Enhancing Diffusion Models for Inverse Problems with Covariance-Aware Posterior Sampling

En-hui Yang; Shayan Mohajer Hamidi

REVIEW 3 major objections 6 minor 1 cited by

Enhancing Diffusion Models for Inverse Problems with Covariance-Aware Posterior Sampling

T0 review · 3 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read This paper derives a closed-form covariance for the DDPM reverse process and uses a finite-difference Hessian estimate to improve posterior sampling for noisy linear inverse problems, beating mean-only samplers on FFHQ and ImageNet…

desk verdict The covariance identity is correct but standard, and the finite-difference 'Hessian' that carries the method is a time derivative, not a spatial second derivative; the algorithm cannot even evaluate it as written. read the letter →

arxiv 2412.20045 v1 pith:R4INA5JP submitted 2024-12-28 cs.CV cs.AI

Shayan Mohajer Hamidi , En-Hui Yang This is my paper · ORCID

classification cs.CVcs.AI

keywords diffusionmodelsinverseproblemsposteriorsamplingTweedie'sformulaconditionalcovarianceHessianapproximationimagerestorationDDPM

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

The paper tries to establish that diffusion-model posterior sampling for linear inverse problems becomes more accurate when the conditional covariance of $x_0$ given $x_t$ is used alongside the conditional mean. It derives a closed-form expression for $\operatorname{Cov}(x_0|x_t)$ that depends on the Hessian of $\log p_t(x_t)$, then approximates that Hessian with a finite difference of consecutive score-function outputs. This yields CA-DPS, a covariance-aware likelihood approximation that plugs into the reverse diffusion step without extra model evaluations. On FFHQ and ImageNet, CA-DPS improves FID, LPIPS, and SSIM across inpainting, deblurring, and super-resolution compared with mean-only posterior samplers, and it requires no hyperparameter tuning. The added second moment matters because mean-only approximations discard uncertainty that grows with noise level, which is exactly where inverse-problem reconstruction tends to fail.

What carries the argument

The key machinery is the closed-form conditional covariance of Eq. (20), together with the finite-difference Hessian approximation $\tilde{H}_t$ of Eq. (21). Theorem 1 supplies a posterior-covariance identity for any exponential-family conditional model, and applying it to the DDPM forward kernel turns $\operatorname{Cov}(x_0|x_t)$ into a function of $\nabla^2_{x_t}\log p_t(x_t)$. Because a pretrained diffusion model exposes only the score, the paper replaces this Hessian with the diagonal ratio of consecutive score differences along the reverse trajectory; the resulting $\tilde{\Sigma}_t$ enters the likelihood covariance and the guidance gradient. The machinery does the work because it converts a second-order quantity that mean-only samplers discard into per-pixel, direction-dependent step sizes using only quantities the score network already produces.

What would settle it

On a distribution with a known or exactly computable score, compare the diagonal of the exact Hessian $\nabla^2_{x_t}\log p_t(x_t)$ with the finite-difference estimate $\tilde{H}_t$ from Eq. (21) at the timesteps and samples the sampler actually visits; if they diverge strongly there, the covariance driving CA-DPS is not the closed-form covariance the paper derives.

Watch

Extended reading notes

Core claim

The central claim is that the conditional distribution $p(x_0|x_t)$ is better approximated by a Gaussian with the Tweedie mean $\tilde{x}_0$ and covariance $\tilde{\Sigma}_t$ from Eq. (22) than by the delta distribution of DPS or the isotropic Gaussian of PiGDM. The paper derives the exact covariance identity $\operatorname{Cov}(x_0|x_t) = \frac{1-\bar\alpha_t}{\bar\alpha_t}\left(I + (1-\bar\alpha_t)\nabla^2_{x_t}\log p_t(x_t)\right)$ as a corollary of a general exponential-family posterior-covariance theorem, then approximates the unknown Hessian by the finite-difference diagonal $\tilde{H}_t$ from Eq. (21). Substituting this covariance into the Gaussian likelihood $p(y|x_t)\approx \mathcal{N}(A\tilde{x}_0, \sigma^2 I + A\tilde{\Sigma}_t A^\top)$ produces the guidance gradient in Eq. (29), which is the only additional step in the reverse sampler. The paper claims that this construction improves reconstruction without hyperparameter tuning and that its toy Gaussian-mixture experiments place the estimated posterior closer to the true posterior than prior methods do.

Load-bearing premise

The load-bearing premise is that the ratio of the change in the score across consecutive timesteps to the timestep gap approximates the Hessian $\nabla^2_{x_t}\log p_t(x_t)$, even though this ratio is a derivative along one random trajectory rather than a spatial second derivative.

Editorial extensions

If this is right

CA-DPS scales the measurement residual by the estimated covariance, so the guidance step adapts per pixel rather than using a fixed strength.
The method removes the need to hand-select a variance schedule like PiGDM's $r_t^2$, because the covariance comes from the score network itself.
Because the Hessian estimate reuses consecutive score evaluations, the per-iteration cost stays comparable to DPS and PiGDM rather than requiring a second-order score model.
The claimed experimental outcome is consistent gains over DPS, PiGDM, DDRM, MCG, Score-SDE, and ADMM-TV across box and random inpainting, Gaussian and motion deblurring, and 4x super-resolution.
The general exponential-family covariance theorem offers a template for adding second-moment information to any posterior sampler that relies on Tweedie-type means.

Reading between the lines

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

Beyond the paper, one could stress-test the finite-difference estimate on the paper's toy Gaussian mixture by comparing $\tilde{H}_t$ from Eq. (21) with the exact Hessian computed by automatic differentiation along sampled reverse trajectories.
A natural extension would average the finite-difference estimate over several timesteps or over multiple reverse trajectories, which could stabilize the covariance early in denoising where the score changes rapidly.
The exponential-family covariance theorem suggests the same mean-plus-covariance construction could apply to other measurement likelihoods, such as Poisson or categorical observations, where an isotropic variance assumption is harder to justify.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: the closed-form covariance formula is correct but not new, and the finite-difference approximation that carries the method is not a Hessian. Algorithm 1 also can't compute it at the point it's needed. The experiments are extensive and code is provided, but they test end-to-end posterior distance, not the covariance approximation, so they don't rescue the theory.

The paper's genuine contribution is the attempt to approximate the reverse-process covariance from a pretrained score network without second-order scores. That's a reasonable goal, and the authors cite the relevant higher-order Tweedie literature. The derivation of Theorem 1 and Corollary 1 is algebraically sound. The experimental comparison covers FFHQ and ImageNet across five tasks, and the toy-posterior study is a good idea.

The problem is load-bearing. Equation (21) defines a tilde-H as the difference of score vectors at consecutive timesteps divided by the timestep. That is a finite difference in time along a reverse trajectory, not a finite difference in space at fixed t. It has no business approximating the Hessian of log pt(xt). The units are already wrong: the true Hessian has units 1/signal-squared, while the time-difference has units 1/(signal·time). For a Gaussian prior with p(x0)=N(0,I), the true Hessian is -I, but Eq (21) is a noisy trajectory increment. Everything downstream—tilde-Sigma, the Gaussian approximation to p(x0|xt), and the likelihood gradient in Eq (29)—inherits this error. The central claim that CA-DPS uses the true posterior covariance is unsupported.

There is also an execution problem: in Algorithm 1, tilde-H requires the score at xt-1, but xt-1 is only produced later in the same iteration. As written, the algorithm is not well-defined. A lagged version would be a different estimator, and it still wouldn't be a Hessian.

On novelty: the covariance identity is essentially Tweedie's second-moment identity, which the authors acknowledge appears in earlier work. The only new piece is the finite-difference scheme, and that piece is invalid. The toy experiments show CA-DPS often gets lower SW distance, but they compare full posterior samples; they never check whether tilde-H approximates the Hessian, so they don't test the paper's mechanism.

Recommendation: I would not send this to review in its current form. The theoretical foundation is broken, and no amount of empirical improvement can justify the claim that the method approximates the true posterior covariance. The authors could salvage the paper by reframing the update as a heuristic and removing the covariance-interpretation claims, or by deriving an actual spatial Hessian approximation from the pretrained model. As presented, it's a reject.

Referee Report

3 major / 6 minor

Summary. The paper proposes covariance-aware diffusion posterior sampling (CA-DPS), a method for solving noisy linear inverse problems with pretrained DDPMs. The theoretical part derives a closed-form expression for the conditional covariance Cov(x0|xt) from an exponential-family Tweedie identity (Theorem 1, Eq. (20)). The algorithmic part replaces the Hessian of log pt(xt) in that expression with a finite-difference quantity tilde_H_t (Eq. (21)) and uses it to build a Gaussian approximation of p(x0|xt), leading to a likelihood gradient (Eq. (29)) and a modified reverse-time update (Algorithm 1). Experiments on FFHQ and ImageNet report improved FID, LPIPS, and SSIM over DPS, PiGDM, DDRM, MCG, PnP-ADMM, Score-SDE, and ADMM-TV, and a toy Gaussian-mixture experiment reports lower sliced Wasserstein distance to the true posterior.

Significance. If the Hessian approximation were valid, the paper would provide a useful and elegant contribution: the covariance identity (20) is algebraically sound, it correctly identifies the dependence of the reverse covariance on score curvature, and the method would add no learned hyperparameters. The paper also ships a code link and a toy setup with an exact posterior. However, the central algorithmic step, Eq. (21), identifies a time-difference of scores with a spatial Hessian, and this identification is false. Because the likelihood approximation and the posterior update inherit that error, the claimed covariance-awareness of CA-DPS is not established. The paper's empirical gains may be real, but they do not validate the theoretical claim.

major comments (3)

[§4, Eq. (21)] The finite-difference quantity tilde_H_t is not an approximation of the Hessian of log pt with respect to x_t. It is the difference of two score-network evaluations at different times and different states, divided by the time step; at best it estimates a time derivative of the score along one reverse-time trajectory. Its units are inverse signal-time, not inverse signal-squared, and there is no established relationship between this quantity and the curvature of log pt at x_t. In the simple case where p(x0) is a standard normal, the true Hessian is minus the identity for every t, while the right-hand side of Eq. (21) is a random trajectory increment that is not minus the identity. Since Eqs. (22), (27), and (29) all rely on Eq. (21), the covariance estimate and the likelihood gradient of CA-DPS are unsupported.
[Algorithm 1, lines 7-8] Equation (21) is not computable at the point where it is used. At iteration t, the method needs the score at x_{t-1}, but x_{t-1} is the output of line 8 and is not yet available at the beginning of the iteration; line 7 produces an intermediate vector, not the corrected x_{t-1}. If the authors intend to evaluate the score at the intermediate vector, that requires an additional score-network forward pass per step and changes the stated complexity, and the pseudocode does not say so. As written, the update is not well-defined.
[Sec. 5.3] The toy experiment does not test the load-bearing assumption. It compares end-to-end sliced Wasserstein distances between posterior samples, which measure the quality of the overall sampler, not whether tilde_H_t approximates the Hessian. No diagnostic is provided that compares tilde_H_t with a reference Hessian or with the true covariance, which is known in the toy model. Consequently, the empirical improvements do not support the claim that CA-DPS is superior because of its covariance approximation; they could arise from the modified gradient step even if Eq. (21) has no Hessian interpretation.

minor comments (6)

[Introduction and Sec. 4] The Hessian is written as the Hessian of log p_t(y|x_t) in two places, but the score available from a DDPM is the gradient of log p_t(x_t); the conditional notation appears to be a typo and should be corrected throughout.
[Table 1] In the FFHQ Inpaint (Box) row for CA-DPS, '0862' should likely read '0.862'; some other entries also omit the leading zero.
[Corollary] The corollary after Theorem 1 is not numbered, although the abstract and introduction refer to it as Corollary 1; please number it consistently.
[Sec. 5.1] The sentence 'CA-DPS emerges as the front-runner in three of them' is ambiguous about which three items are meant, as the paragraph lists metrics rather than a countable set of three.
[Sec. 8] In the proof of Theorem 1, the tensor contraction notation is used before its definition in Eq. (18), and the expression used in the proof is not obviously the same operation as the one later defined; please unify the notation.
[Appendix tables] Tables 3 and 4 in the appendix are not referenced or discussed in the main text; please add a pointer or remove them.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the covariance derivation is self-contained, and the finite-difference Hessian approximation is an unvalidated ansatz, not a circular reduction.

full rationale

The paper's central derivation chain is: Theorem 1 (exponential-family covariance identity) -> Corollary Eq. (20) -> finite-difference approximation Eq. (21) -> covariance estimate Eq. (22) -> likelihood approximation Eq. (24) -> posterior gradient Eq. (29). The covariance identity in Theorem 1 is derived directly from the exponential-family factorization of p(y|eta) and the marginal p(y), as shown in the Section 8 proof; it does not assume the target covariance. The DDPM corollary Eq. (20) follows by substituting the explicit Gaussian perturbation kernel (Eq. 13) into Theorem 1. Eq. (21) introduces Htilde_t as an approximation to the unavailable Hessian; it is an independently defined finite difference of score estimates, not a parameter fitted to benchmark data and not defined in terms of the posterior covariance it is used to predict. The experimental protocol uses fixed pre-trained score networks and standardized measurement operators, so no fitted value is renamed as a prediction. There are no load-bearing self-citations by the present authors. The main weaknesses—Eq. (21) is a time-difference of score vectors rather than a spatial second derivative, and Algorithm 1 appears to reference x_{t-1} before it is computed at line 8—are correctness and rigor concerns, not circularity under the stated criteria. Because no step reduces by construction to its own input, the circularity score is 0.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the closed-form covariance identity (Eq. 20), which is mathematically derived, and on the ad hoc finite-difference approximation of the Hessian (Eq. 21). The latter is the load-bearing assumption that lacks justification. No new physical or statistical entities are introduced.

free parameters (1)

Finite-difference time step = 1 discrete diffusion step
The step size in Eq. (21) is fixed by the DDPM discretization (t-(t-1)=1), but it is a hand-chosen quantity that the covariance estimate depends on; no sensitivity analysis is provided.

assumptions (4)

standard math The transition density p(xt|x0) is Gaussian with mean sqrt(bar_alpha(t)) x0 and covariance (1-bar_alpha(t)) I.
This is the standard DDPM/VP-SDE forward process assumption, stated in Eq. (11).
domain assumption The pretrained diffusion model provides an accurate estimate of the score nabla_xt log pt(xt).
The method relies on the network score being a good proxy for the true score, which is common but not guaranteed for all t.
ad hoc to paper The finite-difference quantity in Eq. (21) approximates the Hessian nabla^2_xt log pt(xt).
This is the central assumption of the proposed covariance approximation; it is not derived or empirically justified in the paper, and it is mathematically dubious.
ad hoc to paper The Hessian is approximated as a diagonal matrix.
The paper assumes a diagonal H_tilde_t without discussing the error introduced by ignoring off-diagonal terms of the true Hessian.

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Cite this review

Pith. "Pith review of Enhancing Diffusion Models for Inverse Problems with Covariance-Aware Posterior Sampling." pith.science (2026). https://pith.science/paper/R4INA5JP

@misc{pith2026241220045,
  author       = {Pith},
  title        = {Pith review of: Enhancing Diffusion Models for Inverse Problems with Covariance-Aware Posterior Sampling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R4INA5JP}},
  note         = {Machine review of arXiv:2412.20045}
}

read the original abstract

Inverse problems exist in many disciplines of science and engineering. In computer vision, for example, tasks such as inpainting, deblurring, and super resolution can be effectively modeled as inverse problems. Recently, denoising diffusion probabilistic models (DDPMs) are shown to provide a promising solution to noisy linear inverse problems without the need for additional task specific training. Specifically, with the prior provided by DDPMs, one can sample from the posterior by approximating the likelihood. In the literature, approximations of the likelihood are often based on the mean of conditional densities of the reverse process, which can be obtained using Tweedie formula. To obtain a better approximation to the likelihood, in this paper we first derive a closed form formula for the covariance of the reverse process. Then, we propose a method based on finite difference method to approximate this covariance such that it can be readily obtained from the existing pretrained DDPMs, thereby not increasing the complexity compared to existing approaches. Finally, based on the mean and approximated covariance of the reverse process, we present a new approximation to the likelihood. We refer to this method as covariance-aware diffusion posterior sampling (CA-DPS). Experimental results show that CA-DPS significantly improves reconstruction performance without requiring hyperparameter tuning. The code for the paper is put in the supplementary materials.

Figures

Figures reproduced from arXiv: 2412.20045 by the authors.

**Figure 1.** Qualitative results on FFHQ dataset. structed from the benchmark methods. To this end, we randomly select five images from the FFHQ test dataset and corrupt them using the measurement methods discussed in Section 5.1, with the exception that we apply 8× superresolution for better visualization. The images are depicted in [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

**Figure 2.** Visualizing the first two dimensions of the esti [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Qualitative results on FFHQ dataset [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Qualitative results on FFHQ dataset [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]

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Forward citations

Cited by 1 Pith paper

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Reviewed August 10, 2026 · model on record in the stance chip above.

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write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...

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