# Recovery Analysis for Plug-and-Play Priors using the Restricted Eigenvalue Condition

arXiv: https://arxiv.org/abs/2106.03668 · HTML: https://mlsanity.com/paper/2106.03668/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- Builds on 46 of its 85 references that are in the map.
- First posted on arXiv 7 Jun 2021.

## Builds on (most cited first, 30 of 46)

- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [Analyzing and Improving the Image Quality of StyleGAN](https://mlsanity.com/paper/1912.04958/) (2019)
- [Progressive Growing of GANs for Improved Quality, Stability, and Variation](https://mlsanity.com/paper/1710.10196/) (2017)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [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)
- [Deep Image Prior](https://mlsanity.com/paper/1711.10925/) (2017)
- [Learning Deep CNN Denoiser Prior for Image Restoration](https://mlsanity.com/paper/1704.03264/) (2017)
- [MoDL: Model Based Deep Learning Architecture for Inverse Problems](https://mlsanity.com/paper/1712.02862/) (2017)
- [PULSE: Self-Supervised Photo Upsampling via Latent Space Exploration of Generative Models](https://mlsanity.com/paper/2003.03808/) (2020)
- [The Little Engine that Could: Regularization by Denoising (RED)](https://mlsanity.com/paper/1611.02862/) (2016)
- [Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks](https://mlsanity.com/paper/1906.04893/) (2019)
- [Deep Learning Techniques for Inverse Problems in Imaging](https://mlsanity.com/paper/2005.06001/) (2020)
- [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)
- [Non-convex Optimization for Machine Learning](https://mlsanity.com/paper/1712.07897/) (2017)
- [Deep Equilibrium Architectures for Inverse Problems in Imaging](https://mlsanity.com/paper/2102.07944/) (2021)
- [Deep Decoder: Concise Image Representations from Untrained Non-convolutional Networks](https://mlsanity.com/paper/1810.03982/) (2018)
- [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)
- [A Review of Convolutional Neural Networks for Inverse Problems in Imaging](https://mlsanity.com/paper/1710.04011/) (2017)
- [Deep Learning Methods for Parallel Magnetic Resonance Image Reconstruction](https://mlsanity.com/paper/1904.01112/) (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)
- [From Denoising to Compressed Sensing](https://mlsanity.com/paper/1406.4175/) (2014)
- [RARE: Image Reconstruction using Deep Priors Learned without Ground Truth](https://mlsanity.com/paper/1912.05854/) (2019)

## Built on by (most cited first, 22 of 22)

- [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)
- [A Neural-Network-Based Convex Regularizer for Inverse Problems](https://mlsanity.com/paper/2211.12461/) (2022)
- [Theoretical Perspectives on Deep Learning Methods in Inverse Problems](https://mlsanity.com/paper/2206.14373/) (2022)
- [Prior Mismatch and Adaptation in PnP-ADMM with a Nonconvex Convergence Analysis](https://mlsanity.com/paper/2310.00133/) (2023)
- [Deep Model-Based Architectures for Inverse Problems under Mismatched Priors](https://mlsanity.com/paper/2207.13200/) (2022)
- [Block Coordinate Plug-and-Play Methods for Blind Inverse Problems](https://mlsanity.com/paper/2305.12672/) (2023)
- [Learning pseudo-contractive denoisers for inverse problems](https://mlsanity.com/paper/2402.05637/) (2024)
- [Learning Sparsity-Promoting Regularizers using Bilevel Optimization](https://mlsanity.com/paper/2207.08939/) (2022)
- [NPN: Non-Linear Projections of the Null-Space for Imaging Inverse Problems](https://mlsanity.com/paper/2510.01608/) (2025)
- [Monotonically Convergent Regularization by Denoising](https://mlsanity.com/paper/2202.04961/) (2022)
- [Convergence and Recovery Guarantees of Unsupervised Neural Networks for Inverse Problems](https://mlsanity.com/paper/2309.12128/) (2023)
- [Radio Map Estimation via Latent Domain Plug-and-Play Denoising](https://mlsanity.com/paper/2501.13472/) (2025)
- [A Recovery Theory for Diffusion Priors: Deterministic Analysis of the Implicit Prior Algorithm](https://mlsanity.com/paper/2509.20511/) (2025)
- [Plug-and-Play Posterior Sampling for Blind Inverse Problems](https://mlsanity.com/paper/2505.22923/) (2025)
- [Neural Inverse Scattering with Score-based Regularization](https://mlsanity.com/paper/2505.14560/) (2025)
- [Moir'eXNet: Adaptive Multi-Scale Demoir'eing with Linear Attention Test-Time Training and Truncated Flow Matching Prior](https://mlsanity.com/paper/2506.15929/) (2025)
- [Weakly Convex Ridge Regularization for 3D Non-Cartesian MRI Reconstruction](https://mlsanity.com/paper/2603.27158/) (2026)
- [From the Gradient-Step Denoiser to the Proximal Denoiser and their associated convergent Plug-and-Play algorithms](https://mlsanity.com/paper/2509.09793/) (2025)
- [ScoreField: Neural Inverse Scattering with Score-Based Generative Priors](https://mlsanity.com/paper/2608.02937/) (2026)
- [Plug-and-Play Methods Provably Converge Even with Improperly Trained Denoisers: Convergence by Architectural Design](https://mlsanity.com/paper/2609.33087/) (2026)

## Abstract (arXiv)

The plug-and-play priors (PnP) and regularization by denoising (RED) methods have become widely used for solving inverse problems by leveraging pre-trained deep denoisers as image priors. While the empirical imaging performance and the theoretical convergence properties of these algorithms have been widely investigated, their recovery properties have not previously been theoretically analyzed. We address this gap by showing how to establish theoretical recovery guarantees for PnP/RED by assuming that the solution of these methods lies near the fixed-points of a deep neural network. We also present numerical results comparing the recovery performance of PnP/RED in compressive sensing against that of recent compressive sensing algorithms based on generative models. Our numerical results suggest that PnP with a pre-trained artifact removal network provides significantly better results compared to the existing state-of-the-art methods.