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REVIEW 4 major objections 4 minor 28 references

Conditional Mutual Information Based Diffusion Posterior Sampling for Solving Inverse Problems

T0 review · 4 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper claims that maximizing the conditional mutual information between the reconstruction and the measurement at each reverse diffusion step yields higher-quality solutions to linear inverse problems, and supports this with a…

desk verdict The CMI guidance term is y-agnostic, so the paper's information-theoretic story doesn't survive contact with its own equations, though the empirical plug-in shows consistent small gains. read the letter →

arxiv 2501.02880 v1 pith:KK4UAWIK submitted 2025-01-06 cs.LG stat.ML

Shayan Mohajer Hamidi , En-Hui Yang This is my paper · ORCID
classification cs.LGstat.ML
keywords conditionalmutualinformationdiffusionmodelsinverseproblemsposteriorsamplingimageinpaintingdeblurringsuper-resolutionHutchinsontraceestimation
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 solve inverse problems—recovering an image from a masked, blurred, or downsampled measurement—by steering the reverse process with a likelihood term, but that likelihood is intractable and standard approximations drift away from the true posterior. This paper argues that the right corrective is information-theoretic: at each reverse step, reshape the intermediate latent so that the final reconstruction and the measurement share as much conditional information as possible, quantified by $\mathrm{I}(\boldsymbol{x}_0;\boldsymbol{y}\mid\boldsymbol{x}_t)$. Under the Gaussian approximation of $p(\boldsymbol{x}_0\mid\boldsymbol{x}_t)$ that is standard in this literature, the paper derives a closed form for this conditional mutual information and for its gradient with respect to $\boldsymbol{x}_t$. The gradient is computed efficiently with Hutchinson-style trace estimation, avoiding an explicit third-order derivative of the score, and is added as a one-line correction to four existing samplers: DPS, $\Pi$GDM, MCG, and DSG. On FFHQ and ImageNet, the corrected solvers improve FID, LPIPS, and SSIM across inpainting, deblurring, and super-resolution tasks.

What carries the argument

The central object is the conditional mutual information $\mathrm{I}(\boldsymbol{x}_0;\boldsymbol{y}\mid\boldsymbol{x}_t)$ and its gradient $\nabla_{\boldsymbol{x}_t}\mathrm{I}(\boldsymbol{x}_0;\boldsymbol{y}\mid\boldsymbol{x}_t)$, evaluated at each reverse diffusion step. The closed form of Theorem 1 turns this information quantity into a determinant ratio of the posterior covariance $\boldsymbol{\Sigma}_{\mathrm{post}}$ and the measurement-updated covariance $\boldsymbol{\Sigma}_{\mathrm{post},y}$; the gradient of Theorem 2 expands it into trace terms that would require $\nabla_{\boldsymbol{x}_t}^{3}\log p_t(\boldsymbol{x}_t)$. Hutchinson's trace estimation replaces that third-order derivative with Hessian-vector products along $r$ random directions, so the correction costs only a few vector-Jacobian products per step rather than an explicit $O(d^3)$ tensor, and can be inserted as one line into existing samplers.

What would settle it

Run a reverse trajectory on a problem with a known Gaussian prior and linear operator, estimate $\mathrm{I}(\boldsymbol{x}_0;\boldsymbol{y}\mid\boldsymbol{x}_t)$ empirically from many posterior samples before and after a CMI step, and check that the step increases the estimate; a decrease would falsify the claim that the closed-form gradient maximizes the true conditional mutual information.

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Extended reading notes

Core claim

The paper's central claim is that maximizing $\mathrm{I}(\boldsymbol{x}_0;\boldsymbol{y}\mid\boldsymbol{x}_t)$ at every reverse step keeps the intermediate latents informative about the measurement, so the final reconstruction is simultaneously faithful to the data and consistent with the diffusion prior. Theorem 1 shows that under the assumption $p(\boldsymbol{x}_0\mid\boldsymbol{x}_t)\sim\mathcal{N}(\boldsymbol{\mu}_{\mathrm{post}},\boldsymbol{\Sigma}_{\mathrm{post}})$ with $\boldsymbol{\Sigma}_{\mathrm{post}}=\frac{1-\bar{\alpha}_t}{\bar{\alpha}_t}\left(\boldsymbol{I}+(1-\bar{\alpha}_t)\nabla_{\boldsymbol{x}_t}^{2}\log p_t(\boldsymbol{x}_t)\right)$, the conditional mutual information equals half the log ratio of the determinants of $\boldsymbol{\Sigma}_{\mathrm{post}}$ and $\left(\boldsymbol{\Sigma}_{\mathrm{post}}^{-1}+\boldsymbol{A}^{\top}\boldsymbol{\Sigma}_n^{-1}\boldsymbol{A}\right)^{-1}$. Theorem 2 differentiates that quantity with respect to $\boldsymbol{x}_t$, producing trace terms that involve the third-order derivative of $\log p_t(\boldsymbol{x}_t)$; Hutchinson's trace estimation reduces each such term to Hessian-vector products along random directions. The resulting CMI gradient is appended to DPS, $\Pi$GDM, MCG, and DSG as a single corrective line, and the experiments report consistent improvements in FID, LPIPS, and SSIM on FFHQ and ImageNet across random and box inpainting, Gaussian and motion deblurring, and super-resolution.

Load-bearing premise

The Gaussian approximation of $p(\boldsymbol{x}_0\mid\boldsymbol{x}_t)$ with a Hessian-based covariance must be accurate at every reverse step; if real image posteriors deviate from it, the computed CMI gradient is miscalibrated and may not actually increase the true conditional mutual information.

Editorial extensions

If this is right

  • DPS, $\Pi$GDM, MCG, and DSG all improve on FID, LPIPS, or SSIM in the large majority of the ten dataset-plus-measurement settings reported in Table I.
  • Because the CMI gradient is a one-line plugin that reuses the existing score network, any future diffusion posterior sampler can adopt the same information-theoretic correction without task-specific retraining.
  • The largest relative gains appear where the base method is weakest, such as MCG on Gaussian deblurring and super-resolution, suggesting the correction mainly repairs measurement consistency rather than prior fidelity.
  • The per-step overhead is dominated by a small number of Hessian-vector products, so the method remains practical on a single GPU for 256$\times$256 images.
  • The method inherits the Gaussian posterior assumption, so its guarantees are conditional on that approximation holding at every time step.

Reading between the lines

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

  • A natural ablation the paper does not run is to replace the CMI gradient with a random vector of the same norm; if the image-quality gains persist, the extra gradient step rather than the information-theoretic direction would be the active ingredient.
  • The same determinant-ratio machinery could be re-derived for a non-Gaussian or learned model of $p(\boldsymbol{x}_0\mid\boldsymbol{x}_t)$; comparing the closed-form gradient against a sample-based estimate of the true CMI would localize where the Gaussian assumption breaks.
  • Because the measurement operator enters only through $\boldsymbol{A}^{\top}\boldsymbol{\Sigma}_n^{-1}\boldsymbol{A}$, the formula generalizes immediately to any linear Gaussian forward model, including color-channel coupling or correlated measurement noise, which the experiments do not explore.
  • The objective suggests a principled criterion for stopping or step-size selection: the CMI value itself could serve as a per-step measure of how much measurement information remains, something the paper leaves implicit.
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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

4 major / 4 minor

Summary. The paper proposes to improve diffusion-model-based solvers for linear inverse problems by adding a gradient step that maximizes the conditional mutual information I(x0; y | xt) during each reverse diffusion step. Under a Gaussian assumption on p(x0|xt), the authors derive a closed-form expression for this CMI (Theorem 1, Eqs. (6)-(7)) and its gradient with respect to xt (Theorem 2, Eqs. (13)-(14)). Because the exact third-order score derivative is expensive, they propose a Hutchinson trace-estimation approximation (Section III-C) and integrate the resulting 'CMI step' into existing methods such as DPS, PiGDM, MCG, and DSG (Algorithm 1). Experiments on FFHQ and ImageNet report consistent improvements in FID, LPIPS, and SSIM over the baselines for inpainting, deblurring, and super-resolution. The authors conclude that maximizing CMI ensures the reconstructed signal retains maximal information about the measurement.

Significance. If the proposed correction genuinely improved posterior sampling in a measurement-dependent way, it would be a simple and broadly applicable plugin for diffusion-based inverse problem solvers. The paper has several genuine strengths: the derivations in Theorems 1 and 2 are standard Gaussian/matrix identities and are largely correct; the Hutchinson-based avoidance of explicit third-order tensors is a sensible computational idea; and the empirical study covers multiple tasks, datasets, and existing solvers, with mostly consistent improvements. However, the central conceptual claim is not supported by the mathematics: the CMI objective as defined in Eqs. (6)-(7) is independent of the observed measurement y. This does not by itself disprove the empirical value of the added gradient, but it means the method is not 'measurement-aware' in the sense claimed, and the information-theoretic justification needs substantial revision. There are also important reproducibility gaps in the description of the efficient gradient computation and the experimental setup.

major comments (4)
  1. [Section III-A, Eqs. (6)-(7), and Algorithm 1 line 8] The objective I(x0; y | xt) as defined is independent of the realized measurement y. Eq. (7a) depends only on xt through the score Hessian, and Eq. (7b) depends only on A and the noise covariance Σ_n; no term in Eqs. (6)-(7) contains y. Consequently, the gradient ∇xt I in Theorem 2 and the CMI correction in Algorithm 1 line 8 are identical for every observed y that shares the same forward operator A and Σ_n. The CMI step therefore cannot 'retain information about the measurement y' in any observation-dependent sense; at best it maximizes the expected information gain averaged over the distribution of y. This undermines the paper's central claim and its information-theoretic motivation. The empirical improvements may still indicate that the added term is a useful y-independent regularizer, but the paper must be reframed accordingly, and the claim that the method conditions on the specific measurement must be removed or tested against a y-independent control.
  2. [Section III-C and Algorithm 1] The efficient approximation is not described at a level that permits reproduction. The estimator in Eq. (22) requires computing v_i^T Σ_post^{-1} [∇^3 log p]_{k,:,:} v_i for each k. This vector over k is not simply '∇^3 log p(x_t) v_i' as stated in the text; it requires the action of the transpose Jacobian of the function f(x) = ∇^2 log p(x) v_i on the vector Σ_post^{-1} v_i, or an equivalent double differentiation with Σ_post^{-1} handled carefully. The paper does not explain how Σ_post^{-1} and Σ_post,y^{-1} are applied in high-dimensional image spaces without forming d×d matrices, nor does it specify the number r of Hutchinson samples or the choice and schedule of the step size η_t. These details are essential because the experimental results depend on them.
  3. [Assumption 1 and Eq. (7a)] The covariance Σ_post is not derived in this paper but imported from Refs. [11] and [20]. Assumption 1 as stated only says p(x0|xt) is Gaussian, but the specific covariance formula in Eq. (7a) is an additional, strong approximation. Moreover, for arbitrary xt the matrix I + (1 - ᾱ_t) ∇^2 log p_t(x_t) need not be positive semidefinite, in which case log det(Σ_post) and Σ_post^{-1} in Theorems 1 and 2 are not well-defined. The paper should state the conditions under which the covariance is positive definite and describe any regularization used in the experiments.
  4. [Section IV, Table I] The empirical evaluation lacks error bars, statistical significance tests, and sensitivity analyses for η_t and r, and no code is provided. More importantly, because the CMI gradient is independent of y, the reported improvements over DPS, PiGDM, MCG, and DSG cannot be attributed to measurement conditioning without an ablation that replaces the CMI term with a matched-scale y-independent perturbation. Such a control is necessary to support the paper's information-theoretic interpretation of the gains.
minor comments (4)
  1. [References] Reference [12] appears to be a duplicate of reference [9], and reference [17] duplicates reference [3]; the citation list should be cleaned up.
  2. [Section III-B, notation] The definitions of ⊙1 and ⊙2 in Section III-B are understandable but hard to follow in Eq. (14b); an index-based version or a simple example would improve readability.
  3. [Algorithm 1] The paper repeatedly states that the method adds 'just one line of code', but the CMI step requires Hessian-vector products, Hutchinson sampling, and solving or applying inverse covariance matrices; the 'one line' characterization is misleading and should be softened.
  4. [Section IV-B] The qualitative section says four images are randomly selected, but Figure 1 shows only a subset or composite; the text should describe what is displayed more precisely.

Circularity Check

1 steps flagged · score 2.0 of 10

No real circularity; only a minor self-citation dependency in the covariance formula, with external support and benchmark-verified central claim.

  1. other [Section III-A, Assumption 1 and Eq. (9) (proof of Theorem 1)]
    "First, following the works of [6], [8], [10], [11], [20], we introduce the following assumption: Assumption 1. The conditional distribution p(x0|xt) follows a multivariate normal distribution... As per the assumption in the theorem, the covariance Sigma_post is given by [11], [20]"

    The CMI closed form in Eq. (6) and every gradient term in Theorem 2 depend on Sigma_post, and Eq. (9) is attributed to the authors' own prior paper [11] as well as to external [20]. To that extent the information-theoretic objective is inherited from the authors' prior covariance model rather than re-derived from first principles within this paper. This is a self-citation dependency, but not a circular reduction: Assumption 1 is explicitly stated, the same covariance formula is independently available in [20], the subsequent entropy calculation is a standard Gaussian/Kalman derivation, and the claimed performance gain is an external benchmark comparison rather than a quantity fitted from those benchmarks.

full rationale

The paper's central derivation is not circular. Theorem 1 starts from an explicitly stated Gaussian assumption on p(x0|xt) and computes the CMI through conditional differential entropies; Theorem 2 differentiates that closed form using standard matrix calculus, and the Hutchinson estimator is a standard unbiased trace approximation. No parameter is fitted to the reported FID/LPIPS/SSIM numbers, and Algorithm 1's CMI step is a fixed correction applied on top of DPS, so the improvements in Table I are an external benchmark result rather than a prediction forced by construction. The only self-citation dependency is Eq. (9), where the covariance Sigma_post is referenced to the authors' own prior work [11] in addition to the external [20]; because [20] independently supplies the same covariance model and the assumption is stated openly, this is a minor citation dependency, not load-bearing circularity. The supplied skeptic observation that Eq. (6) is independent of the realized measurement y is a correctness/interpretation criticism of the 'measurement-aware' narrative, not a circularity: it shows the objective may be a data-independent regularizer, but it does not show that the derivation assumes its conclusion. Overall circularity score is therefore low.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests primarily on Assumption 1 (Gaussian posterior), which is an approximation from prior work, and on the correctness of the Hutchinson approximation of the third-order derivative. The paper introduces no new entities, but the free parameters (step sizes, number of Hutchinson samples) are not specified, affecting reproducibility.

free parameters (3)
  • CMI gradient step size eta_t = not specified
    Step size for the CMI gradient update in Algorithm 1, which must be tuned by the user. The paper does not report its value or schedule.
  • Number of Hutchinson samples r = not specified
    The number of random vectors used to approximate the trace in Section III-C. This affects the accuracy and computational cost of the gradient approximation.
  • DPS step size zeta_t = not specified
    The step size for the baseline DPS gradient term, inherited from DPS and not reported in this paper.
assumptions (5)
  • domain assumption Assumption 1: p(x0|xt) is Gaussian with covariance Sigma_post = (1-alpha_bar_t)/alpha_bar_t * (I + (1-alpha_bar_t) Hessian log pt(xt))
    This is a strong modeling assumption about the posterior distribution of the clean image given the noisy latent. It is borrowed from prior work [11], [20] and is not derived in this paper. The CMI closed form and gradient depend on this covariance.
  • domain assumption The score network s_theta accurately approximates the true score gradient nabla log pt(xt)
    All diffusion sampling methods rely on the pretrained score model. The paper uses standard pretrained models without addressing approximation error.
  • domain assumption The reverse SDE in Eq. (4), using the sum of score and likelihood gradient, correctly samples from the posterior
    This is the standard formulation for conditional diffusion, but the likelihood term is intractable and approximated in practice, which is precisely the problem the paper aims to mitigate.
  • standard math Gaussian entropy formula and Gauss-Markov (Kalman) update are valid
    Used in Theorem 1 to compute the conditional entropy and the posterior covariance. These are standard results.
  • standard math Hutchinson's trace estimation is valid
    Used in Section III-C to approximate the trace involving the third-order derivative without computing the full tensor.

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Pith. "Pith review of Conditional Mutual Information Based Diffusion Posterior Sampling for Solving Inverse Problems." pith.science (2026). https://pith.science/paper/KK4UAWIK

@misc{pith2026250102880,
  author       = {Pith},
  title        = {Pith review of: Conditional Mutual Information Based Diffusion Posterior Sampling for Solving Inverse Problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KK4UAWIK}},
  note         = {Machine review of arXiv:2501.02880}
}
abstract

Inverse problems are prevalent across various disciplines in science and engineering. In the field of computer vision, tasks such as inpainting, deblurring, and super-resolution are commonly formulated as inverse problems. Recently, diffusion models (DMs) have emerged as a promising approach for addressing noisy linear inverse problems, offering effective solutions without requiring additional task-specific training. Specifically, with the prior provided by DMs, one can sample from the posterior by finding the likelihood. Since the likelihood is intractable, it is often approximated in the literature. However, this approximation compromises the quality of the generated images. To overcome this limitation and improve the effectiveness of DMs in solving inverse problems, we propose an information-theoretic approach. Specifically, we maximize the conditional mutual information $\mathrm{I}(\boldsymbol{x}_0; \boldsymbol{y} | \boldsymbol{x}_t)$, where $\boldsymbol{x}_0$ represents the reconstructed signal, $\boldsymbol{y}$ is the measurement, and $\boldsymbol{x}_t$ is the intermediate signal at stage $t$. This ensures that the intermediate signals $\boldsymbol{x}_t$ are generated in a way that the final reconstructed signal $\boldsymbol{x}_0$ retains as much information as possible about the measurement $\boldsymbol{y}$. We demonstrate that this method can be seamlessly integrated with recent approaches and, once incorporated, enhances their performance both qualitatively and quantitatively.

Figures

Figures reproduced from arXiv: 2501.02880 by the authors.

Figure 1
Figure 1. Qualitative results on FFHQ dataset. As observed, CMI+DPS produces reconstructions that more closely resemble the ground truth than DPS. V. CONCLUSION In this work, we have addressed the challenges of solving inverse problems using DMs by proposing a method that maximizes the conditional mutual information I(x0; y|xt). This approach ensures that intermediate signals in the reverse path are generated to retain maximu… view at source ↗

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Reference graph

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