{"id":"605dd8bc-58fb-4830-a88c-da01a6f3650a","arxiv_id":"2411.09850","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"DPS-CM improves diffusion posterior sampling for inverse problems by generating a denoised reverse-measurement trajectory and using it in the likelihood gradient, yielding better restoration in experiments.","lead":"This paper proposes DPS-CM, a diffusion-model method for restoring degraded images that feeds the sampler a crafted noisy version of the measurement instead of the original sharp measurement. On deblurring, super-resolution, and inpainting benchmarks it reports quality gains over existing zero-shot solvers.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. 10's point-estimate swap ignores the nonzero variance of y0|y_t; under the Gaussian likelihood the resulting gradient is not the gradient of the intended log-expectation, so the 'less-biased posterior' rationale is unsupported.","rationale":"The central claim has two parts: a mechanism (crafted measurements reduce posterior-estimate bias via Eq. 10) and an empirical outcome (better restoration). The empirical outcome is supported by broad tables, ablations, and public code, and I do not dispute that DPS-CM often beats DPS on the reported tasks. The load-bearing concern is the mechanism. The reader identified the Eq. 10 point-estimate swap as an unproven concentration step; I agree and sharpen it: under the paper's own Gaussian likelihood, the swap is quantitatively wrong unless the posterior variance of y0 given y_t is zero. The exact expectation yields covariance σ²I+(1−ᾱ_t)I, so the gradient scale is 1/(σ²+1−ᾱ_t), not 1/σ². Since 1−ᾱ_t is large at early timesteps, DPS-CM's guidance is not the gradient of the log-marginal likelihood it claims to approximate. This does not refute the empirical method: ζ_t, ω_t, and μ are tuned per task, and the μ=0 versus μ=0.5 ablation suggests the crafted term contributes. But it means the paper's theoretical justification, 'less biased posterior estimate,' is unsupported, and the contribution is best described as an effective heuristic. A rerun with the exact expectation or a Monte Carlo estimate of the outer expectation would settle whether the point swap is essential or incidental. Because the empirical claims are broad but variance-free (100 samples, no error bars) and the derivation is questionable, the CONDITIONAL verdict is appropriate; I would not move it.","tokens_in":17869,"tokens_out":8393,"duration_ms":88202,"concrete_test":"Independently derive Eq. 10 under the paper's Gaussian measurement model with q(y0|y_t)=N(ŷ0,(1−ᾱ_t)I): the exact log-marginal gradient is (σ²+1−ᾱ_t)^{-1}∇_{x_t}||ŷ0−A(x̂0)||², not the point-estimate gradient. Then re-run the FFHQ Gaussian-deblur row of Table 1 with this corrected gradient, keeping ζ_t, ω_t, and μ otherwise fixed. If PSNR/FID shifts materially, the point swap is the active ingredient; if not, the contribution should be reframed as an effective two-loss heuristic.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3.2's derivation of Eq. 10 is the load-bearing step. The paper replaces E_{y0∼p(y0|y_t)}[p(y0|x̂0)] with p(ŷ0|x̂0), where ŷ0=E[y0|y_t]. For the Gaussian measurement model p(y0|x0)=N(y0;Ax0,σ²I) and the VP-SDE posterior q(y0|y_t)=N(ŷ0,(1−ᾱ_t)I), the exact expectation is N(ŷ0;Ax̂0,(σ²+1−ᾱ_t)I), so its log-gradient is proportional to (σ²+1−ᾱ_t)^{-1}∇_{x_t}||ŷ0−A(x̂0)||², not σ^{-2} as implicitly assumed by Eq. 12. At early timesteps 1−ᾱ_t≈1 and dominates σ²=0.0025, so the guidance magnitude is off by a large time-varying factor that a constant ζ_t cannot absorb. Thus Eq. 12 is not the gradient of the claimed log-marginal likelihood, and the advertised 'less-biased posterior estimate' is not established by the derivation. The empirical gains could instead come from the added (1−μ)||y−A(x̂0)||² DPS term and per-task tuning of {ζ_t,ω_t,μ} in Appendix B.1. The method may still work as a heuristic, but the theoretical rationale should be downgraded unless the point swap is justified or the exact expectation is used.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":18258,"tokens_out":14430,"duration_ms":151822,"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":[{"comment":"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.","section":"Section 3.2, Eq. (9)"},{"comment":"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.","section":"Section 3.2, Eq. (10) and Algorithm 1, line 10"},{"comment":"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.","section":"Section 3.2, Step 1, and Appendix D"}],"minor_comments":[{"comment":"The parentheses in the gradient term for the Poisson case are mismatched; the expression should close the gradient and the ∥·∥_Λ norm consistently.","section":"Appendix A, Algorithm 2, line 10"},{"comment":"There are small typos: 'DiffDIR' should be 'DiffPIR' in Table 8, and 'crated measurements' in Appendix D should be 'crafted measurements'.","section":"Table 8 and Appendix D"},{"comment":"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.","section":"Figure 1"},{"comment":"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.","section":"Tables 1-5"},{"comment":"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.","section":"Section 3.2 and Algorithm 1"},{"comment":"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.","section":"Section 4.2"}],"recommendation":"major_revision","confidential_remarks":"The manuscript has a solid empirical component and is likely to interest the inverse-problem and diffusion-sampling community. The main issue is the theoretical derivation in Section 3.2: the two algebraic steps discussed in the major comments are not innocuous, and the 'less-biased posterior' interpretation should be corrected or downgraded to a heuristic statement. Because the algorithm is clearly specified, the code is released, and the empirical results are broad, these problems are repairable within the scope of a major revision; I do not recommend rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: DPS-CM is a real, reproducible improvement over DPS on the tasks tested, and the crafted-measurement idea is new and worth borrowing. The theory in Section 3.2, however, does not back the claim that the method gives a less-biased posterior estimate. Treat it as a heuristic with a strong empirical record, not as a justified derivation.\n\nWhat's new: running a second diffusion reverse process on the measurement y to produce a crafted trajectory {y_t}, then plugging the Tweedie mean \\hat{y}_0 = E[y0|y_t] into the DPS likelihood term. I checked the citations—DPS, DDRM, DiffPIR, OPC, FPS-SMC, LGD-MC don't do this. The algorithm is clearly specified, the code is public, and the experiments cover six inverse problems on FFHQ and ImageNet, with a reasonable set of baselines including recent ones. The gains over DPS are consistent, roughly 1-6 dB PSNR, and the ablation against DPSyt supports the claim that the crafted trajectory, not just any noisy measurement, is what helps. That is a solid empirical package.\n\nThe soft spot is exactly the one the stress-test note flags. Eq. 10 replaces an expectation over p(y0|y_t) with a single point \\hat{y}_0 = E[y0|y_t]. For the Gaussian measurement model, the exact expectation is a Gaussian with covariance (σ^2 + 1-\\barα_t)I, so the log-gradient scale is (σ^2 + 1-\\barα_t)^{-1}, not the σ^{-2} implicit in Eq. 12. At early timesteps 1-\\barα_t ≈ 1, so the magnitude is off by a large time-varying factor that a constant ζ_t cannot absorb. The paper hasn't justified the swap, and the 'less-biased posterior' claim is therefore not established. In addition, no error bars or significance tests support the word 'significantly,' and the hyperparameters {ζ_t, ω_t, μ} are tuned per task and dataset, which weakens the plug-and-play claim. These are real limitations, but none of them kills the method. The empirical pattern is clear, and the heuristic could well be valuable.\n\nWho this is for: anyone working on zero-shot inverse problems with diffusion priors who wants a cheap, well-tested alternative to DPS and doesn't mind about 2x compute. The paper deserves a serious referee. I would send it to review, with a request to either justify the point swap or use the exact expectation, add measures of variance, and discuss hyperparameter sensitivity. As is, the right summary is: empirically useful, theoretically unproven, and honest about its limitations.","headline":"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.","tokens_in":18775,"tokens_out":4636,"would_cite":true,"duration_ms":42493,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Replacing the clean measurement with a crafted noisy one reduces diffusion-based restoration error.","keywords":["diffusion posterior sampling","inverse problems","crafted measurements","image restoration","deblurring","super-resolution","inpainting","Poisson noise"],"falsifier":"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.","tokens_in":17676,"feed_emoji":"🖼️","tokens_out":11946,"duration_ms":103177,"temperature":0.7,"pith_summary":"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.","feed_headline":"Crafted noisy measurements sharpen diffusion image restoration","feed_subtitle":"It beats DPS on deblurring, super-resolution, and inpainting by cutting early posterior error.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the DPS posterior approximation $p(y|x_t)\\approx p(y|\\hat{x}_0)$ that DPS-CM modifies, and the baseline it is compared against throughout.","marker":"[Chung et al., 2022a]"},{"why":"Defines the DDPM forward and reverse processes that both the measurement and image trajectories follow.","marker":"[Ho et al., 2020]"},{"why":"Provides the reverse-time SDE formulation of conditional diffusion sampling that Eq. 5 builds on.","marker":"[Song et al., 2020]"},{"why":"Underlies Tweedie's formula used to compute the posterior means $\\hat{x}_0$ and $\\hat{y}_0$.","marker":"[Efron, 2011]"},{"why":"DDRM is a spectral-space baseline that avoids the likelihood-gradient approximation and is used for comparison.","marker":"[Kawar et al., 2022]"},{"why":"OPC is an improved posterior-estimate baseline that constructs optimal posterior covariance, against which DPS-CM's gains are measured.","marker":"[Peng et al., 2024]"},{"why":"LGD-MC is a Monte-Carlo posterior-estimate baseline used in the ablation comparing augmented likelihood estimates.","marker":"[Song et al., 2023b]"},{"why":"FPS-SMC is a filtering-based posterior sampler that augments the measurement trajectory, used as an ablation and efficiency comparison.","marker":"[Dou & Song, 2023]"},{"why":"DiffPIR is a plug-and-play restoration baseline that alternates denoising and data consistency, used in the comparisons.","marker":"[Zhu et al., 2023]"}],"fun_headline_variants":["Crafted noise improves diffusion inverse problem solving","Diffusion posterior sampling gets a crafted measurement boost","How crafted noisy measurements fix diffusion restoration errors","Better diffusion recovery via crafted measurement alignment","DPS-CM: Crafted measurements refine diffusion posterior sampling"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Crafted noise improves diffusion inverse problem solving","Diffusion posterior sampling gets a crafted measurement boost","How crafted noisy measurements fix diffusion restoration errors","Better diffusion recovery via crafted measurement alignment","DPS-CM: Crafted measurements refine diffusion posterior sampling"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000506,"raw_usage":{"total_tokens":2532,"prompt_tokens":1073,"completion_tokens":1459,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":689,"completion_tokens_details":{"reasoning_tokens":1389}},"tokens_in":689,"tokens_out":1459,"duration_ms":11625,"temperature":1.0,"reasoning_tokens":1389,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T20:13:49.543594+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}