# Plug-and-play ISTA converges with kernel denoisers

arXiv: https://arxiv.org/abs/2004.03145 · HTML: https://mlsanity.com/paper/2004.03145/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Apr 2020.

## Built on by (most cited first, 12 of 12)

- [Learning to Optimize: A Primer and A Benchmark](https://mlsanity.com/paper/2103.12828/) (2021)
- [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)
- [Fixed-Point and Objective Convergence of Plug-and-Play Algorithms](https://mlsanity.com/paper/2104.10348/) (2021)
- [MAP Estimation with Denoisers: Convergence Rates and Guarantees](https://mlsanity.com/paper/2507.15397/) (2025)
- [Provably Convergent Plug & Play Linearized ADMM, applied to Deblurring Spatially Varying Kernels](https://mlsanity.com/paper/2210.10605/) (2022)
- [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)
- [Video Restoration with a Deep Plug-and-Play Prior](https://mlsanity.com/paper/2209.02854/) (2022)
- [On the Contractivity of Plug-and-Play Operators](https://mlsanity.com/paper/2309.16899/) (2023)
- [Plug-and-Play Methods Provably Converge Even with Improperly Trained Denoisers: Convergence by Architectural Design](https://mlsanity.com/paper/2609.33087/) (2026)

## Abstract (arXiv)

Plug-and-play (PnP) method is a recent paradigm for image regularization, where the proximal operator (associated with some given regularizer) in an iterative algorithm is replaced with a powerful denoiser. Algorithmically, this involves repeated inversion (of the forward model) and denoising until convergence. Remarkably, PnP regularization produces promising results for several restoration applications. However, a fundamental question in this regard is the theoretical convergence of the PnP iterations, since the algorithm is not strictly derived from an optimization framework. This question has been investigated in recent works, but there are still many unresolved problems. For example, it is not known if convergence can be guaranteed if we use generic kernel denoisers (e.g. nonlocal means) within the ISTA framework (PnP-ISTA). We prove that, under reasonable assumptions, fixed-point convergence of PnP-ISTA is indeed guaranteed for linear inverse problems such as deblurring, inpainting and superresolution (the assumptions are verifiable for inpainting). We compare our theoretical findings with existing results, validate them numerically, and explain their practical relevance.