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On Maximum-a-Posteriori estimation with Plug & Play priors and stochastic gradient descent

Rémi Laumont, Valentin de Bortoli, Andrés Almansa, Julie Delon, Alain Durmus, Marcelo Pereyra

TL;DR

This work analyzes MAP estimation for imaging inverse problems using Plug & Play priors defined by denoisers, establishing a rigorous framework for the smooth posterior $p_\varepsilon(x|y)$ and its convergence to the true posterior as $\varepsilon\to 0$. It proves convergence of the proposed PnP-SGD algorithm under milder denoiser assumptions than prior work, and demonstrates competitive performance against other PnP schemes on denoising, deblurring, and inpainting. The results show that PnP-SGD can provide reliable MAP estimates with manageable bias from imperfect denoisers, offering a practical, theoretically grounded approach for implicit priors in imaging. The methods and insights are particularly relevant for robust, scalable Bayesian inference in high-dimensional, nonconvex imaging problems, with implications for uncertainty-aware reconstruction and modular integration of learned priors.

Abstract

Bayesian methods to solve imaging inverse problems usually combine an explicit data likelihood function with a prior distribution that explicitly models expected properties of the solution. Many kinds of priors have been explored in the literature, from simple ones expressing local properties to more involved ones exploiting image redundancy at a non-local scale. In a departure from explicit modelling, several recent works have proposed and studied the use of implicit priors defined by an image denoising algorithm. This approach, commonly known as Plug & Play (PnP) regularisation, can deliver remarkably accurate results, particularly when combined with state-of-the-art denoisers based on convolutional neural networks. However, the theoretical analysis of PnP Bayesian models and algorithms is difficult and works on the topic often rely on unrealistic assumptions on the properties of the image denoiser. This papers studies maximum-a-posteriori (MAP) estimation for Bayesian models with PnP priors. We first consider questions related to existence, stability and well-posedness, and then present a convergence proof for MAP computation by PnP stochastic gradient descent (PnP-SGD) under realistic assumptions on the denoiser used. We report a range of imaging experiments demonstrating PnP-SGD as well as comparisons with other PnP schemes.

On Maximum-a-Posteriori estimation with Plug & Play priors and stochastic gradient descent

TL;DR

This work analyzes MAP estimation for imaging inverse problems using Plug & Play priors defined by denoisers, establishing a rigorous framework for the smooth posterior and its convergence to the true posterior as . It proves convergence of the proposed PnP-SGD algorithm under milder denoiser assumptions than prior work, and demonstrates competitive performance against other PnP schemes on denoising, deblurring, and inpainting. The results show that PnP-SGD can provide reliable MAP estimates with manageable bias from imperfect denoisers, offering a practical, theoretically grounded approach for implicit priors in imaging. The methods and insights are particularly relevant for robust, scalable Bayesian inference in high-dimensional, nonconvex imaging problems, with implications for uncertainty-aware reconstruction and modular integration of learned priors.

Abstract

Bayesian methods to solve imaging inverse problems usually combine an explicit data likelihood function with a prior distribution that explicitly models expected properties of the solution. Many kinds of priors have been explored in the literature, from simple ones expressing local properties to more involved ones exploiting image redundancy at a non-local scale. In a departure from explicit modelling, several recent works have proposed and studied the use of implicit priors defined by an image denoising algorithm. This approach, commonly known as Plug & Play (PnP) regularisation, can deliver remarkably accurate results, particularly when combined with state-of-the-art denoisers based on convolutional neural networks. However, the theoretical analysis of PnP Bayesian models and algorithms is difficult and works on the topic often rely on unrealistic assumptions on the properties of the image denoiser. This papers studies maximum-a-posteriori (MAP) estimation for Bayesian models with PnP priors. We first consider questions related to existence, stability and well-posedness, and then present a convergence proof for MAP computation by PnP stochastic gradient descent (PnP-SGD) under realistic assumptions on the denoiser used. We report a range of imaging experiments demonstrating PnP-SGD as well as comparisons with other PnP schemes.
Paper Structure (27 sections, 3 theorems, 40 equations, 7 figures, 3 tables, 4 algorithms)

This paper contains 27 sections, 3 theorems, 40 equations, 7 figures, 3 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Computation of the MAP solution.
  3. Deep learning approaches.
  4. Plug & Play (PnP) approaches.
  5. Contributions.
  6. A survey of Plug & Play methods in imaging
  7. Plug & Play MAP estimators using proximal splitting
  8. Consensus equilibirum / fixed point interpretation.
  9. Assumptions on algorithm parameters.
  10. AMP algorithms.
  11. Plug & Play MAP estimators using gradient descent
  12. PnP maximum-a-posteriori estimation: analysis and computation
  13. Analysis of maximum-a-posteriori estimation with PnP priors
  14. PnP-SGD and convergence
  15. Experimental study
  16. ...and 12 more sections

Key Result

proposition thmcounterproposition

Assume Hassum:post and that $p \in \mathrm{C}^1(\mathbb{R}^d, \left(0,+\infty\right))$ with $\normLigne{p}_\infty + \normLigne{\nabla p}_\infty< +\infty$. Then for any compact set $\mathsf{K}$, $\mathsf{S}_{\mathsf{K}}^\star \subset \mathsf{S}_{\mathsf{K}}$ with $\mathsf{S}_{\mathsf{K}} = \{x \in \m

Figures (7)

  • Figure 1: Dataset (part 1): First three images in our dataset, and examples of degraded images for the three inverse problems considered in this paper. For denoising, we add a Gaussian noise with variance $\sigma^2 = (30/255)^2$. For deblurring, the operator ${\mathbf{A}}$ correponds to a $9 \times 9$ uniform blur operator, and we add Gaussian noise with variance $\sigma^2=(1/255)^2$. For inpainting, we hide $80\%$ of the pixels.
  • Figure 2: Dataset (part 2): Last three images in our dataset, and examples of degraded images for the three inverse problems considered in this paper. For denoising, we add a Gaussian noise with variance $\sigma^2 = (30/255)^2$. For deblurring, the operator ${\mathbf{A}}$ correponds to a $9 \times 9$ uniform blur operator, and we add Gaussian noise with variance $\sigma^2=(1/255)^2$. For inpainting, we hide $80\%$ of the pixels.
  • Figure 3: Plug & Play denoising for $\sigma^2=(30/255)^2$ with the prior implicit in $D_\varepsilon$ for $\varepsilon=(5/255)^2$ and different values of the regularization parameter $\alpha$. This table shows means and standard deviations for $\mathrm{PSNR}$ and $\mathrm{SSIM}$ values over K=10 independent noise realizations for each of the six images. Initialization plays a very minor role in this case and all algorithms achieve similar (nearly optimal) performance for $\alpha=0.25$.
  • Figure 4: Plug & Play denoising for $\sigma^2=(30/255)^2$, $\varepsilon=(5/255)^2$ with $\alpha=0.25$.
  • Figure 5: Plug & Play deblurring. Image are blurred with a $9\times 9$ uniform kernel, a Gaussian noise of standard deviation $\sigma^2=(1/255)^2$ is added. The denoiser $D_\varepsilon$ is trained at $\varepsilon=(5/255)^2$. The plots shows mean and standard deviation values of $\mathrm{PSNR}$ and $\mathrm{SSIM}$ over K=10 independent noise realizations for each of the six images and different values of the regularization parameter $\alpha$. Initialization plays a very minor role in this case and all algorithms achieve similar (nearly optimal) performance for $\alpha=0.3$, except for FBS which requires a larger (sub-optimal) $\alpha$ to converge.
  • ...and 2 more figures

Theorems & Definitions (6)

  • proposition thmcounterproposition
  • proof
  • proposition thmcounterproposition
  • proof
  • proposition thmcounterproposition
  • proof