Plug-and-Play Posterior Sampling under Mismatched Measurement and Prior Models
This work addresses a theoretical gap for researchers in Bayesian imaging, providing a precise characterization of sensitivity to model mismatches, though it is incremental as it builds on existing PnP-ULA methods.
The paper tackles the lack of theoretical analysis for the plug-and-play unadjusted Langevin algorithm (PnP-ULA) under mismatched measurement and prior models in imaging inverse problems, proposing a posterior-L2 pseudometric to quantify an explicit error bound and validating it numerically on tasks like Gaussian mixture models and image deblurring.
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.