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REVIEW 4 major objections 5 minor 43 references

Plug-and-Play Posterior Sampling for Blind Inverse Problems

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Blind-PnPDM claims that blind inverse problems—where both the target image and the measurement operator are unknown—can be solved by posterior sampling that alternates between two Gaussian denoising steps, each driven by a pre-trained…

desk verdict Clean empirical extension to blind PnP sampling, but the posterior-sampling claim doesn't survive a close read of the split-Gibbs target. read the letter →

arxiv 2505.22923 v1 pith:AN5PMHOA submitted 2025-05-28 eess.IV cs.CV

Anqi Li , Weijie Gan , Ulugbek S. Kamilov This is my paper
classification eess.IVcs.CV
keywords blindinverseproblemsplug-and-playmethodsdiffusionmodelsposteriorsamplingsplitGibbsimagedeblurringkernelestimationGaussiandenoising
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Blind-PnPDM targets blind inverse problems, where the observation is produced by an unknown measurement operator acting on an unknown image, and recasts recovery as posterior sampling instead of point estimation. The method places two pre-trained diffusion models in a plug-and-play sampler, one acting as the image prior and one as the prior over the operator parameters, and alternates between sampling the image given the current operator and sampling the operator given the current image. Each conditional sampling step is split into a Gaussian likelihood update and a denoising prior update, so the joint estimation problem becomes a sequence of Gaussian denoising subproblems. On blind image deblurring with Gaussian and motion kernels, the paper reports that this scheme outperforms Pan-DCP, DeblurGANv2, BlindDPS, and GibbsDDRM in PSNR, SSIM, and LPIPS, while also producing kernel estimates that look closer to the ground truth.

What carries the argument

The load-bearing mechanism is an alternating split-Gibbs sampler with two pre-trained diffusion-model priors. In the image conditional, an auxiliary variable $z$ is sampled from a Gaussian distribution combining the data-fidelity term $\|y-A(\theta)z\|_2^2/2$ with the coupling $\|x^{(k)}-z\|^2/(2\rho_x^2)$, and then the prior step samples $x\sim\exp(-g(x)-\|x-z\|^2/(2\rho_x^2))$ by running one reverse pass of an Elucidated Diffusion Model (EDM) from the noise level $\sigma(t^*)=\rho_x$ down to zero. The operator conditional uses the same construction with a second diffusion model $D_\beta$ and its own auxiliary variable $v$ and coupling strength $\rho_\theta$. Annealing schedules that decrease $\rho_x$ and $\rho_\theta$ over the $K$ iterations are included to accelerate mixing and reduce the chance of getting stuck, and the two diffusion priors are the only learned components.

What would settle it

Run Blind-PnPDM many times on the same blurred input with different random seeds and initializations and examine the spread of the reconstructed images and kernels. If all runs collapse to nearly the same point instead of producing a distribution of plausible image/kernel pairs consistent with the observation, the alternating chain is not sampling the joint posterior and the posterior-sampling interpretation fails.

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Extended reading notes

Core claim

The central claim is that the joint posterior $p(x,\theta\mid y)$ for the unknown image $x$ and unknown operator parameters $\theta$ can be sampled by alternating two conditional Gibbs steps, each split into a likelihood step and a diffusion prior step. Auxiliary variables $z$ and $v$ are coupled to $x$ and $\theta$ through quadratic penalties, making the likelihood steps Gaussian sampling problems in the measurement model and the prior steps exactly the task of denoising $z$ and $v$. Blind-PnPDM implements each prior step as a single reverse pass of a pre-trained EDM diffusion model initialized at the noise level corresponding to the coupling strength, which is the PnPDM recipe transplanted to the blind setting. The experimental claim is that on blind deblurring this alternating denoising scheme beats existing blind-deblurring baselines—including diffusion-based blind solvers—on PSNR, SSIM, and LPIPS, with reconstructed kernels that are more consistent with the ground truth.

Load-bearing premise

The load-bearing premise is that one reverse pass of a pre-trained diffusion model faithfully samples the denoising posterior for both the image and the blur kernel, so the alternating chain converges to the joint posterior; this is assumed, not proved or diagnosed, in the blind setting.

Editorial extensions

If this is right

  • On blind image deblurring with Gaussian and motion kernels, Blind-PnPDM reports higher PSNR and SSIM and lower LPIPS than Pan-DCP, DeblurGANv2, BlindDPS, and GibbsDDRM.
  • The framework needs no per-problem calibration or retraining: the same two pre-trained diffusion priors are inserted through the denoising steps, and only the likelihood model changes with the forward operator.
  • Because the operator prior lives over the parameter vector $\theta$, the same alternating sampler extends to other blind inverse problems with low-dimensional operator parameterizations, such as blind super-resolution or parallel MRI.
  • Treating recovery as sampling rather than optimization means repeated runs with different randomness can in principle produce a spread of plausible image/kernel pairs, giving a form of uncertainty information about the solution.
  • The kernel estimates produced by the method are visually closer to the ground-truth kernels than those of the diffusion-based baselines, not just the images.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A test the paper does not run: compare Blind-PnPDM against an oracle that uses the true kernel distribution as the prior. Matching the oracle would confirm the gains come from the sampling scheme; a gap would point at the kernel diffusion prior's approximation error.
  • The same two-prior alternating construction could be pointed at deconvolution microscopy or blind inpainting by swapping only the likelihood step's measurement model, since the prior steps are agnostic to the forward operator.
  • If the joint posterior is really being sampled, the framework offers calibrated uncertainty over blur kernels, which would be useful for downstream tasks like motion estimation or optical aberration correction; this uncertainty interpretation is implicit but unquantified in the paper.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper proposes Blind-PnPDM, an alternating plug-and-play posterior sampling method for blind inverse problems in which both the image and the measurement operator are unknown. Two diffusion models are used as priors, one for the image and one for the forward-model parameters, within a Split Gibbs Sampler (SGS) framework. Each conditional update is further decomposed into a likelihood step and a prior step, with the prior steps implemented as EDM reverse passes. The method is validated on blind image deblurring with Gaussian and motion kernels, reporting quantitative gains over Pan-DCP, DeblurGANv2, BlindDPS, and GibbsDDRM in PSNR, SSIM, and LPIPS. The central claim is that Algorithm 1 samples from the joint posterior p(x, theta | y) of Eq. (5).

Significance. Blind inverse problems are an important and under-served area for PnP sampling methods, and the idea of using two diffusion priors in an alternating denoising scheme is timely and potentially useful. The reported quantitative improvements over strong diffusion-based baselines (GibbsDDRM, BlindDPS) are encouraging, and the paper addresses a real gap in the PnP literature. However, the manuscript's central theoretical claim—that Algorithm 1 samples from the joint posterior—is not supported by the equations as written, and the empirical evaluation lacks error bars, multiple runs, and code. These issues are load-bearing for the paper's main claims, although the method may well be salvageable as an approximate/heuristic sampling scheme with additional analysis and diagnostics.

major comments (4)
  1. [Section 2.2, Eqs. (6)-(9), Algorithm 1] The paper claims that Algorithm 1 samples from the joint posterior p(x, theta | y) of Eq. (5), but this does not follow from the equations. The x-block is a split-Gibbs block with augmented target pi(x,z | theta) proportional to exp(-f(z; theta, y) - g(x) - ||x - z||^2/(2 rho_x^2)); integrating out z gives a stationary marginal for x proportional to exp(-g(x)) times the Gaussian-smoothed likelihood integral exp(-f(z; theta, y)) N(x; z, rho_x^2 I) dz, which is not the exp(-f(x; theta, y) - g(x)) stated in Eq. (6). The theta-block in Eq. (9) has the same structure. The annealing schedules in Section 3 stop at rho_x = 0.1 and rho_theta = 0.05, so the smoothing bias is not removed, and no bias bound or diagnostic is provided. The empirical results may remain valid as a heuristic, but the posterior-sampling claim does not follow from the presented equations.
  2. [Section 2.2, LikelihoodStep_theta] The displayed update for v^(k) is dimensionally inconsistent: v is in R^b, but the quadratic term uses A(theta^(k)) v. In the blind deblurring model A(theta) x = theta * x, so A(theta^(k)) v is not defined when v is a kernel parameter. If the intended expression is A(v) x^(k+1), then the printed equation ignores the current image x^(k+1) and contradicts the conditional target in Eq. (7). This must be corrected because it directly affects the reproducibility of Algorithm 1 and the meaning of the theta-block.
  3. [Section 2.1, PriorStepx and PriorStep_theta] The implementation of Eq. (8) and Eq. (9) by a single EDM reverse pass from t* with sigma(t*) = rho is inherited from PnPDM, but no argument is given in this paper that such a pass samples the conditional density exp(-g(x) - ||x - z||^2/(2 rho_x^2)) or the analogous kernel density in the blind setting. The original PnPDM justification is not automatically transferred here because both priors are now conditioned on auxiliary variables from different modalities, and the paper provides no empirical diagnostic (e.g., conditional sample quality or chain mixing) to support the approximation. This is load-bearing: if the prior steps are not accurate, the chain is not even targeting the augmented distributions described by Eqs. (8)-(9).
  4. [Section 3, Table 1] The quantitative comparison is based on a single run with no error bars, no multiple seeds, and no ablation of the coupling schedules rho_x and rho_theta. For a sampling method, single-sample metrics are particularly sensitive to randomness in the diffusion reverse passes; without such information, the reported margins (e.g., PSNR 27.42 vs 25.94 for motion blur) cannot be distinguished from sampling noise. Please report mean and standard deviation over multiple seeds and include an ablation of the annealing schedule, or at least a sensitivity analysis for rho_x and rho_theta.
minor comments (5)
  1. [Section 2.1, Eqs. (6) and (7)] Equations (6) and (7) use '=' to equate a distribution with an unnormalized density; the right-hand sides are proportional to the conditional posteriors. Please use 'proportional to' notation consistently throughout.
  2. [Section 3] The text refers to the baseline 'DeblurGAN' while Table 1 lists 'DeblurGANv2'; please align the names in the text, table, and reference list.
  3. [Figures 1 and 2] Figures 1 and 2 are referenced but not included in the manuscript; the captions mention squares at the top displaying estimated kernels, but the qualitative claims cannot be evaluated without the figures.
  4. [Section 4, Conclusion] The claim that Blind-PnPDM is 'the first PnP sampling method for blind inverse problems' is strong given references [28] and [30], which relate to block-coordinate PnP for blind inverse problems and split-Gibbs sampling with deep priors; please soften or clarify the novelty claim.
  5. [Section 3, kernel diffusion model training] The kernel diffusion model is described only as trained for '5M steps with a small U-Net'; the architecture, optimizer, learning rate, and training schedule are not specified, which limits reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the method is an application of SGS/PnPDM to blind inverse problems, and the kernel train/test overlap is a benchmark weakness rather than a construction-level circular step.

full rationale

The paper's derivation chain is not circular. The joint posterior in Eq. (5) is defined independently, and the Gibbs conditionals in Eqs. (6)-(7) are standard decompositions. The likelihood/prior splitting in Eqs. (8)-(9) follows the Split Gibbs Sampler framework and PnPDM, which are cited external works rather than self-citations. The claim that a single EDM reverse pass from t* approximates the Gaussian-smoothed prior step is inherited from PnPDM; this is an assumption or approximation, not a reduction of the target to the algorithm's inputs. The skeptic's concern that the SGS stationary distribution is a rho-smoothed posterior rather than Eq. (5) is a substantive soundness issue, but it is not circularity: the paper does not define the posterior in terms of the algorithm's output, nor does it fit a parameter and then relabel that fit as a prediction. The only self-supporting element is the evaluation: the kernel diffusion model is trained on 100k blur kernels generated following [39] and tested on kernels generated following [39]. This is a benchmark weakness that biases blind-generalization claims, but it does not make any derived quantity equivalent to its input by construction. No load-bearing self-citation, uniqueness theorem imported from the authors, or ansatz smuggled through self-citation appears in the paper.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central algorithm introduces no new physical entities. Its load-bearing elements are two learned models, the hand-chosen schedules, and the approximate identification of diffusion inference with Gaussian prior sampling. The paper trades on prior components (PnPDM, BlindDPS, GibbsDDRM) and contributes the combination plus an empirical evaluation.

free parameters (3)
  • Image coupling schedule rho_x = max(0.9^k * 0.3, 0.1)
    Hand-chosen annealing schedule controlling coupling between x and z; no sensitivity analysis is given.
  • Kernel coupling schedule rho_theta = max(0.9^k * 0.1, 0.05)
    Hand-chosen annealing schedule for the theta sampling step; no sensitivity analysis is given.
  • Number of iterations K = 30
    Selected by hand; no convergence diagnostic or burn-in analysis is provided.
assumptions (4)
  • ad hoc to paper A single EDM reverse pass from t* samples or approximates the conditional prior exp(-g(x) - ||x-z||^2 / (2 rho_x^2)) in Eq. (8).
    This is the central algorithmic identification inherited from PnPDM and used for both x and theta; it is stated without proof in Section 2.2.
  • domain assumption The alternating Gibbs chain with the specified annealing schedules converges to the joint posterior within K=30 iterations.
    No mixing or convergence analysis is provided; the method relies on standard SGS behavior.
  • domain assumption The kernel diffusion prior D_beta trained on synthetic kernels represents the test kernels.
    Section 3 trains on 100k generated kernels and tests on kernels generated following [39], so the prior matches the test distribution.
  • domain assumption The image diffusion prior D_alpha approximates the true image prior at the relevant noise levels.
    Standard assumption for diffusion-based PnP; no calibration is shown.

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Cite this review

Pith. "Pith review of Plug-and-Play Posterior Sampling for Blind Inverse Problems." pith.science (2026). https://pith.science/paper/AN5PMHOA

@misc{pith2026250522923,
  author       = {Pith},
  title        = {Pith review of: Plug-and-Play Posterior Sampling for Blind Inverse Problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AN5PMHOA}},
  note         = {Machine review of arXiv:2505.22923}
}
read the original abstract

We introduce Blind Plug-and-Play Diffusion Models (Blind-PnPDM) as a novel framework for solving blind inverse problems where both the target image and the measurement operator are unknown. Unlike conventional methods that rely on explicit priors or separate parameter estimation, our approach performs posterior sampling by recasting the problem into an alternating Gaussian denoising scheme. We leverage two diffusion models as learned priors: one to capture the distribution of the target image and another to characterize the parameters of the measurement operator. This PnP integration of diffusion models ensures flexibility and ease of adaptation. Our experiments on blind image deblurring show that Blind-PnPDM outperforms state-of-the-art methods in terms of both quantitative metrics and visual fidelity. Our results highlight the effectiveness of treating blind inverse problems as a sequence of denoising subproblems while harnessing the expressive power of diffusion-based priors.

Figures

Figures reproduced from arXiv: 2505.22923 by the authors.

Figure 1
Figure 1. Illustration of the results obtained from several well-known methods for blind image deblurring [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Illustration of the results obtained from several well-known methods for blind image deblurring [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.