# Fixed-Point and Objective Convergence of Plug-and-Play Algorithms arXiv: https://arxiv.org/abs/2104.10348 ยท HTML: https://mlsanity.com/paper/2104.10348/ ## Key facts - Cited by 19 papers in the map (data through 14 Oct 2026). - Builds on 29 of its 65 references that are in the map. - First posted on arXiv 21 Apr 2021. ## Builds on (most cited first, 29 of 29) - [Beyond a Gaussian Denoiser: Residual Learning of Deep CNN for Image Denoising](https://mlsanity.com/paper/1608.03981/) (2016) - [Learning Deep CNN Denoiser Prior for Image Restoration](https://mlsanity.com/paper/1704.03264/) (2017) - [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) - [Deep Plug-and-Play Super-Resolution for Arbitrary Blur Kernels](https://mlsanity.com/paper/1903.12529/) (2019) - [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) - [DeepRED: Deep Image Prior Powered by RED](https://mlsanity.com/paper/1903.10176/) (2019) - [Denoising Prior Driven Deep Neural Network for Image Restoration](https://mlsanity.com/paper/1801.06756/) (2018) - [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) - [One Network to Solve Them All --- Solving Linear Inverse Problems using Deep Projection Models](https://mlsanity.com/paper/1703.09912/) (2017) - [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) - [Provable Convergence of Plug-and-Play Priors with MMSE denoisers](https://mlsanity.com/paper/2005.07685/) (2020) - [Deep Mean-Shift Priors for Image Restoration](https://mlsanity.com/paper/1709.03749/) (2017) - [Block Coordinate Regularization by Denoising](https://mlsanity.com/paper/1905.05113/) (2019) - [Algorithmic Guarantees for Inverse Imaging with Untrained Network Priors](https://mlsanity.com/paper/1906.08763/) (2019) - [Poisson Inverse Problems by the Plug-and-Play scheme](https://mlsanity.com/paper/1511.02500/) (2015) - [Performance Analysis of Plug-and-Play ADMM: A Graph Signal Processing Perspective](https://mlsanity.com/paper/1809.00020/) (2018) - [Plug-and-play ISTA converges with kernel denoisers](https://mlsanity.com/paper/2004.03145/) (2020) - [On the Proof of Fixed-Point Convergence for Plug-and-Play ADMM](https://mlsanity.com/paper/1910.14325/) (2019) - [Multi-resolution Data Fusion for Super-Resolution Electron Microscopy](https://mlsanity.com/paper/1612.00874/) (2016) - [Monotone Operator Theory in Convex Optimization](https://mlsanity.com/paper/1802.02694/) (2018) - [A New Recurrent Plug-and-Play Prior Based on the Multiple Self-Similarity Network](https://mlsanity.com/paper/1907.11793/) (2019) - [Linearized ADMM and Fast Nonlocal Denoising for Efficient Plug-and-Play Restoration](https://mlsanity.com/paper/1901.06110/) (2019) ## Built on by (most cited first, 19 of 19) - [Gradient Step Denoiser for convergent Plug-and-Play](https://mlsanity.com/paper/2110.03220/) (2021) - [Proximal Denoiser for Convergent Plug-and-Play Optimization with Nonconvex Regularization](https://mlsanity.com/paper/2201.13256/) (2022) - [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) - [Provably Convergent Plug-and-Play Quasi-Newton Methods](https://mlsanity.com/paper/2303.07271/) (2023) - [On Maximum-a-Posteriori estimation with Plug & Play priors and stochastic gradient descent](https://mlsanity.com/paper/2201.06133/) (2022) - [Convergent regularization in inverse problems and linear plug-and-play denoisers](https://mlsanity.com/paper/2307.09441/) (2023) - [Good Things Come in Pairs: Paired Autoencoders for Inverse Problems](https://mlsanity.com/paper/2505.06549/) (2025) - [MAP Estimation with Denoisers: Convergence Rates and Guarantees](https://mlsanity.com/paper/2507.15397/) (2025) - [Unsupervised approaches based on optimal transport and convex analysis for inverse problems in imaging](https://mlsanity.com/paper/2311.08972/) (2023) - [Nonasymptotic Convergence Rates for Plug-and-Play Methods With MMSE Denoisers](https://mlsanity.com/paper/2510.27211/) (2025) - [Beyond MMSE: Enhancing PnP Restoration with ProxiMAP](https://mlsanity.com/paper/2605.16396/) (2026) - [Self-supervised Deep Hyperspectral Inpainting with the Sparsity and Low-Rank Considerations](https://mlsanity.com/paper/2306.08128/) (2023) - [On the Contractivity of Plug-and-Play Operators](https://mlsanity.com/paper/2309.16899/) (2023) - [Low-resolution Prior Equilibrium Network for CT Reconstruction](https://mlsanity.com/paper/2401.15663/) (2024) - [Self-supervised Deep Hyperspectral Inpainting with the Plug and Play and Deep Image Prior Models](https://mlsanity.com/paper/2501.08195/) (2025) - [Plug-and-Play with 2.5D Artifact Reduction Prior for Fast and Accurate Industrial Computed Tomography Reconstruction](https://mlsanity.com/paper/2506.14719/) (2025) - [Solving Imaging Inverse Problems Using Plug-and-Play Denoisers: Regularization and Optimization Perspectives](https://mlsanity.com/paper/2509.03475/) (2025) - [Enhanced Denoising and Convergent Regularisation Using Tweedie Scaling](https://mlsanity.com/paper/2503.05956/) (2025) ## Abstract (arXiv) A standard model for image reconstruction involves the minimization of a data-fidelity term along with a regularizer, where the optimization is performed using proximal algorithms such as ISTA and ADMM. In plug-and-play (PnP) regularization, the proximal operator (associated with the regularizer) in ISTA and ADMM is replaced by a powerful image denoiser. Although PnP regularization works surprisingly well in practice, its theoretical convergence -- whether convergence of the PnP iterates is guaranteed and if they minimize some objective function -- is not completely understood even for simple linear denoisers such as nonlocal means. In particular, while there are works where either iterate or objective convergence is established separately, a simultaneous guarantee on iterate and objective convergence is not available for any denoiser to our knowledge. In this paper, we establish both forms of convergence for a special class of linear denoisers. Notably, unlike existing works where the focus is on symmetric denoisers, our analysis covers non-symmetric denoisers such as nonlocal means and almost any convex data-fidelity. The novelty in this regard is that we make use of the convergence theory of averaged operators and we work with a special inner product (and norm) derived from the linear denoiser; the latter requires us to appropriately define the gradient and proximal operators associated with the data-fidelity term. We validate our convergence results using image reconstruction experiments.