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

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)
5 Downloads (Pure)


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) regularization, 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.

Original languageEnglish
Pages (from-to)140-163
Number of pages24
JournalJournal of Mathematical Imaging and Vision
Issue number1
Early online date18 Jan 2023
Publication statusPublished - Jan 2023


  • Bayesian imaging
  • Deblurring
  • Denoising
  • Inpainting
  • Inverse problems
  • Plug and play
  • Stochastic gradient descent

ASJC Scopus subject areas

  • Statistics and Probability
  • Modelling and Simulation
  • Condensed Matter Physics
  • Computer Vision and Pattern Recognition
  • Geometry and Topology
  • Applied Mathematics


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