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

arXiv: https://arxiv.org/abs/2103.04715 · HTML: https://mlsanity.com/paper/2103.04715/

## Key facts

- Cited by 97 papers in the map (data through 14 Oct 2026).
- Builds on 45 of its 99 references that are in the map.
- First posted on arXiv 8 Mar 2021.

## Builds on (most cited first, 30 of 45)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Beyond a Gaussian Denoiser: Residual Learning of Deep CNN for Image Denoising](https://mlsanity.com/paper/1608.03981/) (2016)
- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [FFDNet: Toward a Fast and Flexible Solution for CNN based Image Denoising](https://mlsanity.com/paper/1710.04026/) (2017)
- [Learning Deep CNN Denoiser Prior for Image Restoration](https://mlsanity.com/paper/1704.03264/) (2017)
- [Stochastic Gradient Hamiltonian Monte Carlo](https://mlsanity.com/paper/1402.4102/) (2014)
- [Compressed Sensing using Generative Models](https://mlsanity.com/paper/1703.03208/) (2017)
- [Breaking the Curse of Dimensionality with Convex Neural Networks](https://mlsanity.com/paper/1412.8690/) (2014)
- [The Little Engine that Could: Regularization by Denoising (RED)](https://mlsanity.com/paper/1611.02862/) (2016)
- [Plug-and-Play ADMM for Image Restoration: Fixed Point Convergence and Applications](https://mlsanity.com/paper/1605.01710/) (2016)
- [On instabilities of deep learning in image reconstruction - Does AI come at a cost?](https://mlsanity.com/paper/1902.05300/) (2019)
- [Theoretical guarantees for approximate sampling from smooth and log-concave densities](https://mlsanity.com/paper/1412.7392/) (2014)
- [Plug-and-Play Methods Provably Converge with Properly Trained Denoisers](https://mlsanity.com/paper/1905.05406/) (2019)
- [Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1507.05021/) (2015)
- [Trainable Nonlinear Reaction Diffusion: A Flexible Framework for Fast and Effective Image Restoration](https://mlsanity.com/paper/1508.02848/) (2015)
- [Underdamped Langevin MCMC: A non-asymptotic analysis](https://mlsanity.com/paper/1707.03663/) (2017)
- [A Sufficient Condition for Convergences of Adam and RMSProp](https://mlsanity.com/paper/1811.09358/) (2018)
- [Learning Proximal Operators: Using Denoising Networks for Regularizing Inverse Imaging Problems](https://mlsanity.com/paper/1704.03488/) (2017)
- [Regularization by Denoising: Clarifications and New Interpretations](https://mlsanity.com/paper/1806.02296/) (2018)
- [DeepISP: Towards Learning an End-to-End Image Processing Pipeline](https://mlsanity.com/paper/1801.06724/) (2018)
- [Efficient Bayesian computation by proximal Markov chain Monte Carlo: when Langevin meets Moreau](https://mlsanity.com/paper/1612.07471/) (2016)
- [Solving Linear Inverse Problems Using the Prior Implicit in a Denoiser](https://mlsanity.com/paper/2007.13640/) (2020)
- [Plug-and-Play Priors for Bright Field Electron Tomography and Sparse Interpolation](https://mlsanity.com/paper/1512.07331/) (2015)
- [Plug and play methods for magnetic resonance imaging (long version)](https://mlsanity.com/paper/1903.08616/) (2019)
- [From Denoising to Compressed Sensing](https://mlsanity.com/paper/1406.4175/) (2014)
- [Learning Maximally Monotone Operators for Image Recovery](https://mlsanity.com/paper/2012.13247/) (2020)
- [Plug-and-Play Unplugged: Optimization Free Reconstruction using Consensus Equilibrium](https://mlsanity.com/paper/1705.08983/) (2017)
- [Regularization by Denoising via Fixed-Point Projection (RED-PRO)](https://mlsanity.com/paper/2008.00226/) (2020)

## Built on by (most cited first, 30 of 97)

- [Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2201.11793/) (2022)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [Improving Diffusion Models for Inverse Problems using Manifold Constraints](https://mlsanity.com/paper/2206.00941/) (2022)
- [Denoising Diffusion Models for Plug-and-Play Image Restoration](https://mlsanity.com/paper/2305.08995/) (2023)
- [SNIPS: Solving Noisy Inverse Problems Stochastically](https://mlsanity.com/paper/2105.14951/) (2021)
- [Inversion by Direct Iteration: An Alternative to Denoising Diffusion for Image Restoration](https://mlsanity.com/paper/2303.11435/) (2023)
- [Score-Based Diffusion Models as Principled Priors for Inverse Imaging](https://mlsanity.com/paper/2304.11751/) (2023)
- [Gradient Step Denoiser for convergent Plug-and-Play](https://mlsanity.com/paper/2110.03220/) (2021)
- [Soft Diffusion: Score Matching for General Corruptions](https://mlsanity.com/paper/2209.05442/) (2022)
- [Principled Probabilistic Imaging using Diffusion Models as Plug-and-Play Priors](https://mlsanity.com/paper/2405.18782/) (2024)
- [Multiscale Structure Guided Diffusion for Image Deblurring](https://mlsanity.com/paper/2212.01789/) (2022)
- [Provably Robust Score-Based Diffusion Posterior Sampling for Plug-and-Play Image Reconstruction](https://mlsanity.com/paper/2403.17042/) (2024)
- [Image Denoising: The Deep Learning Revolution and Beyond -- A Survey Paper --](https://mlsanity.com/paper/2301.03362/) (2023)
- [Provable Probabilistic Imaging using Score-Based Generative Priors](https://mlsanity.com/paper/2310.10835/) (2023)
- [Learned reconstruction methods with convergence guarantees](https://mlsanity.com/paper/2206.05431/) (2022)
- [Fast Diffusion EM: a diffusion model for blind inverse problems with application to deconvolution](https://mlsanity.com/paper/2309.00287/) (2023)
- [Denoising: A Powerful Building-Block for Imaging, Inverse Problems, and Machine Learning](https://mlsanity.com/paper/2409.06219/) (2024)
- [Equivariant plug-and-play image reconstruction](https://mlsanity.com/paper/2312.01831/) (2023)
- [Learning multi-scale local conditional probability models of images](https://mlsanity.com/paper/2303.02984/) (2023)
- [Equivariant Bootstrapping for Uncertainty Quantification in Imaging Inverse Problems](https://mlsanity.com/paper/2310.11838/) (2023)
- [Plug-and-Play image restoration with Stochastic deNOising REgularization](https://mlsanity.com/paper/2402.01779/) (2024)
- [LATINO-PRO: LAtent consisTency INverse sOlver with PRompt Optimization](https://mlsanity.com/paper/2503.12615/) (2025)
- [Repulsive Latent Score Distillation for Solving Inverse Problems](https://mlsanity.com/paper/2406.16683/) (2024)
- [Learning Regularization Functionals for Inverse Problems: A Comparative Study](https://mlsanity.com/paper/2510.01755/) (2025)
- [On Maximum-a-Posteriori estimation with Plug & Play priors and stochastic gradient descent](https://mlsanity.com/paper/2201.06133/) (2022)
- [When Scores Learn Geometry: Rate Separations under the Manifold Hypothesis](https://mlsanity.com/paper/2509.24912/) (2025)
- [Posterior-Variance-Based Error Quantification for Inverse Problems in Imaging](https://mlsanity.com/paper/2212.12499/) (2022)
- [The split Gibbs sampler revisited: improvements to its algorithmic structure and augmented target distribution](https://mlsanity.com/paper/2206.13894/) (2022)
- [Prior Mismatch and Adaptation in PnP-ADMM with a Nonconvex Convergence Analysis](https://mlsanity.com/paper/2310.00133/) (2023)
- [A Conditional Normalizing Flow for Accelerated Multi-Coil MR Imaging](https://mlsanity.com/paper/2306.01630/) (2023)

## 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.