic Text-to-Image Diffusion Models with Deep Language Understandingcited by 6,322
  4. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,067
  5. 2021Convex optimizationcited by 2,479
  6. 2022DiffDock: Diffusion Steps, Twists, and Turns for Molecular Dockingcited by 307
  7. 2022Torsional Diffusion for Molecular Conformer Generationcited by 262
  8. 2022Diffusion probabilistic modeling of protein backbones in 3D for the motif-scaffolding problemcited by 229
  9. 2023SE(3) diffusion model with application to protein backbone generationcited by 214
  10. 2022Protein structure generation via folding diffusioncited by 113
  11. 2021Fast and accurate optimization on the orthogonal manifold without retractioncited by 35
  12. 2017Convergence Rate of Riemannian Hamiltonian Monte Carlo and Faster Polytope Volume Computationcited by 32
5 more
  1. 2022Sampling with Riemannian Hamiltonian Monte Carlo in a Constrained Spacecited by 28
  2. 2022Convergence of the Riemannian Langevin Algorithmcited by 23
  3. 2019Estimating Density Models with Truncation Boundaries using Score Matchingcited by 20
  4. 2016Geodesic Walks in Polytopescited by 13
  5. 2020Geomstats: A Python Package for Riemannian Geometry in Machine Learningcited by 6

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 9; 2024: 19; 2025: 17; 2026: 15.

60 papers.

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  1. 2023Solving Inverse Problems with Latent Diffusion Models via Hard Data Consistencycited by 180
  2. 2024Constrained Synthesis with Projected Diffusion Modelscited by 83
  3. 2024Score-based Diffusion Models via Stochastic Differential Equations -- a Technical Tutorialcited by 60
  4. 2023Mirror Diffusion Models for Constrained and Watermarked Generationcited by 38
  5. 2024Diffusion Models as Constrained Samplers for Optimization with Unknown Constraintscited by 33
  6. 2024Contractive Diffusion Probabilistic Modelscited by 29
  7. 2023Metropolis Sampling for Constrained Diffusion Modelscited by 28
  8. 2023Generative Modeling on Manifolds Through Mixture of Riemannian Diffusion Processescited by 21
  9. 2025Training-Free Constrained Generation With Stable Diffusion Modelscited by 20
  10. 2024Constrained Diffusion Models via Dual Trainingcited by 19
  11. 2023Beta Diffusioncited by 17
  12. 2024Generative Modelling of Structurally Constrained Graphscited by 17
38 more
  1. 2025DDAT: Diffusion Policies Enforcing Dynamically Admissible Robot Trajectoriescited by 14
  2. 2024Reflected Flow Matchingcited by 13
  3. 2024Reflected Schr"odinger Bridge for Constrained Generative Modelingcited by 12
  4. 2024Context-Guided Diffusion for Out-of-Distribution Molecular and Protein Designcited by 11
  5. 2023Unbalanced Diffusion Schr"odinger Bridgecited by 9
  6. 2025Constrained Diffusers for Safe Planning and Controlcited by 9
  7. 2024Quantum State Generation with Structure-Preserving Diffusion Modelcited by 8
  8. 2025Mirror Flow Matching with Heavy-Tailed Priors for Generative Modeling on Convex Domainscited by 8
  9. 2024Neural Approximate Mirror Maps for Constrained Diffusion Modelscited by 7
  10. 2023Don't be so negative! Score-based Generative Modeling with Oracle-assisted Guidancecited by 6
  11. 2024Constrained Posterior Sampling: Time Series Generation with Hard Constraintscited by 6
  12. 2024Statistical guarantees for denoising reflected diffusion modelscited by 6
  13. 2025Strictly Constrained Generative Modeling via Split Augmented Langevin Samplingcited by 6
  14. 2025Projected Coupled Diffusion for Test-Time Constrained Joint Generationcited by 6
  15. 2024Geometric Neural Network based on Phase Space for BCI-EEG decodingcited by 5
  16. 2024Equality Constrained Diffusion for Direct Trajectory Optimizationcited by 5
  17. 2024Towards diffusion models for large-scale sea-ice modellingcited by 4
  18. 2026Picasso: Holistic Scene Reconstruction with Physics-Constrained Samplingcited by 4
  19. 2023Accurate and Efficient Structural Ensemble Generation of Macrocyclic Peptides using Internal Coordinate Diffusioncited by 3
  20. 2025Generative Predictive Control: Flow Matching Policies for Dynamic and Difficult-to-Demonstrate Taskscited by 3
  21. 2025How Well do Diffusion Policies Learn Kinematic Constraint Manifolds?cited by 3
  22. 2025Fractional Diffusion Bridge Modelscited by 3
  23. 2024A singular Riemannian Geometry Approach to Deep Neural Networks III. Piecewise Differentiable Layers and Random Walks on n-dimensional Classescited by 2
  24. 2025Dale meets Langevin: A Multiplicative Denoising Diffusion Modelcited by 2
  25. 2026Polynomial Convergence of Riemannian Diffusion Modelscited by 2
  26. 2026Logic-Guided Vector Fields for Constrained Generative Modelingcited by 2
  27. 2026Provably Safe Generative Sampling with Constricting Barrier Functionscited by 2
  28. 2023QonFusion -- Quantum Approaches to Gaussian Random Variables: Applications in Stable Diffusion and Brownian Motioncited by 1
  29. 2024Efficient Fine-Grained Guidance for Diffusion Model Based Symbolic Music Generationcited by 1
  30. 2025Interleaved Gibbs Diffusion: Generating Discrete-Continuous Data with Implicit Constraintscited by 1
  31. 2025Constraint-Aware Flow Matching via Randomized Explorationcited by 1
  32. 2026PolyFlow: Safe and Efficient Polytope-Constrained Flow Matching with Constraint Embedding and Projection-free Updatecited by 1
  33. 2026DiPhon: Diffusion on Graphons for Scalable Graph Generationcited by 1
  34. 2026Motion Planning with Model-Based Diffusion via Constraint Optimization and Adaptive Schedulingcited by 1
  35. 2024Generative modeling assisted simulation of measurement-altered quantum criticalitycited by 0
  36. 2025A collaborative constrained graph diffusion model for the generation of realistic synthetic moleculescited by 0
  37. 2025Constraint-Guided Prediction Refinement via Deterministic Diffusion Trajectoriescited by 0
  38. 2025Neural Stochastic Differential Equations on Compact State Spaces: Theory, Methods, and Application to Suicide Risk Modelingcited by 0

50 most cited of 60 All 60 (JSON)

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  1. 2022Convergence of denoising diffusion models under the manifold hypothesis
  2. 2025Riemannian Denoising Diffusion Probabilistic Models
  3. 2026Sharp Convergence and Sampling Trade-offs for Riemannian Diffusion under Nonnegative Ricci Curvature
  4. 2025Be Tangential to Manifold: Discovering Riemannian Metric for Diffusion Models
  5. 2024Denoising Diffusions with Optimal Transport: Localization, Curvature, and Multi-Scale Complexity
  6. 2024Convergence of Diffusion Models Under the Manifold Hypothesis in High-Dimensions
  7. 2026DiffRGD: An Inference-Time Diffusion Guidance Through Riemannian Gradient Descent
  8. 2024Physics-Informed Diffusion Models