# On Plug-and-Play Regularization using Linear Denoisers

arXiv: https://arxiv.org/abs/2105.05177 · HTML: https://mlsanity.com/paper/2105.05177/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 May 2021.

## Built on by (most cited first, 9 of 9)

- [Gradient Step Denoiser for convergent Plug-and-Play](https://mlsanity.com/paper/2110.03220/) (2021)
- [Convergent regularization in inverse problems and linear plug-and-play denoisers](https://mlsanity.com/paper/2307.09441/) (2023)
- [Radio Map Estimation via Latent Domain Plug-and-Play Denoising](https://mlsanity.com/paper/2501.13472/) (2025)
- [Nonasymptotic Convergence Rates for Plug-and-Play Methods With MMSE Denoisers](https://mlsanity.com/paper/2510.27211/) (2025)
- [On the Contractivity of Plug-and-Play Operators](https://mlsanity.com/paper/2309.16899/) (2023)
- [Gradient Networks](https://mlsanity.com/paper/2404.07361/) (2024)
- [Enhanced Denoising and Convergent Regularisation Using Tweedie Scaling](https://mlsanity.com/paper/2503.05956/) (2025)
- [Trainable Nonexpansive Denoisers for Contractive Image Reconstruction](https://mlsanity.com/paper/2607.23347/) (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)

In plug-and-play (PnP) regularization, the knowledge of the forward model is combined with a powerful denoiser to obtain state-of-the-art image reconstructions. This is typically done by taking a proximal algorithm such as FISTA or ADMM, and formally replacing the proximal map associated with a regularizer by nonlocal means, BM3D or a CNN denoiser. Each iterate of the resulting PnP algorithm involves some kind of inversion of the forward model followed by denoiser-induced regularization. A natural question in this regard is that of optimality, namely, do the PnP iterations minimize some f+g, where f is a loss function associated with the forward model and g is a regularizer? This has a straightforward solution if the denoiser can be expressed as a proximal map, as was shown to be the case for a class of linear symmetric denoisers. However, this result excludes kernel denoisers such as nonlocal means that are inherently non-symmetric. In this paper, we prove that a broader class of linear denoisers (including symmetric denoisers and kernel denoisers) can be expressed as a proximal map of some convex regularizer g. An algorithmic implication of this result for non-symmetric denoisers is that it necessitates appropriate modifications in the PnP updates to ensure convergence to a minimum of f+g. Apart from the convergence guarantee, the modified PnP algorithms are shown to produce good restorations.