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Plug-and-Play Posterior Sampling under Mismatched Measurement and Prior Models
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Proposes a posterior-L2 pseudometric and uses it to bound how far PnP-ULA's invariant distribution moves under a mismatched forward model or denoiser, validated on GMM and deblurring experiments.
Problem
Theoretical understanding of how mismatch between the measurement model and the plug-and-play denoiser prior affects the sampling distribution of PnP-ULA for Bayesian imaging inverse problems is lacking.
Approach
Define a posterior-L2 pseudometric (expectation under the reference sampling distribution of the squared drift difference) and show via Girsanov-type arguments that TV and W1 distances between the invariant distributions of two PnP-ULA chains are bounded by this drift distance plus a discretization term in the step size.
Key results
- Posterior-L2 correlates with W1 between sampling distributions at r=0.98 in the 2D GMM experiment, versus r=0.89 for prior-L2.
- Across 15 grayscale images, the average correlation between posterior-L2 and W1 for DnCNN denoiser depth variants is r=0.9909 (each image r>0.97).
- For forward-model mismatch on CelebA color deblurring, W1 to the reference sampling distribution decreases as the assumed blur sigma approaches the true value 3 (640 at sigma 0 to 341 at sigma 2.5).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Increase the assumed blur-kernel standard deviation toward the true value 3 | PnP-ULA on CelebA validation images · CelebA validation set | Wasserstein distance: 640 -> 642 -> 585 -> 471 -> 341 | Helped |
| Measure denoiser mismatch with posterior-L2 rather than prior-L2 | PnP-ULA sampling distributions · 2D Gaussian Mixture | Correlation with Wasserstein distance: r = 0.89 -> r = 0.98 | Helped |
| Use a more powerful DnCNN denoiser with more layers | PnP-ULA image deblurring · Grayscale image deblurring | Reconstruction quality | Helped |
| Use a 5-layer DnCNN denoiser with 10^5 weights instead of the 5 × 10^5-weight reference denoiser | PnP-ULA image deblurring · Grayscale image deblurring | Similarity of generated samples | No clear effect |
| Use posterior-L2 between denoisers as the mismatch measure | PnP-ULA sampling distributions · Grayscale image deblurring | Average correlation with Wasserstein distance: r = 0.9909 | Helped |
Surprising findings
- Integrating the posterior-L2 over the reference sampling distribution tracks sampling-distribution shifts better than the prior-L2 between denoisers.
Limitations
- Wasserstein bound derived from the TV bound is suboptimal according to the authors.
- Requires assumptions such as Lipschitz denoisers and a bounded/convex constraint set; the posterior-L2 is only a pseudometric.
- Empirical W1 estimates with 1000 samples in 65,536 dimensions are biased, so distances do not go to zero.
Experimental setup
Datasets: CBSD68 CelebA BreCaHAD RxRx1 MetFaces
Benchmarks: CBSD68 CelebA
Metrics: Wasserstein-1 distance posterior-L2 pseudometric prior-L2 pseudometric Pearson correlation PSNR SSIM
Baselines: PnP-ULA with exact MMSE denoiser prior-L2 pseudometric
Builds on: PnP-ULA Langevin dynamics Girsanov theorem DnCNN DRUNet
Closest papers by meaning
- Prior Mismatch and Adaptation in PnP-ADMM with a Nonconvex Convergence Analysis (ICML 2024 · similarity 0.90)
- Provably Robust Score-Based Diffusion Posterior Sampling for Plug-and-Play Image Reconstruction (NeurIPS 2024 · similarity 0.89)
- From stability of Langevin diffusion to convergence of proximal MCMC for non-log-concave sampling (NeurIPS 2025 · similarity 0.88)
- Improving Diffusion Posterior Samplers with Lagged Temporal Corrections for Image Restoration (NeurIPS 2026 · similarity 0.87)
- Divide-and-Conquer Posterior Sampling for Denoising Diffusion priors (NeurIPS 2024 · similarity 0.87)
- Posterior Sampling by Combining Diffusion Models with Annealed Langevin Dynamics (NeurIPS 2025 · similarity 0.87)
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