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Bayesian imaging using Plug & Play priors: when Langevin meets Tweedie

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

TL;DR

This work develops a formal Bayesian framework for inference with Plug & Play priors in imaging, defining regularised oracle models and proving convergence guarantees for two Langevin-based algorithms, PnP-ULA and PPnP-ULA, under mild assumptions on denoisers and likelihoods. By leveraging Tweedie’s identity to connect MMSE denoisers with posterior gradients, the authors establish well-posed, Lipschitz-continuous posteriors and derive non-asymptotic bounds on convergence and bias toward the oracle model. The approach is validated on canonical tasks like non-blind image deblurring and inpainting, showing competitive MMSE point estimates and rich uncertainty visualisation, while outlining practical guidelines for denoiser selection, step-size, and tail regularisation. Overall, the paper provides a principled, convergent pathway to Bayesian inference with implicit PnP priors, enabling uncertainty quantification in high-dimensional imaging problems with learned denoisers.

Abstract

Since the seminal work of Venkatakrishnan et al. in 2013, Plug & Play (PnP) methods have become ubiquitous in Bayesian imaging. These methods derive Minimum Mean Square Error (MMSE) or Maximum A Posteriori (MAP) estimators for inverse problems in imaging by combining an explicit likelihood function with a prior that is implicitly defined by an image denoising algorithm. The PnP algorithms proposed in the literature mainly differ in the iterative schemes they use for optimisation or for sampling. In the case of optimisation schemes, some recent works guarantee the convergence to a fixed point, albeit not necessarily a MAP estimate. In the case of sampling schemes, to the best of our knowledge, there is no known proof of convergence. There also remain important open questions regarding whether the underlying Bayesian models and estimators are well defined, well-posed, and have the basic regularity properties required to support these numerical schemes. To address these limitations, this paper develops theory, methods, and provably convergent algorithms for performing Bayesian inference with PnP priors. We introduce two algorithms: 1) PnP-ULA (Unadjusted Langevin Algorithm) for Monte Carlo sampling and MMSE inference; and 2) PnP-SGD (Stochastic Gradient Descent) for MAP inference. Using recent results on the quantitative convergence of Markov chains, we establish detailed convergence guarantees for these two algorithms under realistic assumptions on the denoising operators used, with special attention to denoisers based on deep neural networks. We also show that these algorithms approximately target a decision-theoretically optimal Bayesian model that is well-posed. The proposed algorithms are demonstrated on several canonical problems such as image deblurring, inpainting, and denoising, where they are used for point estimation as well as for uncertainty visualisation and quantification.

Bayesian imaging using Plug & Play priors: when Langevin meets Tweedie

TL;DR

This work develops a formal Bayesian framework for inference with Plug & Play priors in imaging, defining regularised oracle models and proving convergence guarantees for two Langevin-based algorithms, PnP-ULA and PPnP-ULA, under mild assumptions on denoisers and likelihoods. By leveraging Tweedie’s identity to connect MMSE denoisers with posterior gradients, the authors establish well-posed, Lipschitz-continuous posteriors and derive non-asymptotic bounds on convergence and bias toward the oracle model. The approach is validated on canonical tasks like non-blind image deblurring and inpainting, showing competitive MMSE point estimates and rich uncertainty visualisation, while outlining practical guidelines for denoiser selection, step-size, and tail regularisation. Overall, the paper provides a principled, convergent pathway to Bayesian inference with implicit PnP priors, enabling uncertainty quantification in high-dimensional imaging problems with learned denoisers.

Abstract

Since the seminal work of Venkatakrishnan et al. in 2013, Plug & Play (PnP) methods have become ubiquitous in Bayesian imaging. These methods derive Minimum Mean Square Error (MMSE) or Maximum A Posteriori (MAP) estimators for inverse problems in imaging by combining an explicit likelihood function with a prior that is implicitly defined by an image denoising algorithm. The PnP algorithms proposed in the literature mainly differ in the iterative schemes they use for optimisation or for sampling. In the case of optimisation schemes, some recent works guarantee the convergence to a fixed point, albeit not necessarily a MAP estimate. In the case of sampling schemes, to the best of our knowledge, there is no known proof of convergence. There also remain important open questions regarding whether the underlying Bayesian models and estimators are well defined, well-posed, and have the basic regularity properties required to support these numerical schemes. To address these limitations, this paper develops theory, methods, and provably convergent algorithms for performing Bayesian inference with PnP priors. We introduce two algorithms: 1) PnP-ULA (Unadjusted Langevin Algorithm) for Monte Carlo sampling and MMSE inference; and 2) PnP-SGD (Stochastic Gradient Descent) for MAP inference. Using recent results on the quantitative convergence of Markov chains, we establish detailed convergence guarantees for these two algorithms under realistic assumptions on the denoising operators used, with special attention to denoisers based on deep neural networks. We also show that these algorithms approximately target a decision-theoretically optimal Bayesian model that is well-posed. The proposed algorithms are demonstrated on several canonical problems such as image deblurring, inpainting, and denoising, where they are used for point estimation as well as for uncertainty visualisation and quantification.

Paper Structure

This paper contains 41 sections, 22 theorems, 122 equations, 18 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Bayesian inference in imaging inverse problems
  3. Bayesian computation in imaging inverse problems
  4. Machine learning and Plug & Play approaches in imaging inverse problems
  5. Contributions summary
  6. Bayesian inference with Plug & Play priors: theory methods and algorithms
  7. Bayesian modelling and analysis with Plug & Play priors
  8. Bayesian computation with Plug & Play priors
  9. Theoretical analysis
  10. Notation
  11. Convergence of PnP-ULA
  12. Convergence guarantees for PPnP-ULA
  13. Experimental study
  14. Implementation guidelines and parameter setting
  15. Choice of the denoiser
  16. ...and 26 more sections

Key Result

Proposition 1

Assume assum:post. Then, for any $\varepsilon > 0$ and $y \in \mathbb{R}^m$, the following hold:

Figures (18)

  • Figure 1: Original images used for the deblurring and inpainting experiments.
  • Figure 2: Images of \ref{['fig:original_images']}, blurred using a $9 \times 9$-box-filter operator and corrupted by an additive Gaussian white noise with standard deviation $\sigma=1/255$.
  • Figure 3: Images of \ref{['fig:original_images']}, with 80% missing pixels.
  • Figure 4: Marginal posterior standard deviation of the unobserved pixels for the inpainting problem. Uncertainty is located around edges and in textured areas.
  • Figure 5: Evolution of the $L_2$ distance between the final MMSE estimate and the samples generated by PnP-ULA for the inpainting problem after the burn-in phase. Samples randomly oscillate around the MMSE. It means that they are uncorrelated. For the images Cameraman, Simpson or Bridge, we note a change of range for the $L_2$ distance. It could be interpreted as a mode switching as our posterior is likely not log-concave.
  • ...and 13 more figures

Theorems & Definitions (44)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Proposition 5
  • proof
  • ...and 34 more