# On Maximum-a-Posteriori estimation with Plug & Play priors and stochastic gradient descent arXiv: https://arxiv.org/abs/2201.06133 ยท HTML: https://mlsanity.com/paper/2201.06133/ ## Key facts - Cited by 14 papers in the map (data through 14 Oct 2026). - Builds on 35 of its 81 references that are in the map. - First posted on arXiv 16 Jan 2022. ## Builds on (most cited first, 30 of 35) - [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014) - [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) - [Convex Optimization: Algorithms and Complexity](https://mlsanity.com/paper/1405.4980/) (2014) - [Noise2Noise: Learning Image Restoration without Clean Data](https://mlsanity.com/paper/1803.04189/) (2018) - [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) - [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) - [Plug-and-Play Methods Provably Converge with Properly Trained Denoisers](https://mlsanity.com/paper/1905.05406/) (2019) - [Trainable Nonlinear Reaction Diffusion: A Flexible Framework for Fast and Effective Image Restoration](https://mlsanity.com/paper/1508.02848/) (2015) - [A Sufficient Condition for Convergences of Adam and RMSProp](https://mlsanity.com/paper/1811.09358/) (2018) - [Gradient Step Denoiser for convergent Plug-and-Play](https://mlsanity.com/paper/2110.03220/) (2021) - [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) - [Bayesian imaging using Plug & Play priors: when Langevin meets Tweedie](https://mlsanity.com/paper/2103.04715/) (2021) - [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) - [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) - [An Online Plug-and-Play Algorithm for Regularized Image Reconstruction](https://mlsanity.com/paper/1809.04693/) (2018) - [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) - [Scalable Plug-and-Play ADMM with Convergence Guarantees](https://mlsanity.com/paper/2006.03224/) (2020) - [Unrolled Optimization with Deep Priors](https://mlsanity.com/paper/1705.08041/) (2017) - [Provable Convergence of Plug-and-Play Priors with MMSE denoisers](https://mlsanity.com/paper/2005.07685/) (2020) - [Scalable Bayesian uncertainty quantification in imaging inverse problems via convex optimization](https://mlsanity.com/paper/1803.00889/) (2018) ## Built on by (most cited first, 14 of 14) - [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) - [Plug-and-Play image restoration with Stochastic deNOising REgularization](https://mlsanity.com/paper/2402.01779/) (2024) - [Thin On-Sensor Nanophotonic Array Cameras](https://mlsanity.com/paper/2308.02797/) (2023) - [Deep Model-Based Super-Resolution with Non-uniform Blur](https://mlsanity.com/paper/2204.10109/) (2022) - [Solving Linear Inverse Problems using Higher-Order Annealed Langevin Diffusion](https://mlsanity.com/paper/2305.05014/) (2023) - [Learning pseudo-contractive denoisers for inverse problems](https://mlsanity.com/paper/2402.05637/) (2024) - [MAP Estimation with Denoisers: Convergence Rates and Guarantees](https://mlsanity.com/paper/2507.15397/) (2025) - [Non-Log-Concave and Nonsmooth Sampling via Langevin Monte Carlo Algorithms](https://mlsanity.com/paper/2305.15988/) (2023) - [Provably Convergent Plug & Play Linearized ADMM, applied to Deblurring Spatially Varying Kernels](https://mlsanity.com/paper/2210.10605/) (2022) - [Beyond MMSE: Enhancing PnP Restoration with ProxiMAP](https://mlsanity.com/paper/2605.16396/) (2026) - [Video Restoration with a Deep Plug-and-Play Prior](https://mlsanity.com/paper/2209.02854/) (2022) - [Plug-and-Play Algorithm Convergence Analysis From The Standpoint of Stochastic Differential Equation](https://mlsanity.com/paper/2404.13866/) (2024) - [Beyond sparse denoising in frames: minimax estimation with a scattering transform](https://mlsanity.com/paper/2510.19612/) (2025) ## Abstract (arXiv) Bayesian methods to solve imaging inverse problems usually combine an explicit data likelihood function with a prior distribution that explicitly models expected properties of the solution. Many kinds of priors have been explored in the literature, from simple ones expressing local properties to more involved ones exploiting image redundancy at a non-local scale. In a departure from explicit modelling, several recent works have proposed and studied the use of implicit priors defined by an image denoising algorithm. This approach, commonly known as Plug & Play (PnP) regularisation, can deliver remarkably accurate results, particularly when combined with state-of-the-art denoisers based on convolutional neural networks. However, the theoretical analysis of PnP Bayesian models and algorithms is difficult and works on the topic often rely on unrealistic assumptions on the properties of the image denoiser. This papers studies maximum-a-posteriori (MAP) estimation for Bayesian models with PnP priors. We first consider questions related to existence, stability and well-posedness, and then present a convergence proof for MAP computation by PnP stochastic gradient descent (PnP-SGD) under realistic assumptions on the denoiser used. We report a range of imaging experiments demonstrating PnP-SGD as well as comparisons with other PnP schemes.