Posterior sampling has been shown to be a powerful Bayesian approach for solving imaging inverse problems. The recent plug-and-play unadjusted Langevin algorithm (PnP-ULA) has emerged as a promising method for Monte Carlo sampling and minimum mean squared error (MMSE) estimation by combining physical measurement models with deep-learning priors specified using image denoisers. However, the intricate relationship between the sampling distribution of PnP-ULA and the mismatched data-fidelity and denoiser has not been theoretically analyzed. We address this gap by proposing a posterior-L2 pseudometric and using it to quantify an explicit error bound for PnP-ULA under mismatched posterior distribution. We numerically validate our theory on several inverse problems such as sampling from Gaussian mixture models and image deblurring. Our results suggest that the sensitivity of the sampling distribution of PnP-ULA to a mismatch in the measurement model and the denoiser can be precisely characterized.
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This content will become publicly available on December 10, 2024
Convergence of Nonconvex PNP-ADMM with MMSE Denoisers
Plug-and-Play Alternating Direction Method of Multipliers (PnP-ADMM) is a widely-used algorithm for solving inverse problems by integrating physical measurement models and convolutional neural network (CNN) priors. PnP-ADMM has been theoretically proven to converge for convex data-fidelity terms and nonexpansive CNNs. It has however been observed that PnP-ADMM often empirically converges even for expansive CNNs. This paper presents a theoretical explanation for the observed stability of PnP-ADMM based on the interpretation of the CNN prior as a minimum mean-squared error (MMSE) denoiser. Our explanation parallels a similar argument recently made for the iterative shrinkage/thresholding algorithm variant of PnP (PnP-ISTA) and relies on the connection between MMSE denoisers and proximal operators. We also numerically evaluate the performance gap between PnP-ADMM using a nonexpan-sive DnCNN denoiser and expansive DRUNet denoiser, thus motivating the use of expansive CNNs.
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- Award ID(s):
- 2043134
- NSF-PAR ID:
- 10504927
- Publisher / Repository:
- IEEE
- Date Published:
- Journal Name:
- IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing
- ISBN:
- 979-8-3503-4452-3
- Page Range / eLocation ID:
- 511 to 515
- Format(s):
- Medium: X
- Location:
- Herradura, Costa Rica
- Sponsoring Org:
- National Science Foundation
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