mlsanityData to 7 Oct 2026

Bayesian imaging using Plug & Play priors: when Langevin meets Tweedie

arXiv 2103.04715 ↗First posted 8 Mar 2021stat.ME cs.CV eess.IVcited by 96

Not on the map. Why?

Cited by 28 in Diffusion models, but placed by its own abstract, which didn’t fit.

Abstract (arXiv)

Since the seminal work of Venkatakrishnan et al. in 2013, Plug & Play (PnP) methods have become ubiquitous in Bayesian imaging. These methods derive Minimum Mean Square Error (MMSE) or Maximum A Posteriori (MAP) estimators for inverse problems in imaging by combining an explicit likelihood function with a prior that is implicitly defined by an image denoising algorithm. The PnP algorithms proposed in the literature mainly differ in the iterative schemes they use for optimisation or for sampling. In the case of optimisation schemes, some recent works guarantee the convergence to a fixed point, albeit not necessarily a MAP estimate. In the case of sampling schemes, to the best of our knowledge, there is no known proof of convergence. There also remain important open questions regarding whether the underlying Bayesian models and estimators are well defined, well-posed, and have the basic regularity properties required to support these numerical schemes. To address these limitations, this paper develops theory, methods, and provably convergent algorithms for performing Bayesian inference with PnP priors. We introduce two algorithms: 1) PnP-ULA (Unadjusted Langevin Algorithm) for Monte Carlo sampling and MMSE inference; and 2) PnP-SGD (Stochastic Gradient Descent) for MAP inference. Using recent results on the quantitative convergence of Markov chains, we establish detailed convergence guarantees for these two algorithms under realistic assumptions on the denoising operators used, with special attention to denoisers based on deep neural networks. We also show that these algorithms approximately target a decision-theoretically optimal Bayesian model that is well-posed. The proposed algorithms are demonstrated on several canonical problems such as image deblurring, inpainting, and denoising, where they are used for point estimation as well as for uncertainty visualisation and quantification.

Key facts. Cited by 96 papers in the map (data through 7 Oct 2026). Builds on 45 of its 99 references that are in the map. First posted on arXiv 8 Mar 2021.

Builds on

45 of 99 references.

References · 45

  1. 2020Denoising Diffusion Probabilistic Modelscited by 20,851
  2. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,067
  3. 2018Spectral Normalization for Generative Adversarial Networkscited by 1,618
  4. 2016Beyond a Gaussian Denoiser: Residual Learning of Deep CNN for Image Denoisingcited by 1,352
  5. 2016Optimization Methods for Large-Scale Machine Learningcited by 1,140
  6. 2017FFDNet: Toward a Fast and Flexible Solution for CNN based Image Denoisingcited by 509
  7. 2017Learning Deep CNN Denoiser Prior for Image Restorationcited by 407
  8. 2014Stochastic Gradient Hamiltonian Monte Carlocited by 341
  9. 2017Compressed Sensing using Generative Modelscited by 337
  10. 2014Breaking the Curse of Dimensionality with Convex Neural Networkscited by 329
  11. 2016The Little Engine that Could: Regularization by Denoising (RED)cited by 303
  12. 2016Plug-and-Play ADMM for Image Restoration: Fixed Point Convergence and Applicationscited by 244
33 more
  1. 2019On instabilities of deep learning in image reconstruction - Does AI come at a cost?cited by 211
  2. 2014Theoretical guarantees for approximate sampling from smooth and log-concave densitiescited by 206
  3. 2019Plug-and-Play Methods Provably Converge with Properly Trained Denoiserscited by 197
  4. 2015Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithmcited by 195
  5. 2015Trainable Nonlinear Reaction Diffusion: A Flexible Framework for Fast and Effective Image Restorationcited by 194
  6. 2017Underdamped Langevin MCMC: A non-asymptotic analysiscited by 131
  7. 2018A Sufficient Condition for Convergences of Adam and RMSPropcited by 124
  8. 2017Learning Proximal Operators: Using Denoising Networks for Regularizing Inverse Imaging Problemscited by 107
  9. 2018Regularization by Denoising: Clarifications and New Interpretationscited by 102
  10. 2018DeepISP: Towards Learning an End-to-End Image Processing Pipelinecited by 88
  11. 2016Efficient Bayesian computation by proximal Markov chain Monte Carlo: when Langevin meets Moreaucited by 83
  12. 2015Plug-and-Play Priors for Bright Field Electron Tomography and Sparse Interpolationcited by 78
  13. 2020Solving Linear Inverse Problems Using the Prior Implicit in a Denoisercited by 78
  14. 2019Plug and play methods for magnetic resonance imaging (long version)cited by 75
  15. 2014From Denoising to Compressed Sensingcited by 67
  16. 2020Learning Maximally Monotone Operators for Image Recoverycited by 62
  17. 2017Plug-and-Play Unplugged: Optimization Free Reconstruction using Consensus Equilibriumcited by 58
  18. 2020Regularization by Denoising via Fixed-Point Projection (RED-PRO)cited by 55
  19. 2020Learned convex regularizers for inverse problemscited by 54
  20. 2020Total Deep Variation for Linear Inverse Problemscited by 53
  21. 2013Proximal Markov chain Monte Carlo algorithmscited by 46
  22. 2021Stochastic Image Denoising by Sampling from the Posterior Distributioncited by 45
  23. 2017Unrolled Optimization with Deep Priorscited by 40
  24. 2020Provable Convergence of Plug-and-Play Priors with MMSE denoiserscited by 39
  25. 2017Deep Mean-Shift Priors for Image Restorationcited by 34
  26. 2018Scalable Bayesian uncertainty quantification in imaging inverse problems via convex optimizationcited by 22
  27. 2019On the well-posedness of Bayesian inverse problemscited by 20
  28. 2017Asymptotic Bias of Stochastic Gradient Searchcited by 19
  29. 2017Uncertainty quantification for radio interferometric imaging: I. proximal MCMC methodscited by 13
  30. 2014Approximate Models and Robust Decisionscited by 10
  31. 2019Neumann Networks for Inverse Problems in Imagingcited by 9
  32. 2017A Bayesian Hyperprior Approach for Joint Image Denoising and Interpolation, with an Application to HDR Imagingcited by 7
  33. 2018Scene-Adapted Plug-and-Play Algorithm with Guaranteed Convergence: Applications to Data Fusion in Imagingcited by 5

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 3; 2022: 12; 2023: 27; 2024: 23; 2025: 21; 2026: 10.

96 papers.

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  1. 2022Denoising Diffusion Restoration Modelscited by 951
  2. 2021Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modelingcited by 594
  3. 2022Improving Diffusion Models for Inverse Problems using Manifold Constraintscited by 501
  4. 2023Denoising Diffusion Models for Plug-and-Play Image Restorationcited by 335
  5. 2021SNIPS: Solving Noisy Inverse Problems Stochasticallycited by 191
  6. 2023Inversion by Direct Iteration: An Alternative to Denoising Diffusion for Image Restorationcited by 164
  7. 2023Score-Based Diffusion Models as Principled Priors for Inverse Imagingcited by 126
  8. 2021Gradient Step Denoiser for convergent Plug-and-Playcited by 107
  9. 2022Soft Diffusion: Score Matching for General Corruptionscited by 101
  10. 2024Principled Probabilistic Imaging using Diffusion Models as Plug-and-Play Priorscited by 93
  11. 2022Multiscale Structure Guided Diffusion for Image Deblurringcited by 76
  12. 2024Provably Robust Score-Based Diffusion Posterior Sampling for Plug-and-Play Image Reconstructioncited by 73
38 more
  1. 2023Image Denoising: The Deep Learning Revolution and Beyond -- A Survey Paper --cited by 71
  2. 2023Provable Probabilistic Imaging using Score-Based Generative Priorscited by 51
  3. 2022Learned reconstruction methods with convergence guaranteescited by 46
  4. 2023Fast Diffusion EM: a diffusion model for blind inverse problems with application to deconvolutioncited by 36
  5. 2024Denoising: A Powerful Building-Block for Imaging, Inverse Problems, and Machine Learningcited by 35
  6. 2023Equivariant plug-and-play image reconstructioncited by 26
  7. 2023Learning multi-scale local conditional probability models of imagescited by 21
  8. 2023Equivariant Bootstrapping for Uncertainty Quantification in Imaging Inverse Problemscited by 17
  9. 2024Plug-and-Play image restoration with Stochastic deNOising REgularizationcited by 17
  10. 2024Repulsive Latent Score Distillation for Solving Inverse Problemscited by 15
  11. 2022On Maximum-a-Posteriori estimation with Plug & Play priors and stochastic gradient descentcited by 14
  12. 2025LATINO-PRO: LAtent consisTency INverse sOlver with PRompt Optimizationcited by 14
  13. 2025When Scores Learn Geometry: Rate Separations under the Manifold Hypothesiscited by 13
  14. 2022Posterior-Variance-Based Error Quantification for Inverse Problems in Imagingcited by 12
  15. 2025Learning Regularization Functionals for Inverse Problems: A Comparative Studycited by 12
  16. 2022The split Gibbs sampler revisited: improvements to its algorithmic structure and augmented target distributioncited by 11
  17. 2023A Conditional Normalizing Flow for Accelerated Multi-Coil MR Imagingcited by 10
  18. 2023Prior Mismatch and Adaptation in PnP-ADMM with a Nonconvex Convergence Analysiscited by 10
  19. 2022Bayesian Inversion for Nonlinear Imaging Models using Deep Generative Priorscited by 9
  20. 2023Learning from small data sets: Patch-based regularizers in inverse problems for image reconstructioncited by 9
  21. 2024Regularization by denoising: Bayesian model and Langevin-within-split Gibbs samplingcited by 9
  22. 2024Zero-Shot Adaptation for Approximate Posterior Sampling of Diffusion Models in Inverse Problemscited by 9
  23. 2023Block Coordinate Plug-and-Play Methods for Blind Inverse Problemscited by 8
  24. 2023Plug-and-Play Posterior Sampling under Mismatched Measurement and Prior Modelscited by 8
  25. 2022PnP-ReG: Learned Regularizing Gradient for Plug-and-Play Gradient Descentcited by 7
  26. 2023Bayesian Posterior Perturbation Analysis with Integral Probability Metricscited by 7
  27. 2023Noise-Free Sampling Algorithms via Regularized Wasserstein Proximalscited by 7
  28. 2024Taming Score-Based Diffusion Priors for Infinite-Dimensional Nonlinear Inverse Problemscited by 7
  29. 2024Wasserstein Gradient Flows of MMD Functionals with Distance Kernel and Cauchy Problems on Quantile Functionscited by 7
  30. 2025Deep Equilibrium models for Poisson Imaging Inverse problems via Mirror Descentcited by 7
  31. 2025Landing with the Score: Riemannian Optimization through Denoisingcited by 7
  32. 2023Restoration based Generative Modelscited by 6
  33. 2023Model-corrected learned primal-dual models for fast limited-view photoacoustic tomographycited by 6
  34. 2023Solving Linear Inverse Problems using Higher-Order Annealed Langevin Diffusioncited by 6
  35. 2024Do Bayesian imaging methods report trustworthy probabilities?cited by 6
  36. 2024Empirical Bayesian image restoration by Langevin sampling with a denoising diffusion implicit priorcited by 6
  37. 2024FiRe: Fixed-points of Restoration Priors for Solving Inverse Problemscited by 6
  38. 2024Plug-and-Play Priors as a Score-Based Methodcited by 5

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