mlsanityData to 14 Oct 2026

Closed-Form Diffusion Models

arXiv 2310.12395 ↗First posted 19 Oct 2023cs.LG stat.MLcited by 46

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

Score-based generative models (SGMs) sample from a target distribution by iteratively transforming noise using the score function of the perturbed target. For any finite training set, this score function can be evaluated in closed form, but the resulting SGM memorizes its training data and does not generate novel samples. In practice, one approximates the score by training a neural network via score-matching. The error in this approximation promotes generalization, but neural SGMs are costly to train and sample, and the effective regularization this error provides is not well-understood theoretically. In this work, we instead explicitly smooth the closed-form score to obtain an SGM that generates novel samples without training. We analyze our model and propose an efficient nearest-neighbor-based estimator of its score function. Using this estimator, our method achieves competitive sampling times while running on consumer-grade CPUs.

Key facts. Cited by 46 papers in the map (data through 14 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 43 of its 63 references that are in the map. First posted on arXiv 19 Oct 2023.

Builds on

43 of 63 references.

Closest ancestors · 19

Same direction, newest first

  1. 2023Generalization in diffusion models arises from geometry-adaptive harmonic representations4 Oct
  2. 2023Fast Diffusion Model12 Jun
  3. 2023On the Generalization of Diffusion Model24 May
  4. 2023Iterative α-(de)Blending: a Minimalist Deterministic Diffusion Model5 May
  5. 2023Diffusion Models are Minimax Optimal Distribution Estimators3 Mar
  6. 2022Score-based Generative Modeling Secretly Minimizes the Wasserstein Distance13 Dec
  7. 2022Convergence of score-based generative modeling for general data distributions26 Sep
  8. 2022Convergence of denoising diffusion models under the manifold hypothesis10 Aug
  9. 2022Classifier-Free Diffusion Guidance26 Jul
  10. 2022Score-Based Generative Models Detect Manifolds2 Jun
  11. 2022Elucidating the Design Space of Diffusion-Based Generative Models1 Jun
  12. 2022Fast Sampling of Diffusion Models with Exponential Integrator29 Apr
7 more
  1. 2022Pseudo Numerical Methods for Diffusion Models on Manifolds20 Feb
  2. 2021High-Resolution Image Synthesis with Latent Diffusion Models20 Dec
  3. 2021Score-based Generative Modeling in Latent Space10 Jun
  4. 2021Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling1 Jun
  5. 2021Diffusion Models Beat GANs on Image Synthesis11 May
  6. 2020Denoising Diffusion Implicit Models6 Oct
  7. 2020Denoising Diffusion Probabilistic Models19 Jun

Other references · 24

  1. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,058
  2. 20233D Gaussian Splatting for Real-Time Radiance Field Renderingcited by 6,226
  3. 2023Adding Conditional Control to Text-to-Image Diffusion Modelscited by 5,503
  4. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,101
  5. 2022Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flowcited by 3,700
  6. 2014Deep Learning Face Attributes in the Wildcited by 3,693
  7. 2022DreamFusion: Text-to-3D using 2D Diffusioncited by 3,025
  8. 2016Improved Techniques for Training GANscited by 2,970
  9. 2017Billion-scale similarity search with GPUscited by 2,497
  10. 2022Video Diffusion Modelscited by 2,305
  11. 2021Mip-NeRF 360: Unbounded Anti-Aliased Neural Radiance Fieldscited by 2,032
  12. 2022Imagen Video: High Definition Video Generation with Diffusion Modelscited by 1,778
12 more
  1. 2018On the Spectral Bias of Neural Networkscited by 1,341
  2. 2018Demystifying MMD GANscited by 1,293
  3. 2021Diffusion Probabilistic Models for 3D Point Cloud Generationcited by 735
  4. 2022Equivariant Diffusion for Molecule Generation in 3Dcited by 677
  5. 2022DiffDock: Diffusion Steps, Twists, and Turns for Molecular Dockingcited by 313
  6. 2022Novel View Synthesis with Diffusion Modelscited by 300
  7. 2022Structure-based Drug Design with Equivariant Diffusion Modelscited by 217
  8. 2022Continuous diffusion for categorical datacited by 214
  9. 2023Inversion by Direct Iteration: An Alternative to Denoising Diffusion for Image Restorationcited by 165
  10. 2022Statistical Efficiency of Score Matching: The View from Isoperimetrycited by 64
  11. 2022Can Push-forward Generative Models Fit Multimodal Distributions?cited by 41
  12. 2021DEANN: Speeding up Kernel-Density Estimation using Approximate Nearest Neighbor Searchcited by 8

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2023: 1; 2024: 7; 2025: 19; 2026: 19.

46 papers.

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By area:

  1. 2024An analytic theory of creativity in convolutional diffusion modelscited by 104
  2. 2024Optimal score estimation via empirical Bayes smoothingcited by 61
  3. 2025On the Closed-Form of Flow Matching: Generalization Does Not Arise from Target Stochasticitycited by 50
  4. 2024Towards a Mechanistic Explanation of Diffusion Model Generalizationcited by 42
  5. 2025Memorization and Regularization in Generative Diffusion Modelscited by 40
  6. 2024Denoising: A Powerful Building-Block for Imaging, Inverse Problems, and Machine Learningcited by 35
  7. 2024The Unreasonable Effectiveness of Gaussian Score Approximation for Diffusion Models and its Applicationscited by 32
  8. 2025Locality in Image Diffusion Models Emerges from Data Statisticscited by 25
  9. 2025Minimax Optimality of the Probability Flow ODE for Diffusion Modelscited by 23
  10. 2025Diffusion Models and the Manifold Hypothesis: Log-Domain Smoothing is Geometry Adaptivecited by 20
  11. 2025Selective Underfitting in Diffusion Modelscited by 18
  12. 2024Plug-and-Play image restoration with Stochastic deNOising REgularizationcited by 17
34 more
  1. 2025Does Generation Require Memorization? Creative Diffusion Models using Ambient Diffusioncited by 17
  2. 2023The Hidden Linear Structure in Score-Based Models and its Applicationcited by 15
  3. 2025On the Interpolation Effect of Score Smoothing in Diffusion Modelscited by 15
  4. 2025Flow Map Distillation Without Datacited by 14
  5. 2025Density Ratio Estimation with Conditional Probability Pathscited by 13
  6. 2026Making Reconstruction FID Predictive of Diffusion Generation FIDcited by 13
  7. 2025Resolving Memorization in Empirical Diffusion Model for Manifold Data in High-Dimensional Spacescited by 8
  8. 2025From Navigation to Refinement: Revealing the Two-Stage Nature of Flow-based Diffusion Models through Oracle Velocitycited by 8
  9. 2026The Geometry of Noise: Why Diffusion Models Don't Need Noise Conditioningcited by 8
  10. 2025Geometric Regularity in Deterministic Sampling Dynamics of Diffusion-based Generative Modelscited by 4
  11. 2026A Kinetic Energy Perspective of Flow Matchingcited by 3
  12. 2026Is Flow Matching Just Trajectory Replay for Sequential Data?cited by 3
  13. 2026Momentum Guidance: Plug-and-Play Guidance for Flow Modelscited by 3
  14. 2026Closed-form conditional diffusion models for data assimilationcited by 3
  15. 2026An exact information theory of generalization phase transitions in Bayesian diffusion modelscited by 3
  16. 2025Bigger Isn't Always Better: Towards a General Prior for Medical Image Reconstructioncited by 2
  17. 2025Transport Novelty Distance: A Distributional Metric for Evaluating Material Generative Modelscited by 2
  18. 2025On The Hidden Biases of Flow Matching Samplerscited by 2
  19. 2026An analytic theory of convolutional neural network inverse problems solverscited by 2
  20. 2026Diffusion Models Memorize in Training -- and Generalize in Inferencecited by 2
  21. 2026Beyond MMSE: Enhancing PnP Restoration with ProxiMAPcited by 2
  22. 2025Sensitivity Analysis for Diffusion Modelscited by 1
  23. 2025Is Your Conditional Diffusion Model Actually Denoising?cited by 1
  24. 2026Categorical Reparameterization with Denoising Diffusion modelscited by 1
  25. 2026Efficient and Training-Free Single-Image Diffusion Modelscited by 1
  26. 2026Transformers as In-Context Samplers: From Closed-Form Diffusion to Estimation-Free Samplingcited by 1
  27. 2024Operator-Informed Score Matching for Markov Diffusion Modelscited by 0
  28. 2025Demystifying Transition Matching: When and Why It Can Beat Flow Matchingcited by 0
  29. 2026Fast and Scalable Analytical Diffusioncited by 0
  30. 2026Follow the Mean: Reference-Guided Flow Matchingcited by 0
  31. 2026Training-Free Generative Sampling via Moment-Matched Score Smoothingcited by 0
  32. 2026Filtered Posterior Mean Collections: A Unified Framework for Analytical Models of Diffusion Generalizationcited by 0
  33. 2026Training-Free Imitation Learning with Closed-Form Diffusion Policiescited by 0
  34. 2026Revisiting Spectral Representations in Generative Diffusion Modelscited by 0

Similar, not cited

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  1. 2022Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions
  2. 2023Improved Convergence of Score-Based Diffusion Models via Prediction-Correction
  3. 2024A Good Score Does not Lead to A Good Generative Model
  4. 2024Nearest Neighbour Score Estimators for Diffusion Generative Models
  5. 2026Non-asymptotic Convergence of Stochastic Gradient Descent in Score-based Generative Models
  6. 2024A Score-Based Density Formula, with Applications in Diffusion Generative Models
  7. 2022Convergence for score-based generative modeling with polynomial complexity
  8. 2025Algorithm- and Data-Dependent Generalization Bounds for Diffusion Models