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REVIEW 3 major objections 5 minor 37 references

Enhancing and Accelerating Diffusion-Based Inverse Problem Solving through Measurements Optimization

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper claims that repeatedly fitting the measurement with Stochastic Gradient Langevin Dynamics between diffusion denoising steps turns 1000–4000-step inverse problem solvers into 50–100-step solvers without sacrificing quality.

desk verdict A fast, well-ablated plug-in for diffusion inverse solvers; the empirical story is convincing enough to referee, but the projection theory is thin and the evaluation shortcuts need tightening. read the letter →

arxiv 2412.03941 v1 pith:4MBMZSGR submitted 2024-12-05 cs.CV cs.AI

Tianyu Chen , Zhendong Wang , Mingyuan Zhou This is my paper · ORCID
classification cs.CVcs.AI
keywords diffusionmodelsinverseproblemsmeasurementsoptimizationstochasticgradientLangevindynamicsplug-and-playmoduleimagerestorationposteriorsamplingphaseretrieval
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

Diffusion models can restore images from corrupted measurements, but current solvers need 1000 to 4000 diffusion evaluations per image because they feed in only a sliver of measurement information at each step. This paper introduces Measurements Optimization (MO), a plug-in module that injects far more measurement information per step: it first runs Stochastic Gradient Langevin Dynamics to fit an image to the measurement, then adds noise and denoises with a pretrained diffusion model to pull the fit back onto the natural-image manifold. Plugging MO into existing solvers such as DPS and Red-diff cuts the required evaluations to 50–100 for most tasks, a 10 to 40-fold speedup, while matching or beating the previous state of the art in quality.

What carries the argument

The load-bearing object is the Measurements Optimization (MO) module, an inner-outer loop wrapped around any diffusion sampler. At each diffusion time step $t$, an inner loop of $N$ stochastic gradient ascent steps of the form $x \leftarrow x + \eta \nabla_x \|y - A(x)\|_2^2 + \sqrt{2\eta}\,\epsilon$ with $\epsilon \sim \mathcal{N}(0, I)$ fits an image to the measurement (this is SGLD); then the outer loop noises the result to level $\sigma(t)$ and feeds it to the mean-predicted denoiser $D_\theta$, which returns the estimate $\hat{x}_0$ that seeds the next step. Theorem 1 (adapted from Section B.3 of EDM) justifies the denoising query: an ideal denoiser trained on a finite dataset returns a likelihood-weighted mixture of training images, so the query projects the SGLD solution onto the training-data manifold, and the Gaussian noise added by SGLD lets the diffusion model overwrite unrecoverable content. The paper uses the EDM schedule $s(t)=1$, $\sigma(t)=t$ with a $\rho=7$ discrete time grid, and sets the SGLD learning rate to $5 \times 10^{-5}$ for all tasks except ImageNet phase retrieval.

What would settle it

Run MO on any task with a known forward operator and compare the measurement residual $\|y - A(\hat{x}_0)\|_2$ immediately before and after the denoising query at several noise levels; if denoising consistently raises the residual above the level SGLD already reached, the manifold projection is discarding measurement information and the alternating scheme cannot converge to a measurement-consistent solution.

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Extended reading notes

Core claim

The central claim is that the high NFE cost of diffusion inverse problem solving is not an inherent property of the diffusion prior but a consequence of injecting too little measurement information per step. Existing methods take a single gradient step on the fidelity loss $\|y - A(\hat{x}_0)\|_2^2$ per diffusion evaluation; MO instead runs an inner SGLD loop that repeatedly minimizes $\|y - A(x)\|_2^2$, then asks the pretrained denoiser $D_\theta(x; \sigma(t))$ to pull the result back onto the data manifold via a noise-add-and-denoise query. The paper asserts that this two-step iteration, run at every diffusion time step, achieves state-of-the-art or near-state-of-the-art results on five linear and three nonlinear tasks on FFHQ 256 and ImageNet 256, with at most 100 function evaluations per image in all cases except ImageNet phase retrieval (1000). The headline numbers are DPS-MO reaching 28.71 dB PSNR on FFHQ HDR at 100 NFEs versus DAPS's 27.12 dB at 4000 NFEs, and Red-diff-MO raising Red-diff's phase-retrieval PSNR from 15.60 to 27.34 dB at a tenth of the NFEs.

Load-bearing premise

The whole scheme rests on the assumption that a real pretrained diffusion denoiser, when queried by adding noise and denoising, reliably returns an estimate that stays on the natural-image manifold while still carrying the measurement information the SGLD steps just encoded; if that projection corrupts or discards the measurement fit, the alternating loop cannot converge to a correct solution.

Editorial extensions

If this is right

  • DPS-MO reaches state-of-the-art or near-state-of-the-art quality on eight linear and nonlinear tasks using no more than 100 NFEs, except for ImageNet phase retrieval which uses 1000 NFEs (Tables 1–4).
  • MO is a plug-and-play module: integrating it into DPS yields DPS-MO and into Red-diff yields Red-diff-MO, and both improve on their base methods; Red-diff-MO raises phase-retrieval PSNR from 15.60 to 27.34 dB at 100 NFEs versus 1000 (Table 5).
  • The module reduces wall-clock time and memory overhead relative to DPS: DPS-MO produces an inpainted image in 5.3 seconds versus over 60 seconds for DPS and Red-diff, with memory comparable to Red-diff (Table 7).
  • The choice of inner optimizer matters: SGLD clearly outperforms Adam for inpainting, and the authors attribute this to the Gaussian noise SGLD adds, which the diffusion model can then use to redraw masked or corrupted content (Table 6).
  • Refreshing the SGLD solution at every diffusion step with an initialization from the previous denoised output is essential; reusing a single SGLD solution across all steps degrades LPIPS/PSNR substantially (Table 9).

Reading between the lines

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

  • Because MO only needs gradient access to the forward operator, the same inner SGLD loop should extend to other differentiable measurement models such as magnetic resonance or computed tomography reconstructions without retraining; the paper demonstrates the generality only through its eight tasks.
  • The fresh Gaussian noise injected by SGLD at every inner step suggests MO is sampling from an approximate posterior rather than merely minimizing a loss, so the module could be repurposed to generate diverse measurement-consistent samples beyond the four-run phase-retrieval and inpainting illustrations.
  • A direct stress test would be to run MO under larger measurement noise or with fewer SGLD steps; the paper fixes Gaussian noise at $\sigma_n = 0.05$ and tunes the step count per task, so the claimed 10–40-fold speedup under realistic noise levels remains untested.
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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 / 5 minor

Summary. The paper proposes Measurements Optimization (MO), a plug-and-play module for diffusion-based inverse problem solving. At each diffusion step, MO runs several SGLD steps on the measurement loss and then adds noise to the resulting image and denoises it with a pretrained diffusion model, aiming to project the solution back onto the image manifold. The module is integrated into DPS and Red-diff, yielding DPS-MO and Red-diff-MO. The authors report state-of-the-art or near-state-of-the-art performance on five linear and three nonlinear tasks on FFHQ-256 and ImageNet-256, typically with 50-100 NFEs (with ImageNet phase retrieval requiring 1000 NFEs). They also provide ablations over sampling schedules, optimizer choice, SGLD step count, and memory/time comparisons.

Significance. If the empirical claims hold, the paper offers a practically valuable 10-40x speedup over existing diffusion-based inverse problem solvers, with a simple module that is orthogonal to the underlying sampler and can be inserted into existing methods. The paper is thorough in task coverage, clearly specifies the algorithm, and provides extensive qualitative results. The significance is moderated, however, by the weak theoretical support for the core manifold-projection mechanism and by evaluation choices that do not quantify uncertainty and involve substantial per-task hyperparameter tuning. The claimed SOTA gains would be more convincing with variance-aware reporting and a direct check of measurement consistency after the denoising step.

major comments (3)
  1. [§5.1–5.2, Tables 1–4] The claim that the noising/denoising step "projects the optimization solution back onto the training data manifold" is not supported by Theorem 1. The theorem, adapted from EDM §B.3, characterizes the optimal denoiser for an empirical distribution as a weighted average of training images; a weighted average is not an element of the training set, so the theorem does not establish a manifold projection in the sense used in the paper. Moreover, the result assumes an ideal denoiser and a finite training set, and the paper provides no evidence that a real diffusion denoiser at finite σ preserves the measurement information introduced by the SGLD steps. I recommend adding a direct check of the measurement residual after the denoising step, e.g., ||y − A(Dθ(xSGLD_t + σ(t)ε, σ(t)))||, and an ablation that isolates the effect of the projection (e.g., skipping it or replacing it with a simple proximal step).
  2. [§5.1–5.2, Tables 1–4] All quantitative comparisons are reported as point estimates (means over 100 images) with no variance or confidence intervals, and for nonlinear tasks the metrics are the best over four independent samples. The headline difference, e.g., DPS-MO 28.71 dB vs. DAPS 27.12 dB on FFHQ HDR, could be within run-to-run variability, especially under best-of-four selection. Please report means ± standard deviations (or confidence intervals) for at least the key DPS-MO and best-baseline comparisons, and clarify whether the best-of-four selection is applied to baselines as well.
  3. [§5.2, Table 8, Appendix 8.3] Many hyperparameters are tuned per task and dataset: NFE, σmax, σmin, and the number of SGLD steps N_SGLD are chosen to "achieve the best performance" via grid search, and σmax is set to 1 (rather than the default 80) for ImageNet SR and random inpainting. This makes the 50–100 NFE claim conditional on substantial per-task calibration and weakens the "plug-and-play" generality. Please state the selection procedure (e.g., a validation split) and report sensitivity of the main results to these hyperparameters, or at least discuss the cost of tuning relative to the baselines.
minor comments (5)
  1. [Appendix 8.4] The SGLD update in Algorithm 1 uses a single learning rate η, but Appendix 8.4 defines a time-dependent schedule η_i with p=2, r=0.01 and a loss scaled by 1/(2τ²) with τ=0.01. The algorithm and main text should be aligned with the implementation for reproducibility.
  2. [§5.3] The section title contains a typo: "Performace" should be "Performance."
  3. [Table 10 caption] The caption says "100 and 100 NFEs" but should presumably read "1000 and 1000 NFEs"; the same issue appears in Table 12. Please check all captions for this typo.
  4. [Figures 3 and 5] The axis labels contain "uni00A0" artifacts (non-breaking space characters) that should be removed before camera-ready.
  5. [§4, references] The claim that SGLD "guarantees asymptotic convergence to global minima in non-convex settings" with reference [7] is stronger than what Gelfand and Mitter (1991) prove, which concerns annealed Langevin dynamics under specific conditions. Please soften the claim or cite a more specific SGLD convergence result.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: MO is an empirical plug-and-play algorithm evaluated on external benchmarks, and its only theoretical support, Theorem 1, is an external EDM result with stated assumptions rather than a self-referential reduction.

full rationale

The claimed derivation chain is algorithmic: SGLD minimizes the measurement loss, and a pretrained diffusion denoiser projects the result back toward the data distribution. The reported gains are benchmark comparisons against DAPS, DPS, Red-diff, and other external baselines, not predictions derived from fitted parameters. No parameter is fitted to a subset of data and then renamed as a prediction; the SGLD learning rates, step counts, NFE counts, and noise-schedule endpoints are disclosed hyperparameters selected by grid search in Appendix 8.3 and Table 8, which is standard model selection and does not make the evaluation circular. Theorem 1 is adapted from EDM Section B.3, an external source with no overlapping authors, and its proof in Appendix 7 derives the optimal denoiser for a finite empirical distribution; it does not assume or entail the success of MO, so it is not circular, though it may be an incomplete justification for real diffusion models at finite noise levels. Baselines are taken from [30] under the same task configuration, so the comparison is independent. The manuscript's self-identified limitation, non-differentiable or expensive forward operators in Section 6, is a practical scope limitation and not a circular step. The only substantive concern, namely whether the noising/denoising projection in Algorithm 1 preserves measurement information, is an unresolved robustness or correctness issue, not a case where the output is equivalent to the input by construction. No quotation from the paper exhibits a fitted quantity being reported as a prediction or a claim whose premises already contain its conclusion, so the circularity burden is not met and the score is 0.

Assumptions & free parameters 7 free parameters · 4 assumptions · 0 invented entities

The central method rests on a pretrained diffusion prior and a set of per-task tuned hyperparameters. The main unproved assumption is that the noising/denoising query acts as a valid projection onto the natural image manifold. No new physical entities are postulated.

free parameters (7)
  • SGLD learning rate eta = 5e-5 for all tasks except ImageNet phase retrieval, where 5e-4
    Chosen by hand; controls how aggressively the measurement loss is minimized and is central to MO behavior.
  • Number of SGLD steps N per diffusion step = 50 to 500 depending on task (Table 8)
    Grid-searched per task to achieve the best reported performance; directly determines optimization strength and runtime.
  • Diffusion noise level sigma_max = 80 (default) or 1 for ImageNet SR and ImageNet random inpainting
    Tuned per task; the value 1 was selected to achieve SOTA performance for those two tasks.
  • sigma_min = 0.002, 0.02, or 0.05 per task
    Chosen from a small set per task to avoid overfitting; affects the final denoising noise level.
  • Number of diffusion evaluations (NFE) = 50 or 100 for most tasks; 1000 for ImageNet phase retrieval
    Selected per task as the lowest NFE that maintains SOTA performance; this is part of the paper's efficiency claim.
  • SGLD noise scale tau = 0.01
    Set to match the assumed unknown measurement noise magnitude; scales the gradient in the SGLD update.
  • SGLD schedule shape p and r = p=2, r=0.01
    Used in the per-step learning rate schedule in Appendix 8.4; adopted from DAPS and not tuned in this paper.
assumptions (4)
  • domain assumption Pretrained diffusion denoiser D_theta projects any x onto the data manifold when noise is added and denoised
    Used in Algorithm 1, lines 9-10; justified heuristically via Theorem 1, which assumes a finite training set and an ideal denoiser.
  • domain assumption SGLD, with a finite number of steps and chosen learning rate, finds low-loss solutions for the measurement objective even for non-convex A
    Invoked in Section 4; theoretical guarantees for SGLD only hold asymptotically under regularity conditions that are not verified for the tested forward operators.
  • domain assumption Measurement noise is Gaussian with known scale, y = A(x0) + n, n ~ N(0, sigma_n^2 I)
    Standard Bayesian inverse problem assumption in Section 3; experiments set sigma_n = 0.05 for all tasks.
  • domain assumption Theorem 1 (adapted from EDM, Section B.3) applies to the actual pretrained diffusion models used in experiments
    The theorem assumes a finite training set, sufficient data, and sufficient model capacity; these conditions are not checked for the FFHQ and ImageNet checkpoints.

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Cite this review

Pith. "Pith review of Enhancing and Accelerating Diffusion-Based Inverse Problem Solving through Measurements Optimization." pith.science (2026). https://pith.science/paper/4MBMZSGR

@misc{pith2026241203941,
  author       = {Pith},
  title        = {Pith review of: Enhancing and Accelerating Diffusion-Based Inverse Problem Solving through Measurements Optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4MBMZSGR}},
  note         = {Machine review of arXiv:2412.03941}
}
read the original abstract

Diffusion models have recently demonstrated notable success in solving inverse problems. However, current diffusion model-based solutions typically require a large number of function evaluations (NFEs) to generate high-quality images conditioned on measurements, as they incorporate only limited information at each step. To accelerate the diffusion-based inverse problem-solving process, we introduce \textbf{M}easurements \textbf{O}ptimization (MO), a more efficient plug-and-play module for integrating measurement information at each step of the inverse problem-solving process. This method is comprehensively evaluated across eight diverse linear and nonlinear tasks on the FFHQ and ImageNet datasets. By using MO, we establish state-of-the-art (SOTA) performance across multiple tasks, with key advantages: (1) it operates with no more than 100 NFEs, with phase retrieval on ImageNet being the sole exception; (2) it achieves SOTA or near-SOTA results even at low NFE counts; and (3) it can be seamlessly integrated into existing diffusion model-based solutions for inverse problems, such as DPS \cite{chung2022diffusion} and Red-diff \cite{mardani2023variational}. For example, DPS-MO attains a peak signal-to-noise ratio (PSNR) of 28.71 dB on the FFHQ 256 dataset for high dynamic range imaging, setting a new SOTA benchmark with only 100 NFEs, whereas current methods require between 1000 and 4000 NFEs for comparable performance.

Figures

Figures reproduced from arXiv: 2412.03941 by the authors.

Figure 1
Figure 1. [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Workflow overview illustrating how our MOmodule can be integrated into sampling-based methods (Full algorithm is in Algo￾rithm 2). Solving the MOmodule with measurement y provides substantial information, reducing the NFE requirements for inference. the mean-predicted diffusion model Dθ to obtain xˆ0(xt). For simplicity, we use xˆ0 as an abbreviation for xˆ0(xt) when there is no ambiguity. The full algorithm of MO i… view at source ↗
Figure 4
Figure 4. Inpainting task with 170 × 170 box. Four independent runs are able to genrate different faces and provide diversity. domness leads to greater diversity in generation, as shown in [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figures from the paper (17 more)
Figure 5
Figure 5. Figure 5: Performance comparison with respect to NFEs, showing [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: Visualization and comparison of the intermediate stages [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: LPIPS and SSIM metrics on FFHQ Box Inpainting tasks [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 8
Figure 8. Figure 8: Phase Retrieval 4 [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]
Figure 9
Figure 9. Figure 9: Phase Retrieval Ground Truth Measurement Recovered Ground Truth Measurement Recovered Ground Truth Measurement Recovered Ground Truth Measurement Recovered Ground Truth Measurement Recovered Ground Truth Measurement Recovered [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]
Figure 10
Figure 10. Figure 10: Inpainting (Random) 5 [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]
Figure 11
Figure 11. Figure 11: Inpainting (Random) Ground Truth Measurement Recovered Ground Truth Measurement Recovered Ground Truth Measurement Recovered Ground Truth Measurement Recovered Ground Truth Measurement Recovered Ground Truth Measurement Recovered [PITH_FULL_IMAGE:figures/full_fig_p01…
Figure 12
Figure 12. Figure 12: Inpainting (Box) Ground Truth Measurement Recovered Ground Truth Measurement Recovered Ground Truth Measurement Recovered Ground Truth Measurement Recovered Ground Truth Measurement Recovered Ground Truth Measurement Recovered [PITH_FULL_IMAGE:figures/full_fig_p016_12.png]
Figure 13
Figure 13. Figure 13: Inpainting (Box) 6 [PITH_FULL_IMAGE:figures/full_fig_p016_13.png]
Figure 14
Figure 14. Figure 14: High Dynamic Range Ground Truth Measurement Run 1 Run 2 Run 3 Run 4 Ground Truth Measurement Run 1 Run 2 Run 3 Run 4 Ground Truth Measurement Run 1 Run 2 Run 3 Run 4 [PITH_FULL_IMAGE:figures/full_fig_p017_14.png]
Figure 15
Figure 15. Figure 15: High Dynamic Range 7 [PITH_FULL_IMAGE:figures/full_fig_p017_15.png]
Figure 16
Figure 16. Figure 16: Nonlinear Deblurring Ground Truth Measurement Run 1 Run 2 Run 3 Run 4 Nonlinear blur Ground Truth Measurement Run 1 Run 2 Run 3 Run 4 Nonlinear blur Ground Truth Measurement Run 1 Run 2 Run 3 Run 4 Nonlinear blur [PITH_FULL_IMAGE:figures/full_fig_p018_16.png]
Figure 17
Figure 17. Figure 17: Nonlinear Deblurring 8 [PITH_FULL_IMAGE:figures/full_fig_p018_17.png]
Figure 18
Figure 18. Figure 18: Motion Deblurring Ground Truth Measurement Recovered Ground Truth Measurement Recovered Ground Truth Measurement Recovered Ground Truth Measurement Recovered Ground Truth Measurement Recovered Ground Truth Measurement Recovered [PITH_FULL_IMAGE:figures/full_fig_p019_…
Figure 19
Figure 19. Figure 19: Motion Deblurring Ground Truth Measurement Recovered [PITH_FULL_IMAGE:figures/full_fig_p019_19.png]
Figure 20
Figure 20. Figure 20: Gaussian Deblurring 9 [PITH_FULL_IMAGE:figures/full_fig_p019_20.png]
Figure 21
Figure 21. Figure 21: Gaussian Deblurring 10 [PITH_FULL_IMAGE:figures/full_fig_p020_21.png]

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    Bingliang Zhang, Wenda Chu, Julius Berner, Chenlin Meng, Anima Anandkumar, and Yang Song. Improving diffusion inverse problem solving with decoupled noise annealing. arXiv preprint arXiv:2407.01521, 2024. 1, 2, 5, 6 9

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    The unreasonable effectiveness of deep features as a perceptual metric

    Richard Zhang, Phillip Isola, Alexei A Efros, Eli Shecht- man, and Oliver Wang. The unreasonable effectiveness of deep features as a perceptual metric. In Proceedings of the IEEE conference on computer vision and pattern recogni- tion, pages 586–595, 2018. 5 10 Enhancing and A...

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    1 n nX i=1 N (x; zi, σ2I)∥D(x; σ) − zi∥2 2 # 0 = nX i=1 N (x; zi, σ2I)∇D(x;σ) ∥D(x; σ) − zi∥2 2 0 = nX i=1 N (x; zi, σ2I) [2D(x; σ) − 2zi] 0 =

    Complete Proof of Theorem 1 The proof is adopted from Section B.3 of EDM [11]. We include the proof here for the completeness. Proof 1 (Proof of Theorem 1) Assume the mean- predicted diffusion model is trained on a finite of sample {z1, z2, . . . ,zn}. Then we assume the distr...

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    Task Configuration We use the exactly the same task configuration as in [30] where you can find in Section D.1 in [30]

    Experiment Details 8.1. Task Configuration We use the exactly the same task configuration as in [30] where you can find in Section D.1 in [30]. 8.2. Baseline Implementation The baselines results are reported by [30]. Since we using the same configuration and dataset as [30] fo...

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    SGLD Initialization As described in Algorithm 2 and Algorithm 3, for eachMO, we initialize with ˆx0 from the previous diffusion time step. A natural question arises: what if we solve ∥y − A(x0)∥2 2 only once and reuse the solution to query the diffusion prior, including the no...

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    Additional Results We also include the empirical results for our method under 1000 and 4000 NFEs, as shown in Tables 10, 11, 12, and

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    This suggests that our method saturates at an early stage for most tasks

    These results indicate that increasing NFEs provides only marginal benefits for PSNR and LPIPS, and may even lead to overfitting. This suggests that our method saturates at an early stage for most tasks

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    More Qualitative Samples Here, we present qualitative samples from the FFHQ and ImageNet datasets across various tasks. 3 Method SR (×4) Inpaint (Box) Inpaint (Random)Gaussian deblurringMotion deblurringNFELPIPS↓ PSNR↑ LPIPS↓ PSNR↑ LPIPS↓ PSNR↑ LPIPS↓ PSNR↑ LPIPS↓ PSNR↑ DPS-MO...

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.