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A hyperparameter-free alignment rule sets the likelihood step in diffusion posterior sampling, claiming better perceptual quality with little or no distortion loss, across inverse problems.

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AdaPS adaptively scales likelihood guidance in DDIM posterior sampling via agreement between two surrogates, improving LPIPS/PSNR trade-offs on image restoration tasks without task-specific tuning.

T0 review reviewed 2026-08-03 challenge →

load-bearing objection Empirically promising adaptive guidance sampler, but Eq. 24 has a time-dependent factor error that undercuts the 'derived' step size; send to review with expectation of major revision. the 4 major comments →

arxiv 2511.18471 v3 pith:TJIOWTUP submitted 2025-11-23 cs.CV

Jacobian-Aware Posterior Sampling for Inverse Problems

Liav Hen , Tom Tirer , Raja Giryes , Shady Abu-Hussein This is my paper
classification cs.CV
keywords diffusion modelsinverse problemsposterior samplingadaptive guidanceDDIMimage restorationhyperparameter-freelikelihood approximation
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

AdaPS is a way to set the strength of the data-fidelity (likelihood) update in diffusion-based posterior sampling, a step that previous methods tune per task or fix by schedule. The paper shows that the right step size is the projection coefficient between two complementary surrogates of the intractable likelihood gradient: when the surrogates agree, take a larger step; when they conflict, shrink it. It also identifies a missing scaling factor in the DDIM reformulation that earlier likelihood insertions ignored, so the guidance scales correctly with step count and stochasticity. Across super-resolution and deblurring tasks, the paper reports that AdaPS matches or beats prior samplers on perceptual quality (LPIPS) with minimal or no PSNR loss, without task-specific tuning.

Core claim

Under the linear-Gaussian measurement model and a Gaussian approximation of the intermediate clean-signal posterior, the paper derives a closed-form MAP residual in noise space that approximates the difference between the optimal posterior noise and the unconditional denoiser's noise prediction, without backpropagating through the denoiser. The final update is x_{t-1} = DDIM(x_t) - 2γ_t (⟨d_t, g_t⟩/||g_t||²) g_t, where g_t is any likelihood-direction surrogate, d_t is the Jacobian-free MAP residual, γ_t is the DDIM coefficient, and the factor 2 compensates for the empirically observed ~0.5 cosine alignment between the two surrogates. The paper argues this yields a principled, hyperparameter-

What carries the argument

The load-bearing object is the projection coefficient ⟨d_t, g_t⟩/||g_t||² between two approximations of the intractable likelihood score. d_t is a MAP-based residual computed in the denoised-image space without the denoiser Jacobian, while g_t is a likelihood direction that may include the Jacobian. The DDIM coefficient γ_t multiplies the whole correction, and the factor 2 is a data-agnostic bias correction for the systematic shrinkage caused by imperfect alignment. Splitting magnitude (Jacobian-free) from direction (potentially Jacobian-aware) is what lets the method avoid ill-conditioned step sizes while preserving useful Jacobian information.

Load-bearing premise

The fixed factor-of-two bias correction assumes that throughout the trajectory the expected cosine similarity between the two likelihood surrogates is close to 0.5 in every task, dataset, and noise level; if the true mean alignment differs, the guidance magnitude is systematically wrong.

What would settle it

Measure the per-timestep average of the cosine similarity between the two likelihood surrogates on a task the paper does not test, such as inpainting or phase retrieval at 256px; if it departs from 0.5 by more than roughly 0.1 over the mid-trajectory, the factor-2 rule is miscalibrated. Directly comparing AdaPS with a fixed factor 2 against a version using a per-task fitted factor would settle whether the universal constant is correct.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Posterior samplers built on unconditional diffusion priors no longer need per-task step-size tuning; the alignment rule sets the scale automatically.
  • The missing scaling factor means earlier likelihood insertions were miscalibrated; including it makes guidance scale with step count and stochasticity, so more diffusion steps continue to improve perceptual quality.
  • Splitting magnitude from direction gives a stable compromise that avoids overweighting ill-conditioned directions while keeping the benefit of Jacobian information, without extra computational cost.
  • The same construction extends to nonlinear measurement operators through a scalar Gauss-Newton MAP refinement with fixed hyperparameters.
  • Because the method is hyperparameter-free, it transfers across tasks and noise levels without re-fitting, which prior approaches generally require.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The alignment statistic could be monitored online during sampling; replacing the fixed factor 2 with a moving average of the cosine similarity would make the method adaptive to tasks where the 0.5 assumption fails.
  • The agreement heuristic doubles as a confidence diagnostic: when the two likelihood surrogates strongly disagree, the update is automatically attenuated, which may explain the method's robustness to high measurement noise.
  • If a latent-diffusion version could define both surrogates in latent space, the schedule-robustness demonstrated here would plausibly carry over, provided the alignment statistic remains informative in that space.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The submission is internally inconsistent: the title and abstract describe a "Jacobian-Aware Posterior Sampling (JAPS)" method, while the full text presents "Adaptive Posterior diffusion Sampling (AdaPS)". Taking the full text as the actual contribution, the paper proposes AdaPS, which incorporates a likelihood-score surrogate g_t into the DDIM update (Eq. 15) and selects the guidance coefficient ξ_t by least-squares projection of a posterior-noise residual d_t onto g_t (Eq. 16). d_t is approximated via a MAP estimate of x_0 given (x_t, y) under a Gaussian approximation (Eqs. 23–24), and the update includes an empirical factor-of-two bias correction (Eq. 19). The method is evaluated on super-resolution, Gaussian deblurring, and motion deblurring on CelebA-HQ and ImageNet-256, plus a nonlinear deblurring extension. The experimental sections are extensive, but the derivation contains a concrete algebraic error and an empirically fitted constant that undermine the theoretical claims.

Significance. If the derivation were sound, the contribution would be significant: an adaptive, multi-task guidance scaling that preserves DDIM scheduling, avoids Jacobian computation in the magnitude, and attains a favorable perception–distortion trade-off. The experimental breadth is a strength: two datasets, three tasks, noisy/noiseless settings, ablations over step count, noise, stochasticity, and surrogate pairings, and a cross-validated ΠGDM reimplementation. The reported LPIPS gains (e.g., Table 1, ImageNet motion deblurring: 0.147 for AdaPS-ΠGDM vs 0.205 for DDPG) are meaningful. However, Eq. (24) is algebraically wrong and the factor-of-two correction is an empirical fit; as written, the central claim of a "principled, hyperparameter-free" sampler is not established. The significance is therefore conditional on fixing the derivation and validating the fitted constant.

major comments (4)
  1. [Abstract / Title; §1–§3] The manuscript's title and abstract describe 'Jacobian-Aware Posterior Sampling (JAPS)' with a 'corrected conditional factor', but the body introduces 'Adaptive Posterior diffusion Sampling (AdaPS)' and never defines JAPS. For example, the abstract claims the method 'bridges direct and proximal approaches' and 'leverages the Jacobian's prior knowledge', while §3's derivation uses d_t from an \hat{x}_0-space MAP residual and g_t chosen as DPS or ΠGDM. This is not cosmetic: the reader cannot verify which method is being presented. Harmonize the title, abstract, and body before review can be completed.
  2. [§3.2.2, Eq. (24); also Eq. (34)] Combining Eq. (7), Eq. (9), and Eq. (23) gives ε*_t − ε_θ,t = [(x_t − √ᾱ_t x*_0) − (x_t − √ᾱ_t \hat{x}_0)]/√(1−ᾱ_t) = (√ᾱ_t/√(1−ᾱ_t))(\hat{x}_0 − x*_0). Substituting Eq. (23) yields d_t = (√ᾱ_t/√(1−ᾱ_t)) A^T(AA^T + σ_y²/r_t² I)^{-1}(A\hat{x}_0−y), not the √ᾱ_t√(1−ᾱ_t) in Eq. (24). The error is by a factor 1/(1−ᾱ_t), which diverges as t→0. Because d_t enters ξ*_t, ⟨d_t,g_t⟩, and the update Eq. (19), the implemented update is not the derived one; the missing factor cannot be absorbed into the fixed factor 2. This is a load-bearing inconsistency. The same expression appears in Eq. (34). Please correct the derivation and state whether the implementation uses the corrected factor or the printed one.
  3. [§3.2.1, Eq. (19)] The 'simple, data-agnostic bias correction, scaling by 2≈1/E[⟨d̂_t, ĝ_t⟩] (empirically≈0.5 at mid-trajectory)' is an empirical fit, not a derived constant. No supporting measurement of the alignment statistic is shown, and it may vary with task, noise level, step count, and diffusion time. This contradicts the 'hyperparameter-free' claim and makes the update magnitude partly heuristic. Furthermore, correcting Eq. (24) will change d_t and hence the statistic, so the factor 2 may no longer be valid. Provide evidence of its stability or revise the claims.
  4. [B.1, Eqs. (30)–(33)] The nonlinear extension replaces J(x_0)^T J(x_0) with λ(x_0) I_n (Eq. 30) and fixes λ_damp>0 and K=5 (after Eq. 33). Fixed constants are still hyperparameters; fixing them across tasks does not make the method 'essentially hyperparameter-free.' Since only one nonlinear task (Table 3) is shown, the robustness of these constants is not established. This is secondary, but should be addressed if the nonlinear claim is retained.
minor comments (4)
  1. [§3.2.1 / Supplement C] In §3.2.1, g_t is used in Eq. (16) without the normalization mentioned in Supplement C ('when using any likelihood surrogate for g_t, it is normalized'). Define g_t consistently in the main text.
  2. [Table 1, caption] The caption says 'Values in gray are excluded because they were obtained with a larger number of sampling steps,' but no gray values are distinguishable in the rendered text. Mark excluded entries explicitly.
  3. [Eq. (24) / Eq. (34)] The equivalence between Eq. (24)'s d_t and \tilde{ε}^{MAP}_θ in Eq. (34) is not stated. Align the notation so the surrogate definitions are easy to compare.
  4. [§5, Limitations] The Limitations say the derivation 'leverages the linear–Gaussian setting for clarity and efficiency,' but Eq. (23) depends on the linear-Gaussian form. The statement that extension to nonlinear/non-Gaussian models is possible 'in principle' should be qualified by the approximate nature and the extra constants introduced in B.1.

Circularity Check

0 steps flagged

No significant circularity: the adaptive step is a projection-based combination of two stated surrogates, and the main claims are supported by external comparisons.

full rationale

I walked the derivation chain from Eq. 14 through Eq. 24 and the experimental tables. The least-squares choice xi*_t = <d_t,g_t>/||g_t||^2 in Eq. 16 is literally the projection of the residual d_t onto the direction g_t, so it is not circular: d_t is a MAP residual computed in x0-space (Eq. 23) and g_t is a separate likelihood-score surrogate (DPS/PGDM); the update then moves along g_t with that projected magnitude. The Gaussian approximation p(x0|xt) ~ N(xhat0, r_t^2 I) is explicitly adopted from prior work [4,28], i.e. an imported assumption rather than a derived result, and is not used to prove the experimental outcomes. The fixed factor 2 in Eq. 19 is an empirical calibration ('empirically about 0.5 at mid-trajectory'), which weakens the literal claim of being fully hyperparameter-free but is not a hidden fit to the reported metrics and does not make the prediction follow from the input by construction. Self-citations ([1,11,14,33,34]) are background and baselines, not load-bearing; no uniqueness theorem is imported. I also note an algebraic inconsistency: combining Eqs. 7 and 23 gives d_t = [sqrt(alpha_bar_t)/sqrt(1-alpha_bar_t)] A^T(...)(A xhat0 - y), whereas Eq. 24 has sqrt(alpha_bar_t)*sqrt(1-alpha_bar_t), i.e. a missing 1/(1-alpha_bar_t) factor. This is a correctness/reproducibility concern in the derivation, not a circularity, so it does not change the circularity score. The central empirical claims are evaluated against external baselines and are self-contained.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 0 invented entities

The central derivation rests on standard diffusion-score identities and the Gaussian posterior approximation; the main ad hoc elements are the empirical factor-2 bias correction and the unverified isotropic curvature approximation in the nonlinear extension.

free parameters (3)
  • bias correction factor = 2
    Applied in Eq. 19 to compensate for empirical alignment E[⟨d̂_t, ĝ_t⟩] ≈ 0.5 at mid-trajectory (Section 3.2.1). Not derived; a hand-set constant that directly scales the likelihood update.
  • λ_damp = not reported
    Damping parameter in the nonlinear scalar Gauss–Newton update (Eq. 33); fixed across tasks but value not given in the paper.
  • K = 5
    Number of optimization iterations per diffusion step in the nonlinear extension (Section B.1); chosen once and fixed across tasks.
axioms (5)
  • standard math The score approximation ∇_xt log p_t(x_t) ≈ −ϵ_θ(x_t,t)/√(1−ᾱ_t) (Tweedie) is valid.
    Used in Eq. 8, Section 2.2; standard result from denoising score matching.
  • domain assumption p(x0|xt) ≈ N(x̂0, r_t² I) with r_t² = 1−ᾱ_t
    Eq. 21, Section 3.2.2; Gaussian posterior approximation inherited from ΠGDM [28]; used to derive the MAP surrogate d_t.
  • domain assumption Linear–Gaussian measurement model y = Ax + ε with known σ_y
    Eqs. 1–2, Section 2.1; central derivation in Section 3.2.2 assumes this; used in all main experiments.
  • ad hoc to paper E[⟨d̂_t, ĝ_t⟩] ≈ 0.5 at mid-trajectory and is constant across tasks, times, datasets, noise levels
    Section 3.2.1; justifies the fixed factor 2. No theoretical or empirical evidence beyond a single observation; load-bearing for step-size calibration.
  • ad hoc to paper J^T J ≈ λ(x0) I_n (isotropic curvature) in the nonlinear extension
    Eq. 30, Section B.1; standard scalar Gauss–Newton approximation, but unverified for neural blur operators.

reviewed 2026-08-03 · how reviews work

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Cite this review

Pith. "Pith review of Jacobian-Aware Posterior Sampling for Inverse Problems." pith.science (2026). https://pith.science/paper/TJIOWTUP

@misc{pith2026251118471,
  author       = {Pith},
  title        = {Pith review of: Jacobian-Aware Posterior Sampling for Inverse Problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TJIOWTUP}},
  note         = {Machine review of arXiv:2511.18471}
}
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read the original abstract

Diffusion models provide powerful generative priors for solving inverse problems by sampling from a posterior distribution conditioned on corrupted measurements. Existing methods primarily follow two paradigms: direct methods, which approximate the likelihood term, and proximal methods, which incorporate intermediate solutions satisfying measurement constraints into the sampling process. Under standard Gaussian approximations and locally-linear measurements, we demonstrate that these approaches differ fundamentally in their treatment of the diffusion denoiser's Jacobian within the likelihood term. While this Jacobian encodes critical prior knowledge of the data distribution, training-induced non-idealities can degrade performance in zero-shot settings. In this work, we bridge direct and proximal approaches by proposing a principled Jacobian-Aware Posterior Sampler (JAPS). JAPS leverages the Jacobian's prior knowledge while mitigating its detrimental effects through a corresponding proximal solution, requiring no additional computational cost. Additionally, we integrate our guidance into DDIM sampling, with a corrected conditional factor that has been missing in previous works. Our method enhances reconstruction quality across diverse linear and nonlinear noisy imaging tasks, outperforming existing diffusion-based baselines in perceptual quality while maintaining or improving distortion metrics.

Figures

Figures reproduced from arXiv: 2511.18471 by Liav Hen, Raja Giryes, Shady Abu-Hussein, Tom Tirer.

Figure 1
Figure 1. Figure 1: Results of our method AdaPS compared to selected base￾lines across several noisy image reconstruction tasks. Despite be￾ing simple and hyperparameter-free, AdaPS consistently balances distortion and perceptual quality without any task-specific tuning, particularly under realistic noise levels. On ImageNet-256, AdaPS improves LPIPS over DDPG with almost no PSNR cost, and on CelebA-HQ it yields substantial p… view at source ↗
Figure 2
Figure 2. Figure 2: Schematic overview of our method. We introduce a principled, hyperparameter-free rule for balancing likelihood guidance with the prior in diffusion-based posterior sampling. However, because dt and gt are distinct surrogates of the likelihood update, perfect alignment is unlikely; the projec￾tion coefficient ⟨ ˆdt, gˆt⟩ ≤ 1 therefore systematically shrinks the update step, even when the directions largely … view at source ↗
Figure 3
Figure 3. Figure 3: Qualitative comparison of AdaPS and representative methods. Best viewed in zoom-in. Comparison to ΠGDM. Beyond the likelihood approxi￾mation, [28] introduce a time-decaying multiplicative step size equal to (1 − α¯t). While this choice performs well at 100 sampling steps, it does not account for the diffu￾sion schedule’s discretization (i.e., changing the number of steps). Consequently—and counter-intuitiv… view at source ↗
Figure 5
Figure 5. Figure 5: Qualitative comparison. ΠGDM deteriorates at larger step counts, while AdaPS remains stable and continues improving. Design choices for gt and dt. Beyond our default role split (direction from gt, magnitude from dt), we also evalu- [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Qualitative comparison of DPS and AdaPS-DPS on non-linear deblurring. [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Response to increasing measurement noise level. AdaPS shows enhanced robustness to increased noise compared to ΠGDM [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Sensitivity to stochasticity (η). PSNR/LPIPS for SR×4 as a function of η. AdaPS maintains similar performance across stochasticity levels, whereas ΠGDM is more sensitive to the amount of injected noise. B.3. Different Choices for gt and dt Our method combines two complementary surrogates to regularize posterior updates: the direction is set by gt, and the magnitude by dt, with a correlation-based correctio… view at source ↗
Figure 9
Figure 9. Figure 9: Additional visual results on ImageNet-256 validation set. Best viewed in zoom-in [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗

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This paper was first reviewed by deepseek-v4-flash on August 3, 2026.