Plug-and-Play Posterior Sampling under Mismatched Measurement and Prior Models
Marien Renaud, Jiaming Liu, Valentin de Bortoli, Andr\'es Almansa, Ulugbek S. Kamilov
TL;DR
This paper provides a theoretical analysis of the plug-and-play unadjusted Langevin algorithm (PnP-ULA), quantifying how mismatches in measurement models and denoisers affect its sampling distribution in imaging inverse problems.
Contribution
It introduces a posterior-L2 pseudometric to explicitly bound the error of PnP-ULA under model mismatches, bridging a gap in theoretical understanding.
Findings
The sensitivity of PnP-ULA to mismatched models can be precisely characterized.
The proposed theory is validated on Gaussian mixture models and image deblurring.
Error bounds help understand the impact of model mismatch on sampling accuracy.
Abstract
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…
Peer Reviews
Decision·ICLR 2024 poster
It is great that the paper proves a theoretical result about a relevant topic where most work is highly empirical. The setting is very clear and the derivations seem sound (although I have not checked in great detail.) The authors work under fairly general assumptions and also illustrate their bounds empirically.
- The contribution is somewhat incremental given all the preparatory work in Laumont et al. 2022. Model mismatch is certainly a relevant topic and it is nice to have a paper about it so this is a somewhat subjective statement relative to papers I've reviewed for ICLR this year. - The prose is waffly, with too much hyperbole. An example: "In this section, our focal point resides in the investigation of the profound impact of a drift shift on the invariant distribution of the PnP-ULA Markov chai
The considered problem is relevant, especially in medical imaging, where we want algorithms to be robust to mismatch in the forward model. Inverse problems using diffusion models is also an active area of research and the proposed results could be relevant.
- The main result in Theorem 1 shows that the TV between the stationary distributions of two Markov chains that have different drift functions can be bounded in terms of the proposed ``posterior-$L_2$ pseudometric''. This pseudometric is defined in terms of the expectation of the difference between the two drift functions when samples are drawn from the stationary distribution of one of the drifts. It's not clear at all how this pseudo metric behaves, and whether it is sufficiently small for two
- Sections 1–4 are generally clearly written. The readers can get what the authors try to convey without diving into the mathematical details. - The quantification of the error bound for PnP-ULA under a mismatched posterior distribution is of theoretical importance.
- The section associated with the numerical experiment is hard to dig out. Particularly, it's not easy to understand how and why the proposed setting can be adopted to validate the theoretical corollary. - As claimed by the authors, "our results can be seen as a PnP counterpart of existing results on diffusion models.", which therefore weakens the novelty of this paper. - It seems like the theoretical results drawn rely on "oracle" information that is unavailable in practice. So the practical us
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Taxonomy
TopicsMedical Imaging Techniques and Applications · Statistical Methods and Inference · Gaussian Processes and Bayesian Inference