# Convergent regularization in inverse problems and linear plug-and-play denoisers

arXiv: https://arxiv.org/abs/2307.09441 · HTML: https://mlsanity.com/paper/2307.09441/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 54 references that are in the map.
- First posted on arXiv 18 Jul 2023.

## Builds on (most cited first, 23 of 23)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Deep Convolutional Neural Network for Inverse Problems in Imaging](https://mlsanity.com/paper/1611.03679/) (2016)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [The Little Engine that Could: Regularization by Denoising (RED)](https://mlsanity.com/paper/1611.02862/) (2016)
- [Learned Primal-dual Reconstruction](https://mlsanity.com/paper/1707.06474/) (2017)
- [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)
- [Gradient Step Denoiser for convergent Plug-and-Play](https://mlsanity.com/paper/2110.03220/) (2021)
- [Adversarial Regularizers in Inverse Problems](https://mlsanity.com/paper/1805.11572/) (2018)
- [Regularization by Denoising: Clarifications and New Interpretations](https://mlsanity.com/paper/1806.02296/) (2018)
- [Plug-and-Play Methods for Integrating Physical and Learned Models in Computational Imaging](https://mlsanity.com/paper/2203.17061/) (2022)
- [NETT: Solving Inverse Problems with Deep Neural Networks](https://mlsanity.com/paper/1803.00092/) (2018)
- [A deep convolutional neural network using directional wavelets for low-dose X-ray CT reconstruction](https://mlsanity.com/paper/1610.09736/) (2016)
- [Learned convex regularizers for inverse problems](https://mlsanity.com/paper/2008.02839/) (2020)
- [Learned reconstruction methods with convergence guarantees](https://mlsanity.com/paper/2206.05431/) (2022)
- [A characterization of proximity operators](https://mlsanity.com/paper/1807.04014/) (2018)
- [Convolutional Dictionary Learning: A Comparative Review and New Algorithms](https://mlsanity.com/paper/1709.02893/) (2017)
- [Fixed-Point and Objective Convergence of Plug-and-Play Algorithms](https://mlsanity.com/paper/2104.10348/) (2021)
- [Bilevel parameter learning for higher-order total variation regularisation models](https://mlsanity.com/paper/1508.07243/) (2015)
- [Boosting the Performance of Plug-and-Play Priors via Denoiser Scaling](https://mlsanity.com/paper/2002.11546/) (2020)
- [On Plug-and-Play Regularization using Linear Denoisers](https://mlsanity.com/paper/2105.05177/) (2021)
- [A convex variational model for learning convolutional image atoms from incomplete data](https://mlsanity.com/paper/1812.03077/) (2018)
- [Data-consistent neural networks for solving nonlinear inverse problems](https://mlsanity.com/paper/2003.11253/) (2020)

## Built on by (most cited first, 13 of 13)

- [What's in a Prior? Learned Proximal Networks for Inverse Problems](https://mlsanity.com/paper/2310.14344/) (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 Training of Convex Regularizers using Maximum Likelihood Estimation](https://mlsanity.com/paper/2404.05445/) (2024)
- [Learning sparsity-promoting regularizers for linear inverse problems](https://mlsanity.com/paper/2412.16031/) (2024)
- [LoFi: Neural Local Fields for Scalable Image Reconstruction](https://mlsanity.com/paper/2411.04995/) (2024)
- [Efficient Bayesian Computation Using Plug-and-Play Priors for Poisson Inverse Problems](https://mlsanity.com/paper/2503.16222/) (2025)
- [A Stability Benchmark of Generative Regularizers for Inverse Problems](https://mlsanity.com/paper/2605.10076/) (2026)
- [Solving Imaging Inverse Problems Using Plug-and-Play Denoisers: Regularization and Optimization Perspectives](https://mlsanity.com/paper/2509.03475/) (2025)
- [Efficient Unrolled Networks for Large-Scale 3D Inverse Problems](https://mlsanity.com/paper/2601.02141/) (2026)
- [Enhanced Denoising and Convergent Regularisation Using Tweedie Scaling](https://mlsanity.com/paper/2503.05956/) (2025)
- [Deep regularization networks for inverse problems with noisy operators](https://mlsanity.com/paper/2506.07008/) (2025)
- [A Distributionally Robust Framework for Learned Reconstructions in Inverse Problems](https://mlsanity.com/paper/2606.30230/) (2026)

## Abstract (arXiv)

Plug-and-play (PnP) denoising is a popular iterative framework for solving imaging inverse problems using off-the-shelf image denoisers. Their empirical success has motivated a line of research that seeks to understand the convergence of PnP iterates under various assumptions on the denoiser. While a significant amount of research has gone into establishing the convergence of the PnP iteration for different regularity conditions on the denoisers, not much is known about the asymptotic properties of the converged solution as the noise level in the measurement tends to zero, i.e., whether PnP methods are provably convergent regularization schemes under reasonable assumptions on the denoiser. This paper serves two purposes: first, we provide an overview of the classical regularization theory in inverse problems and survey a few notable recent data-driven methods that are provably convergent regularization schemes. We then continue to discuss PnP algorithms and their established convergence guarantees. Subsequently, we consider PnP algorithms with linear denoisers and propose a novel spectral filtering technique to control the strength of regularization arising from the denoiser. Further, by relating the implicit regularization of the denoiser to an explicit regularization functional, we rigorously show that PnP with linear denoisers leads to a convergent regularization scheme. More specifically, we prove that in the limit as the noise vanishes, the PnP reconstruction converges to the minimizer of a regularization potential subject to the solution satisfying the noiseless operator equation. The theoretical analysis is corroborated by numerical experiments for the classical inverse problem of tomographic image reconstruction.