mlsanityData to 14 Oct 2026

Learning Diffusion Priors from Observations by Expectation Maximization

arXiv 2405.13712 ↗First posted 22 May 2024cs.LG stat.MLcited by 61

On the map: Diffusion models → Inverse problems and image restoration

Abstract (arXiv)

Diffusion models recently proved to be remarkable priors for Bayesian inverse problems. However, training these models typically requires access to large amounts of clean data, which could prove difficult in some settings. In this work, we present DiEM, a novel method based on the expectation-maximization algorithm for training diffusion models from incomplete and noisy observations only. Unlike previous works, DiEM leads to proper diffusion models, which is crucial for downstream tasks. As part of our methods, we propose and motivate an improved posterior sampling scheme for unconditional diffusion models. We present empirical evidence supporting the effectiveness of our approach.

Key facts. Cited by 61 papers in the map (data through 14 Oct 2026). Placed in Inverse problems and image restoration, one of 14 directions in Diffusion models. Builds on 58 of its 109 references that are in the map. First posted on arXiv 22 May 2024.

Builds on

58 of 109 references.

Closest ancestors · 6

Same direction, newest first

  1. 2024Ambient Diffusion Posterior Sampling: Solving Inverse Problems with Diffusion Models Trained on Corrupted Data13 Mar
  2. 2024Improving Diffusion Models for Inverse Problems Using Optimal Posterior Covariance3 Feb
  3. 2023Denoising Diffusion Models for Plug-and-Play Image Restoration15 May
  4. 2023Inversion by Direct Iteration: An Alternative to Denoising Diffusion for Image Restoration20 Mar
  5. 2022Diffusion Posterior Sampling for General Noisy Inverse Problems29 Sep
  6. 2022Denoising Diffusion Restoration Models27 Jan

Other references · 52

  1. 2017Attention Is All You Needcited by 60,361
  2. 2015Deep Residual Learning for Image Recognitioncited by 52,096
  3. 2014Adam: A Method for Stochastic Optimizationcited by 37,240
  4. 2020Denoising Diffusion Probabilistic Modelscited by 21,076
  5. 2019PyTorch: An Imperative Style, High-Performance Deep Learning Librarycited by 20,305
  6. 2015U-Net: Convolutional Networks for Biomedical Image Segmentationcited by 19,713
  7. 2018The Unreasonable Effectiveness of Deep Features as a Perceptual Metriccited by 11,177
  8. 2020Denoising Diffusion Implicit Modelscited by 9,781
  9. 2020Score-Based Generative Modeling through Stochastic Differential Equationscited by 9,122
  10. 2021Diffusion Models Beat GANs on Image Synthesiscited by 8,255
  11. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,058
  12. 2022Scalable Diffusion Models with Transformerscited by 6,054
38 more
  1. 2018A Style-Based Generator Architecture for Generative Adversarial Networkscited by 6,013
  2. 2022Classifier-Free Diffusion Guidancecited by 5,847
  3. 2022Flow Matching for Generative Modelingcited by 5,702
  4. 2016Layer Normalizationcited by 5,669
  5. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,101
  6. 2022Elucidating the Design Space of Diffusion-Based Generative Modelscited by 3,045
  7. 2016Improved Techniques for Training GANscited by 2,970
  8. 2015Variational Inference with Normalizing Flowscited by 2,170
  9. 2016Real-Time Single Image and Video Super-Resolution Using an Efficient Sub-Pixel Convolutional Neural Networkcited by 1,204
  10. 2017Sigmoid-Weighted Linear Units for Neural Network Function Approximation in Reinforcement Learningcited by 950
  11. 2017The Perception-Distortion Tradeoffcited by 587
  12. 2022Fast Sampling of Diffusion Models with Exponential Integratorcited by 532
  13. 2021Solving Inverse Problems in Medical Imaging with Score-Based Generative Modelscited by 521
  14. 2018fastMRI: An Open Dataset and Benchmarks for Accelerated MRIcited by 470
  15. 2023Universal Guidance for Diffusion Modelscited by 437
  16. 2017Learning a Variational Network for Reconstruction of Accelerated MRI Datacited by 342
  17. 2021Robust Compressed Sensing MRI with Deep Generative Priorscited by 286
  18. 2023Practical and Asymptotically Exact Conditional Sampling in Diffusion Modelscited by 232
  19. 2021SNIPS: Solving Noisy Inverse Problems Stochasticallycited by 191
  20. 2023Manifold Preserving Guided Diffusioncited by 157
  21. 2014Statistical guarantees for the EM algorithm: From population to sample-based analysiscited by 133
  22. 2021Noise2Score: Tweedie's Approach to Self-Supervised Image Denoising without Clean Imagescited by 113
  23. 2023Ambient Diffusion: Learning Clean Distributions from Corrupted Datacited by 111
  24. 2023Score-based Data Assimilationcited by 96
  25. 2020Statistical and Topological Properties of Sliced Probability Divergencescited by 85
  26. 2020Faster Wasserstein Distance Estimation with the Sinkhorn Divergencecited by 73
  27. 2021Estimating High Order Gradients of the Data Distribution by Denoisingcited by 64
  28. 2024Consistent Diffusion Meets Tweedie: Training Exact Ambient Diffusion Models with Noisy Datacited by 44
  29. 2023GSURE-Based Diffusion Model Training with Corrupted Datacited by 40
  30. 2019OmniFold: A Method to Simultaneously Unfold All Observablescited by 31
  31. 2023User-defined Event Sampling and Uncertainty Quantification in Diffusion Models for Physical Dynamical Systemscited by 30
  32. 2017Adversarial Variational Optimization of Non-Differentiable Simulatorscited by 22
  33. 2020Neural Empirical Bayes: Source Distribution Estimation and its Applications to Simulation-Based Inferencecited by 21
  34. 2022Posterior samples of source galaxies in strong gravitational lenses with score-based priorscited by 20
  35. 2014Two Modeling Strategies for Empirical Bayes Estimationcited by 16
  36. 2024Sourcerer: Sample-based Maximum Entropy Source Distribution Estimationcited by 9
  37. 2022Strong Lensing Source Reconstruction Using Continuous Neural Fieldscited by 6
  38. 2020Density Deconvolution with Normalizing Flowscited by 4

50 most cited of 52 All 52 (JSON)

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2024: 7; 2025: 27; 2026: 27.

61 papers.

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  1. 2024A Survey on Diffusion Models for Inverse Problemscited by 180
  2. 2025On the Guidance of Flow Matchingcited by 45
  3. 2024Generative Data Assimilation of Sparse Weather Station Observations at Kilometer Scalescited by 27
  4. 2025Distributional Diffusion Models with Scoring Rulescited by 25
  5. 2025Appa: Bending Weather Dynamics with Latent Diffusion Models for Global Data Assimilationcited by 20
  6. 2025Lost in Latent Space: An Empirical Study of Latent Diffusion Models for Physics Emulationcited by 19
  7. 2024How much is a noisy image worth? Data Scaling Laws for Ambient Diffusioncited by 18
  8. 2025Ambient Diffusion Omni: Training Good Models with Bad Datacited by 18
  9. 2025Does Generation Require Memorization? Creative Diffusion Models using Ambient Diffusioncited by 17
  10. 2025Solving Inverse Problems with FLAIRcited by 17
  11. 2025RNE: plug-and-play diffusion inference-time control and energy-based trainingcited by 17
  12. 2025Diffusion models for inverse problemscited by 16
38 more
  1. 2024Free Hunch: Denoiser Covariance Estimation for Diffusion Models Without Extra Costscited by 14
  2. 2024Improving Probabilistic Diffusion Models With Optimal Diagonal Covariance Matchingcited by 13
  3. 2024Frequency-Guided Posterior Sampling for Diffusion-Based Image Restorationcited by 11
  4. 2025LanPaint: Training-Free Diffusion Inpainting with Asymptotically Exact and Fast Conditional Samplingcited by 11
  5. 2025DiffEM: Learning from Corrupted Data with Diffusion Models via Expectation Maximizationcited by 6
  6. 2025Generative Modeling from Black-box Corruptions via Self-Consistent Stochastic Interpolantscited by 6
  7. 2026Reward-Guided Discrete Diffusion via Clean-Sample Markov Chain for Molecule and Biological Sequence Designcited by 6
  8. 2026D-Flow SGLD: Source-Space Posterior Sampling for Scientific Inverse Problems with Flow Matchingcited by 6
  9. 2025Inverse Flow and Consistency Modelscited by 5
  10. 2025Training-Free Data Assimilation with GenCastcited by 5
  11. 2025DAISI: Data Assimilation with Inverse Sampling using Stochastic Interpolantscited by 5
  12. 2025Efficient Zero-Shot Inpainting with Decoupled Diffusion Guidancecited by 5
  13. 2026Self-Supervised Learning from Noisy and Incomplete Datacited by 5
  14. 2026Count Bridges enable Modeling and Deconvolving Transcriptomic Datacited by 4
  15. 2026Training-Free Bayesian Filtering with Generative Emulatorscited by 4
  16. 2026Ambient Diffusion Policy: Imitation Learning from Suboptimal Data in Roboticscited by 4
  17. 2025PriorGuide: Test-Time Prior Adaptation for Simulation-Based Inferencecited by 3
  18. 2026WIND: Weather Inverse Diffusion for Zero-Shot Atmospheric Modelingcited by 3
  19. 2025A Data-Driven Prism: Multi-View Source Separation with Diffusion Model Priorscited by 2
  20. 2025Local MAP Sampling for Diffusion Modelscited by 2
  21. 2025Coupled Data and Measurement Space Dynamics for Enhanced Diffusion Posterior Samplingcited by 2
  22. 2025Latent Refinement via Flow Matching for Training-free Linear Inverse Problem Solvingcited by 2
  23. 2026Ambient Dataloops: Generative Models for Dataset Refinementcited by 2
  24. 2026Predict-Project-Renoise: Sampling Diffusion Models under Hard Constraintscited by 2
  25. 2026Noise-Adaptive Diffusion Sampling for Inverse Problems Without Task-Specific Tuningcited by 2
  26. 2026Exact Posterior Score Estimation for Solving Linear Inverse Problemscited by 2
  27. 2026Pointwise is Pointless? A Multimodal Ablation Study for Precipitation Nowcasting with Graph Neural Networkscited by 2
  28. 2025MAP-based Problem-Agnostic diffusion model for Inverse Problemscited by 1
  29. 2025Diffusion Models for Inverse Problems in the Exponential Familycited by 1
  30. 2026LLM Flow Processes for Text-Conditioned Regressioncited by 1
  31. 2026Riemannian AmbientFlow: Towards Simultaneous Manifold Learning and Generative Modeling from Corrupted Datacited by 1
  32. 2026Outlier-robust Diffusion Posterior Sampling for Bayesian Inverse Problemscited by 1
  33. 2026Fast and Robust Likelihood-Guided Diffusion Posterior Sampling with Amortized Variational Inferencecited by 1
  34. 2026Optimizing Diffusion Priors in Image Reconstruction from a Single Observationcited by 1
  35. 2026ADDA: a Modular Framework for Representing, Simulating and Assimilating Dynamics with End-to-end Differentiabilitycited by 1
  36. 2024Improving Decoupled Posterior Sampling for Inverse Problems using Data Consistency Constraintcited by 0
  37. 2025Exact Evaluation of the Accuracy of Diffusion Models for Inverse Problems with Gaussian Data Distributionscited by 0
  38. 2025Zero-shot Human Pose Estimation using Diffusion-based Inverse solverscited by 0

50 most cited of 61 All 61 (JSON)

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  1. 2024Learning Diffusion Model from Noisy Measurement using Principled Expectation-Maximization Method
  2. 2024Divide-and-Conquer Posterior Sampling for Denoising Diffusion Priors
  3. 2025A Mixture-Based Framework for Guiding Diffusion Models
  4. 2025Unifying and extending Diffusion Models through PDEs for solving Inverse Problems
  5. 2025Solving Inverse Problems via Diffusion-Based Priors: An Approximation-Free Ensemble Sampling Approach
  6. 2024An Expectation-Maximization Algorithm for Training Clean Diffusion Models from Corrupted Observations
  7. 2025A Recovery Theory for Diffusion Priors: Deterministic Analysis of the Implicit Prior Algorithm
  8. 2022Diffusion models as plug-and-play priors