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REVIEW 3 major objections 6 minor 46 references

Enhancing Diffusion Posterior Sampling for Inverse Problems by Integrating Crafted Measurements

T0 review · 3 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Replacing the clean measurement with a crafted noisy one reduces diffusion-based restoration error.

desk verdict A useful empirical improvement over DPS that overclaims its theoretical grounding; the crafted-measurement trick is real, but Eq. 10 doesn't support the less-bias claim. read the letter →

arxiv 2411.09850 v2 pith:PQMSMRGA submitted 2024-11-15 cs.CV cs.AIcs.LG

Shijie Zhou , Huaisheng Zhu , Rohan Sharma , Jiayi Chen , Ruiyi Zhang , Kaiyi Ji , Changyou Chen This is my paper · ORCID
classification cs.CVcs.AIcs.LG
keywords diffusionposteriorsamplinginverseproblemscraftedmeasurementsimagerestorationdeblurringsuper-resolutioninpaintingPoissonnoise
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

Standard diffusion posterior sampling (DPS) solves inverse problems by adding a likelihood gradient that pushes the denoised estimate $A(\hat{x}_0)$ toward the clean measurement $y$. This paper argues that this gradient injects high-frequency detail too early in the reverse process, when the diffusion prior is still forming low-frequency structure, and that the mismatch accumulates into posterior-sampling error. The proposed DPS-CM instead conditions each step on a crafted noisy measurement $y_t$ produced by a second reverse-diffusion trajectory on $y$, using its Tweedie mean $\hat{y}_0$ in a DPS-style gradient. The paper shows this reduces early $\epsilon$-prediction and reconstruction errors and reports consistent gains over DPS and recent baselines on deblurring, super-resolution, inpainting, nonlinear deblurring, and Poisson-noise tasks.

What carries the argument

The mechanism is the crafted measurement trajectory $\{y_t\}_{t=T}^{0}$: a second reverse-diffusion process run on the degraded measurement $y$ using the same pretrained score model, so that $y_t$ carries the same noise level and frequency content as the image trajectory $x_t$. At each timestep Tweedie's formula gives its mean estimate $\hat{y}_0 = E[y_0|y_t]$, and the derivation moves the outer expectation over $p(y_0|y_t)$ inside the log-likelihood and collapses it to the single point $p(\hat{y}_0|\hat{x}_0)$. This yields the approximate posterior gradient of Eq. 12, whose crafted term is weighted against the standard DPS term by $\mu$; the identity of the two diffusion schedules is what makes the guidance adaptively low-frequency early and high-frequency late.

What would settle it

Compute the gradient in Eq. 10 by Monte Carlo: draw $y_0^{(i)}$ from $p(y_0|y_t)$ and compare the average gradient with the single-point gradient $\nabla \log p(\hat{y}_0|\hat{x}_0)$ on fixed $(x_t, y_t)$ pairs during early timesteps of Gaussian deblurring. If the two disagree substantially while DPS-CM still outperforms DPS, the crafted trajectory, not the single-point collapse, is the source of the gain; if they agree, the collapse is harmless but the advertised less-biased posterior has yet to be demonstrated.

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Extended reading notes

Core claim

The central claim is that posterior-sampling error in diffusion-based restoration is reduced by replacing the clean measurement $y$ with a crafted noisy measurement $y_t$ in the likelihood estimate. The authors observe that DPS's likelihood gradient $\nabla_{x_t} \log p(y|\hat{x}_0)$ brings high-frequency information into the early stages, whereas a forward-process noisy measurement matches the early frequency content but carries random noise that corrupts late details. DPS-CM generates $y_t$ by denoising the measurement with another reverse trajectory conditioned on $y$, so the measurement trajectory follows the same low-to-high frequency schedule as the image trajectory. The posterior estimate then substitutes the Tweedie mean $\hat{y}_0 = E[y_0|y_t]$ for $y$, yielding $\nabla_{x_t} \log p(x_t|y_t) \approx s_\theta(x_t,t) - \zeta_t \nabla_{x_t} \|\hat{y}_0 - A(\hat{x}_0)\|^2$, and the method blends this with the original DPS term through a mixing weight $\mu$. Experiments on FFHQ and ImageNet report the best or second-best results across Gaussian and motion deblurring, 4x super-resolution, random and box inpainting, nonlinear deblurring, and Poisson-noise tasks.

Load-bearing premise

The method assumes that averaging over all plausible clean measurements can be replaced by the single most likely one, its denoised mean, without changing which direction the restoration is pushed; the paper gives no error bound for that replacement.

Editorial extensions

If this is right

  • Early-stage posterior-sampling error, measured by $\epsilon$-prediction error and intermediate reconstruction MSE, is smaller for DPS-CM than for DPS.
  • Restoration quality improves on Gaussian and motion deblurring, 4x super-resolution, random and box inpainting, nonlinear deblurring, and Poisson-noise degradation on both FFHQ and ImageNet.
  • Ablation with $\mu$ shows the crafted term and the original DPS term are complementary: $\mu = 0.5$ outperforms either pure version, and pure crafted guidance at $\mu = 1$ still beats DPS at $\mu = 0$.
  • The crafted measurement dominates Monte-Carlo augmented posterior estimates on perceptual metrics (LPIPS, FID) and is roughly 50 percent faster than FPS-SMC.
  • DPS-CM nearly doubles DPS's runtime, but an accelerated variant that disables crafted guidance after $t=400$ keeps most of the gain with over 20 percent faster sampling.

Reading between the lines

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

  • The single-point collapse of $E_{y_0\sim p(y_0|y_t)}[p(y_0|\hat{x}_0)]$ to $p(\hat{y}_0|\hat{x}_0)$ is an unproven step; replacing $\hat{y}_0$ with a Monte Carlo estimate over $p(y_0|y_t)$ would directly test how much of the gain depends on that collapse rather than on the crafted schedule.
  • Because the method reuses the image score model for the measurement trajectory, it should transfer to any inverse problem whose measurement lies on a manifold the pretrained model can denoise; for far modalities such as phase retrieval the paper itself notes a small measurement-modality diffusion model would be needed.
  • The frequency-matching rationale implies the benefit should scale with how aggressively the early schedule suppresses high frequencies; sharp-mask inpainting, where the paper's gains are largest, is consistent with that prediction and could be used to probe the mechanism on new operators.
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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

3 major / 6 minor

Summary. The paper proposes DPS-CM, an unsupervised diffusion posterior sampling method for noisy inverse problems. DPS-CM adds a second reverse diffusion trajectory that 'crafts' a sequence of noisy measurements y_t by denoising the observed measurement y with the same pretrained score model and DPS-style reconstruction guidance. The restoration trajectory x_t is then updated with a weighted combination of the standard DPS consistency gradient for y and a crafted-measurement consistency gradient for ŷ0 = E[y0|y_t]. Experiments on FFHQ and ImageNet cover Gaussian and motion deblurring, 4x super-resolution, random and box inpainting, nonlinear deblurring, and Poisson noise; Tables 1-5 report consistent improvements over DPS and most baselines. The paper claims that this integration introduces a less biased posterior estimate and avoids premature high-frequency signal injection during early sampling.

Significance. If the empirical findings hold, DPS-CM is a simple and practically useful plug-in improvement to DPS that requires no retraining and works across several forward operators and noise types. The strengths of the paper include its clearly specified algorithm, released code, tabulated hyperparameters, broad experimental scope, and comparisons with recent posterior-sampling methods such as FPS-SMC and LGD-MC. The paper's explanatory claim that crafted measurements yield a less biased posterior is not established by the derivation in Section 3.2, which contains two unjustified algebraic steps; the contribution should be reframed as an empirically strong heuristic unless the derivation is repaired.

major comments (3)
  1. [Section 3.2, Eq. (9)] The factorization of ∇_x log p(x_t|y_t) drops the prior density of y0. In the second line, p(x_t|y0) must be expanded as p(y0|x_t)p(x_t)/p(y0). The integral therefore equals p(x_t) E_{y0∼p(y0|y_t)}[p(y0|x_t)/p(y0)], so the gradient of log p(x_t|y_t) contains E[p(y0|x_t)/p(y0)], not E[p(y0|x_t)] as written. The omitted 1/p(y0) depends on y0 and cannot be absorbed as a constant, because it appears inside the expectation. This invalidates the stated identity and, with it, the derivation of Eq. (12) as the gradient of the crafted-measurement posterior.
  2. [Section 3.2, Eq. (10) and Algorithm 1, line 10] The point-estimate swap E_{y0∼p(y0|y_t)}[p(y0|ŷ0? x̂0)] ≈ p(ŷ0|x̂0) ignores the nonzero variance of y0|y_t. For the Gaussian measurement model with noise variance σ²I and the VP-SDE forward posterior variance (1−ᾱ_t)I, the exact expectation is N(ŷ0; A x̂0, (σ² + 1 − ᾱ_t)I), whose log-gradient carries the time-varying coefficient (σ² + 1 − ᾱ_t)^{-1}. Equation (12) instead uses a constant ζ_t, which cannot reproduce this time-varying factor; early in sampling 1−ᾱ_t ≈ 1, so the discrepancy is large. The advertised 'less-biased posterior estimate' is therefore unsupported unless the swap is justified by an error bound, the exact expectation is used, or Eq. (12) is explicitly stated to be a heuristic guidance term.
  3. [Section 3.2, Step 1, and Appendix D] The crafted measurement trajectory is generated with the same pretrained image-domain score model s_θ, and the derivation treats y_t as a forward-noised sample from the measurement distribution. This is an additional modeling assumption, stated informally as x and y lying on close manifolds, but it is not validated experimentally. Appendix D concedes that the construction fails for phase retrieval, which limits the claimed generality of the method. The authors should state this restriction in the main text or provide a concrete criterion or an ablation with a measurement-domain score model to show where the same-model construction is reliable.
minor comments (6)
  1. [Appendix A, Algorithm 2, line 10] The parentheses in the gradient term for the Poisson case are mismatched; the expression should close the gradient and the ∥·∥_Λ norm consistently.
  2. [Table 8 and Appendix D] There are small typos: 'DiffDIR' should be 'DiffPIR' in Table 8, and 'crated measurements' in Appendix D should be 'crafted measurements'.
  3. [Figure 1] The caption should define how the 32 Hz low/high frequency cutoff is chosen and state explicitly which curve corresponds to DPS, DPS_y_t, and DPS-CM, since the text refers to colors that may not be distinguishable in print.
  4. [Tables 1-5] The reported metrics are averages over 100 images without error bars or significance tests; some improvements are small (e.g., FFHQ 4× super-resolution PSNR 27.81 vs. DPS 27.55 in Table 1), so confidence intervals would strengthen the empirical claims.
  5. [Section 3.2 and Algorithm 1] The notation y_t for crafted measurements and y_t for forward-sampled noisy measurements is easy to confuse; using distinct symbols such as y_t^craft and y_t^fwd throughout would improve readability.
  6. [Section 4.2] The sentence claiming 'overall best performance over four metrics' should be qualified, because in Table 1 PSLD has a higher PSNR than DPS-CM on FFHQ Gaussian deblurring (28.45 vs. 27.45), even though DPS-CM is better on the other metrics.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the crafted-measurement construction is self-referential in the sense that it reuses the same pretrained score model, but the reported restoration improvements are evaluated against external data and do not reduce to fitted or self-cited quantities.

full rationale

The central derivation chain is heuristic but not circular. DPS-CM generates a crafted measurement trajectory y_t using the same pretrained score model s_theta that later serves as the restoration prior, and Eq. 10 replaces E_{y0~p(y0|y_t)}[p(y0|x_hat0)] with the point value p(y_hat0|x_hat0). This point-estimate swap is a mathematically uncontrolled approximation, and the resulting update in Eq. 12 may not be the gradient of the claimed log-marginal likelihood; however, that is a correctness or robustness concern rather than circularity, because the approximation is not equivalent to any assumed conclusion by construction. The paper does not fit a parameter to a target quantity and then rename that fit as a prediction, and it does not invoke a uniqueness theorem or load-bearing self-citation. The same-score-model reuse makes the construction self-referential, and the authors themselves note in Appendix D that the approach may fail when the measurement modality differs substantially from the target modality, but the headline results are benchmarked against external measurements and baseline methods using the same prior, so the empirical claims are not forced by the inputs. The lack of concentration or error bounds for the expectation-to-point swap weakens the theoretical rationale for 'less-biased posterior estimation,' but no equation in the paper reduces to its own input, and no prediction is equivalent to a fitted quantity. Accordingly, no circular step meeting the required evidentiary standard is present.

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

The method inherits standard diffusion-model assumptions plus three task-specific assumptions. Its main free parameters are three hand-tuned scalars per task and dataset; no new physical entities are introduced. The crafted measurement trajectory is an algorithmic construction, not an independently evidenced entity.

free parameters (3)
  • ζt (x-trajectory step size) = 1.4 to 3.1 for Gaussian noise; 0.4 to 0.5 for Poisson noise
    Tuned per task and dataset in Table 6; scales the reconstruction-gradient update in Eq.12.
  • ωt (y-trajectory step size) = 6.0 to 19.0 for Gaussian noise; 3.0 to 6.0 for Poisson noise
    Tuned per task and dataset in Table 6; scales the measurement-domain guidance in Eq.11.
  • µ (integration weight) = 0.285, 0.5, or 0.7 depending on task and dataset
    Chosen per task in Table 6; Figure 4 shows a one-task ablation where PSNR peaks at µ=0.5.
assumptions (3)
  • domain assumption Pre-trained score model sθ accurately approximates the true score of the target image distribution.
    Used via Tweedie's formula (Eq.4) to form both x̂0 and ŷ0; inherited from DPS and the diffusion-model literature.
  • ad hoc to paper The same score model sθ trained on images is a valid generative prior for the measurement y.
    Step 1 (Section 3.2) states there is no off-the-shelf model for the measurement modality and reuses sθ; Appendix D concedes this breaks for far-modality tasks like phase retrieval.
  • ad hoc to paper The expectation over p(y0|y_t) can be collapsed to its mean inside the likelihood: E[p(y0|x̂0)] ≈ p(E[y0|y_t]|x̂0).
    Eq.10 performs this swap with no bound or concentration argument; this is the core of the claimed reduction in posterior bias.

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Cite this review

Pith. "Pith review of Enhancing Diffusion Posterior Sampling for Inverse Problems by Integrating Crafted Measurements." pith.science (2026). https://pith.science/paper/PQMSMRGA

@misc{pith2026241109850,
  author       = {Pith},
  title        = {Pith review of: Enhancing Diffusion Posterior Sampling for Inverse Problems by Integrating Crafted Measurements},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PQMSMRGA}},
  note         = {Machine review of arXiv:2411.09850}
}
read the original abstract

Diffusion models have emerged as a powerful foundation model for visual generations. With an appropriate sampling process, it can effectively serve as a generative prior for solving general inverse problems. Current posterior sampling-based methods take the measurement (i.e., degraded image sample) into the posterior sampling to infer the distribution of the target data (i.e., clean image sample). However, in this manner, we show that high-frequency information can be prematurely introduced during the early stages, which could induce larger posterior estimate errors during restoration sampling. To address this observation, we first reveal that forming the log-posterior gradient with the noisy measurement ( i.e., noisy measurement from a diffusion forward process) instead of the clean one can benefit the early posterior sampling. Consequently, we propose a novel diffusion posterior sampling method DPS-CM, which incorporates a Crafted Measurement (i.e., noisy measurement crafted by a reverse denoising process, rather than constructed from the diffusion forward process) to form the posterior estimate. This integration aims to mitigate the misalignment with the diffusion prior caused by cumulative posterior estimate errors. Experimental results demonstrate that our approach significantly improves the overall capacity to solve general and noisy inverse problems, such as Gaussian deblurring, super-resolution, inpainting, nonlinear deblurring, and tasks with Poisson noise, relative to existing approaches. Code is available at: https://github.com/sjz5202/DPS-CM.

Figures

Figures reproduced from arXiv: 2411.09850 by the authors.

Figure 1
Figure 1. (a) mean magnitude ratio between high- and low-frequency area at timestep t; (b) ϵ-prediction error ratio: DPSyt DPS and DPS-CM DPS at timestep t; (c) reconstruction error between target x and intermediate xˆ0(xt) at timestep t. These results are achieved by performing Gaussian Deblurring on 10 FFHQ images. space of the gradient with the low-high cut-off frequency of 32Hz and visualize the mean magnitude ratio betwe… view at source ↗
Figure 2
Figure 2. Qualitative example of Gaussian Deblurring on FFHQ dataset including DPSyt . All measurements are corrupted by additional Gaussian noise with a variance of σy = 0.05. Such observations verify our claim that the gradient of log￾likelihood ∇xt log p (y|xˆ0) involved in DPS introduces more high-frequency signals during early stages which poses negative effects on early-time low-frequency recovery reflecting from the la… view at source ↗
Figure 3
Figure 3. Qualitative results of motion deblurring and inpaint￾ing (box) on ImageNet dataset. All measurements are corrupted by additional Gaussian noise with a variance of σy = 0.05. CM for baselines DPS, DDRM, OPC, DiffPIR, FPS-SMC, and LGD-MC. For PSLD, Stable Diffusion v-1.5 (Rombach et al., 2022) is applied. For ReSample, we use the VQ-4 autoencoder and the FFHQ-LDM with CelebA-LDM from latent diffusion (Rombach et al., … view at source ↗
Figures from the paper (15 more)
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: Additional Qualitative Results on Gaussian Deblur with Gaussian Noise. Measurement Reference DPS DiffPIR DPS-CM [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]
Figure 6
Figure 6. Figure 6: Additional Qualitative Results on Gaussian Deblur with Gaussian Noise. Measurement Reference DPS OPC DiffPIR PSLD ReSample DPS-CM [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: Additional Qualitative Results on Motion Deblurring with Gaussian Noise. Measurement Reference DPS OPC DiffPIR PSLD ReSample DPS-CM [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]
Figure 8
Figure 8. Figure 8: Additional Qualitative Results on super-resolution with Gaussian Noise. 13 [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]
Figure 9
Figure 9. Figure 9: Additional Qualitative Results on Box Inpainting with Gaussian Noise. Measurement Reference DPS OPC DiffPIR PSLD ReSample DPS-CM [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]
Figure 10
Figure 10. Figure 10: Additional Qualitative Results on Random Inpainting with Gaussian Noise. Measurement Reference DPS OPC DPS-CM [PITH_FULL_IMAGE:figures/full_fig_p014_10.png]
Figure 11
Figure 11. Figure 11 [PITH_FULL_IMAGE:figures/full_fig_p014_11.png]
Figure 12
Figure 12. Figure 12: Additional Qualitative Results on Nonlinear Deblurring with Gaussian Noise. Measurement Reference DPS DPS-CM [PITH_FULL_IMAGE:figures/full_fig_p014_12.png]
Figure 13
Figure 13. Figure 13: Additional Qualitative Results on Nonlinear Deblurring with Gaussian Noise. 14 [PITH_FULL_IMAGE:figures/full_fig_p014_13.png]
Figure 14
Figure 14. Figure 14: Additional Qualitative Results on Super-resolution with Gaussian noise compared with FPS-SMC (M = 10). Measurement Reference DPS FPS-SMC DPS-CM [PITH_FULL_IMAGE:figures/full_fig_p015_14.png]
Figure 15
Figure 15. Figure 15: Additional Qualitative Results on Super-resolution with Gaussian Noise. Measurement Reference DPS DPS-CM Measurement Reference DPS DPS-CM [PITH_FULL_IMAGE:figures/full_fig_p015_15.png]
Figure 16
Figure 16. Figure 16: Additional Qualitative Results on Gaussian Deblurring with Poisson noise. Measurement Reference DPS DPS-CM Measurement Reference DPS DPS-CM [PITH_FULL_IMAGE:figures/full_fig_p015_16.png]
Figure 17
Figure 17. Figure 17: Additional Qualitative Results on Motion Deblurring with Poisson noise. Measurement Reference DPS DPS-CM Measurement Reference DPS DPS-CM [PITH_FULL_IMAGE:figures/full_fig_p015_17.png]
Figure 18
Figure 18. Figure 18: Additional Qualitative Results on Super-resolution with Poisson noise. 15 [PITH_FULL_IMAGE:figures/full_fig_p015_18.png]

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    \@lbibitem[] @bibitem@first@sw\@secondoftwo \@lbibitem[#1]#2 \@extra@b@citeb \@ifundefined br@#2\@extra@b@citeb \@namedef br@#2 \@nameuse br@#2\@extra@b@citeb \@ifundefined b@#2\@extra@b@citeb @num @parse #2 @tmp #1 NAT@b@open@#2 NAT@b@shut@#2 \@ifnum @merge>\@ne @bibitem@firs...

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    @open @close @open @close and [1] URL: #1 \@ifundefined chapter * \@mkboth \@ifxundefined @sectionbib * \@mkboth * \@mkboth\@gobbletwo \@ifclassloaded amsart * \@ifclassloaded amsbook * \@ifxundefined @heading @heading NAT@ctr thebibliography [1] @ \@biblabel @NAT@ctr \@bibset...

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.