# Plug-and-Play Algorithm Convergence Analysis From The Standpoint of Stochastic Differential Equation

arXiv: https://arxiv.org/abs/2404.13866 · HTML: https://mlsanity.com/paper/2404.13866/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 33 references that are in the map.
- First posted on arXiv 22 Apr 2024.

## Builds on (most cited first, 14 of 14)

- [Plug-and-Play Image Restoration with Deep Denoiser Prior](https://mlsanity.com/paper/2008.13751/) (2020)
- [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)
- [Regularization by Denoising: Clarifications and New Interpretations](https://mlsanity.com/paper/1806.02296/) (2018)
- [Theoretical Linear Convergence of Unfolded ISTA and its Practical Weights and Thresholds](https://mlsanity.com/paper/1808.10038/) (2018)
- [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)
- [Deep Unfolding: Model-Based Inspiration of Novel Deep Architectures](https://mlsanity.com/paper/1409.2574/) (2014)
- [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)
- [On Maximum-a-Posteriori estimation with Plug & Play priors and stochastic gradient descent](https://mlsanity.com/paper/2201.06133/) (2022)
- [Local Linear Convergence of ISTA and FISTA on the LASSO Problem](https://mlsanity.com/paper/1501.02888/) (2015)

## Abstract (arXiv)

The Plug-and-Play (PnP) algorithm is popular for inverse image problem-solving. However, this algorithm lacks theoretical analysis of its convergence with more advanced plug-in denoisers. We demonstrate that discrete PnP iteration can be described by a continuous stochastic differential equation (SDE). We can also achieve this transformation through Markov process formulation of PnP. Then, we can take a higher standpoint of PnP algorithms from stochastic differential equations, and give a unified framework for the convergence property of PnP according to the solvability condition of its corresponding SDE. We reveal that a much weaker condition, bounded denoiser with Lipschitz continuous measurement function would be enough for its convergence guarantee, instead of previous Lipschitz continuous denoiser condition.