mlsanityData to 7 Oct 2026

Variational Diffusion Models

arXiv 2107.00630 ↗First posted 1 Jul 2021cs.LG stat.MLcited by 1,145

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

Diffusion-based generative models have demonstrated a capacity for perceptually impressive synthesis, but can they also be great likelihood-based models? We answer this in the affirmative, and introduce a family of diffusion-based generative models that obtain state-of-the-art likelihoods on standard image density estimation benchmarks. Unlike other diffusion-based models, our method allows for efficient optimization of the noise schedule jointly with the rest of the model. We show that the variational lower bound (VLB) simplifies to a remarkably short expression in terms of the signal-to-noise ratio of the diffused data, thereby improving our theoretical understanding of this model class. Using this insight, we prove an equivalence between several models proposed in the literature. In addition, we show that the continuous-time VLB is invariant to the noise schedule, except for the signal-to-noise ratio at its endpoints. This enables us to learn a noise schedule that minimizes the variance of the resulting VLB estimator, leading to faster optimization. Combining these advances with architectural improvements, we obtain state-of-the-art likelihoods on image density estimation benchmarks, outperforming autoregressive models that have dominated these benchmarks for many years, with often significantly faster optimization. In addition, we show how to use the model as part of a bits-back compression scheme, and demonstrate lossless compression rates close to the theoretical optimum. Code is available at https://github.com/google-research/vdm .

Key facts. Cited by 1,145 papers in the map (data through 7 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 36 of its 54 references that are in the map. First posted on arXiv 1 Jul 2021.

Builds on

36 of 54 references.

Closest ancestors · 8

Same direction, newest first

  1. 2021Score-based Generative Modeling in Latent Space10 Jun
  2. 2021A Variational Perspective on Diffusion-Based Generative Models and Score Matching5 Jun
  3. 2021Improved Denoising Diffusion Probabilistic Models18 Feb
  4. 2021Maximum Likelihood Training of Score-Based Diffusion Models22 Jan
  5. 2020Score-Based Generative Modeling through Stochastic Differential Equations26 Nov
  6. 2020Denoising Diffusion Implicit Models6 Oct
  7. 2020Denoising Diffusion Probabilistic Models19 Jun
  8. 2020Improved Techniques for Training Score-Based Generative Models16 Jun

Other references · 28

  1. 2015U-Net: Convolutional Networks for Biomedical Image Segmentationcited by 19,604
  2. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,005
  3. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,067
  4. 2013Auto-Encoding Variational Bayescited by 3,980
  5. 2014Sequence to Sequence Learning with Neural Networkscited by 2,974
  6. 2017Progressive Growing of GANs for Improved Quality, Stability, and Variationcited by 2,967
  7. 2020Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domainscited by 2,170
  8. 2016WaveNet: A Generative Model for Raw Audiocited by 1,710
  9. 2018Glow: Generative Flow with Invertible 1x1 Convolutionscited by 1,539
  10. 2019Generating Long Sequences with Sparse Transformerscited by 1,523
  11. 2015LSUN: Construction of a Large-scale Image Dataset using Deep Learning with Humans in the Loopcited by 1,350
  12. 2020DiffWave: A Versatile Diffusion Model for Audio Synthesiscited by 1,339
16 more
  1. 2016Pixel Recurrent Neural Networkscited by 814
  2. 2018Image Transformercited by 770
  3. 2016Improving Variational Inference with Inverse Autoregressive Flowcited by 636
  4. 2020NVAE: A Deep Hierarchical Variational Autoencodercited by 635
  5. 2020WaveGrad: Estimating Gradients for Waveform Generationcited by 620
  6. 2020Efficient Content-Based Sparse Attention with Routing Transformerscited by 469
  7. 2017PixelCNN++: Improving the PixelCNN with Discretized Logistic Mixture Likelihood and Other Modificationscited by 415
  8. 2020Very Deep VAEs Generalize Autoregressive Models and Can Outperform Them on Imagescited by 261
  9. 2019Flow++: Improving Flow-Based Generative Models with Variational Dequantization and Architecture Designcited by 258
  10. 2020Learning Energy-Based Models by Diffusion Recovery Likelihoodcited by 122
  11. 2019Practical Lossless Compression with Latent Variables using Bits Back Codingcited by 85
  12. 2018Generating High Fidelity Images with Subscale Pixel Networks and Multidimensional Upscalingcited by 74
  13. 2019Bit-Swap: Recursive Bits-Back Coding for Lossless Compression with Hierarchical Latent Variablescited by 51
  14. 2021Consistency Regularization for Variational Auto-Encoderscited by 50
  15. 2019HiLLoC: Lossless Image Compression with Hierarchical Latent Variable Modelscited by 36
  16. 2019Compression with Flows via Local Bits-Back Codingcited by 34

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 23; 2022: 129; 2023: 296; 2024: 297; 2025: 246; 2026: 154.

1,145 papers.

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By area:

  1. 2021High-Resolution Image Synthesis with Latent Diffusion Modelscited by 18,694
  2. 2022Photorealistic Text-to-Image Diffusion Models with Deep Language Understandingcited by 6,322
  3. 2022Classifier-Free Diffusion Guidancecited by 5,717
  4. 2023Adding Conditional Control to Text-to-Image Diffusion Modelscited by 5,466
  5. 2022Flow Matching for Generative Modelingcited by 5,426
  6. 2022DreamFusion: Text-to-3D using 2D Diffusioncited by 3,006
  7. 2023Stable Video Diffusion: Scaling Latent Video Diffusion Models to Large Datasetscited by 2,526
  8. 2022Video Diffusion Modelscited by 2,280
  9. 2022DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Stepscited by 1,961
  10. 2022Imagen Video: High Definition Video Generation with Diffusion Modelscited by 1,765
  11. 2021Palette: Image-to-Image Diffusion Modelscited by 1,296
  12. 2022Diffusion Models: A Comprehensive Survey of Methods and Applicationscited by 1,107
38 more
  1. 2022Diffusion Models in Vision: A Surveycited by 1,008
  2. 2022Diffusion-LM Improves Controllable Text Generationcited by 1,006
  3. 2023One-step Diffusion with Distribution Matching Distillationcited by 978
  4. 2023Training Diffusion Models with Reinforcement Learningcited by 939
  5. 2022DPM-Solver++: Fast Solver for Guided Sampling of Diffusion Probabilistic Modelscited by 924
  6. 2024Simple and Effective Masked Diffusion Language Modelscited by 800
  7. 2022On Distillation of Guided Diffusion Modelscited by 721
  8. 2024SiT: Exploring Flow and Diffusion-based Generative Models with Scalable Interpolant Transformerscited by 707
  9. 2022Equivariant Diffusion for Molecule Generation in 3Dcited by 666
  10. 2023ModelScope Text-to-Video Technical Reportcited by 651
  11. 2023Muse: Text-To-Image Generation via Masked Generative Transformerscited by 640
  12. 2024Simplified and Generalized Masked Diffusion for Discrete Datacited by 606
  13. 2022Score Jacobian Chaining: Lifting Pretrained 2D Diffusion Models for 3D Generationcited by 597
  14. 2025Self Forcing: Bridging the Train-Test Gap in Autoregressive Video Diffusioncited by 569
  15. 2022Executing your Commands via Motion Diffusion in Latent Spacecited by 567
  16. 2022LION: Latent Point Diffusion Models for 3D Shape Generationcited by 552
  17. 2021Tackling the Generative Learning Trilemma with Denoising Diffusion GANscited by 515
  18. 2025Dream 7B: Diffusion Large Language Modelscited by 515
  19. 2022Is Conditional Generative Modeling all you need for Decision-Making?cited by 504
  20. 2022DiGress: Discrete Denoising diffusion for graph generationcited by 503
  21. 2022All are Worth Words: A ViT Backbone for Diffusion Modelscited by 498
  22. 2022Diffusion Models for Adversarial Purificationcited by 469
  23. 2023Consistency Trajectory Models: Learning Probability Flow ODE Trajectory of Diffusioncited by 448
  24. 2021Diffusion Autoencoders: Toward a Meaningful and Decodable Representationcited by 447
  25. 2023UniPC: A Unified Predictor-Corrector Framework for Fast Sampling of Diffusion Modelscited by 441
  26. 2022Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptionscited by 440
  27. 2023Directly Fine-Tuning Diffusion Models on Differentiable Rewardscited by 431
  28. 2023Simple diffusion: End-to-end diffusion for high resolution imagescited by 413
  29. 2024From Slow Bidirectional to Fast Autoregressive Video Diffusion Modelscited by 401
  30. 2025Block Diffusion: Interpolating Between Autoregressive and Diffusion Language Modelscited by 385
  31. 2024Your Absorbing Discrete Diffusion Secretly Models the Conditional Distributions of Clean Datacited by 371
  32. 2022PhysDiff: Physics-Guided Human Motion Diffusion Modelcited by 365
  33. 2023I2VGen-XL: High-Quality Image-to-Video Synthesis via Cascaded Diffusion Modelscited by 343
  34. 2022Understanding Diffusion Models: A Unified Perspectivecited by 337
  35. 2022Analytic-DPM: an Analytic Estimate of the Optimal Reverse Variance in Diffusion Probabilistic Modelscited by 331
  36. 2021Score-based diffusion models for accelerated MRIcited by 308
  37. 2021Come-Closer-Diffuse-Faster: Accelerating Conditional Diffusion Models for Inverse Problems through Stochastic Contractioncited by 306
  38. 2023Diffusion Self-Guidance for Controllable Image Generationcited by 305

50 most cited of 1,145 All 1,145 (JSON)

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Close in wording, no citation either way

  1. 2025Diffusion Models: A Mathematical Introduction
  2. 2025Information Theoretic Learning for Diffusion Models with Warm Start
  3. 2024Diffusion Priors for Variational Likelihood Estimation and Image Denoising
  4. 2023The Missing U for Efficient Diffusion Models
  5. 2023A Variational Perspective on Solving Inverse Problems with Diffusion Models
  6. 2024Diffusion Models in Low-Level Vision: A Survey
  7. 2022Convergence of denoising diffusion models under the manifold hypothesis
  8. 2022Diffusion models as plug-and-play priors