Exact Evaluation of the Accuracy of Diffusion Models for Inverse Problems with Gaussian Data Distributions

Bruno Galerne; Emile Pierret

REVIEW 3 major objections 7 minor 31 references

Exact Evaluation of the Accuracy of Diffusion Models for Inverse Problems with Gaussian Data Distributions

T0 review · 3 major / 7 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read For Gaussian data, CGDM—which uses the exact noisy-likelihood covariance—produces conditional samples closest to the true posterior by exact 2-Wasserstein distance.

desk verdict Solid analytic benchmark for conditional diffusion samplers, with a real new covariance formula — but the printed recursions have typos that must be fixed before the W2 curves can be trusted. read the letter →

arxiv 2507.07008 v2 pith:RRP36XEJ submitted 2025-07-09 cs.LG

Emile Pierret , Bruno Galerne This is my paper · ORCID

classification cs.LG

keywords diffusionmodelsinverseproblemsGaussiandatadistributionsconditionalsampling2-WassersteindistanceposteriorbiasCGDMmicrotextures

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 asks a direct question: when a diffusion model is used to sample solutions of a linear inverse problem, does its output distribution actually match the true conditional distribution? To get a definitive answer the authors restrict the prior to a Gaussian with known mean and covariance, where the score is known in closed form. Within this setting they derive every relevant distribution exactly and compare Deep Posterior Sampling (DPS), Pseudo-inverse Guided Diffusion Models (PiGDM), and their own new algorithm CGDM by exact 2-Wasserstein distance. They find that DPS and PiGDM are biased—DPS severely underestimates the noisy-likelihood covariance and becomes unstable unless its $\alpha_{\mathrm{DPS}}$ is reduced—while CGDM, which uses the exact covariance formula, achieves the smallest distance to the true posterior at every diffusion time. The practical payoff is a quantitative benchmark for how faithful conditional diffusion samplers are.

What carries the argument

The load-bearing object is the exact noisy-likelihood covariance $C^{\mathrm{CGDM}}_{v|t} = (1-\alpha_t)A\Sigma\Sigma_t^{-1}A^T + \sigma^2 I$ with $\Sigma_t = \alpha_t\Sigma + (1-\alpha_t)I$. This is the true covariance of $p_t(v|x_t)$ under the Gaussian prior; DPS and PiGDM replace it with $\sigma^2 I$ and $(1-\alpha_t)AA^T+\sigma^2 I$, respectively. The argument's engine is that with a Gaussian prior all forward and backward processes remain Gaussian, so Algorithm 5 recursively propagates exact covariance matrices and the 2-Wasserstein distance between two Gaussians has the closed form in Eq. (51)--(52). For deblurring Gaussian microtextures, Proposition 1 shows the three algorithms' backward covariances share a common Fourier eigenbasis with $\Sigma$, letting the Wasserstein calculation be done frequency by frequency.

What would settle it

Run the three algorithms on an exactly Gaussian prior with the exact score and a linear operator where $\Sigma$ and $A$ do not share an eigenbasis, computing the full-matrix 2-Wasserstein distance of the final backward samples to $p(x_0|v)$; any case where PiGDM or DPS beats CGDM would contradict the claimed ordering, as would a per-channel blur experiment in which the ranking changes.

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Extended reading notes

Core claim

Under the Gaussian assumption, the noisy likelihood $p_t(v|x_t)$ is exactly Gaussian with covariance $C^{\mathrm{CGDM}}_{v|t} = (1-\alpha_t)A\Sigma\Sigma_t^{-1}A^T + \sigma^2 I$ (Eq. 32, 34), where $\Sigma_t = \alpha_t\Sigma + (1-\alpha_t)I$. The paper's central claim is that plugging this exact covariance into the conditional backward step defines a new algorithm, CGDM, whose final samples minimize the exact 2-Wasserstein distance to the true conditional distribution among CGDM, PiGDM, and DPS. DPS, which effectively fixes the covariance to $\sigma^2 I$, systematically underestimates it and produces biased, unstable posterior samples; PiGDM, which uses $(1-\alpha_t)AA^T + \sigma^2 I$, is closer but still omits the prior covariance structure. The exact recursive Gaussian characterizations of the backward processes make this ranking quantitative: DPS errors reach above $10^1$ in the toy example while CGDM stays in the $10^{-3}$--$10^{-2}$ range, and the same ordering persists in deblurring experiments on Gaussian microtextures. CGDM is not exactly equal to the ideal posterior sampler because the backward initialization and noise covariance remain approximate, but its error is far smaller than the alternatives.

Load-bearing premise

The ranking and bias magnitudes are only defined for exactly Gaussian priors with known covariance and exact scores, and the efficient deblurring recipe additionally requires the same blur kernel on every channel.

Editorial extensions

If this is right

For Gaussian deblurring, the exact ordering of conditional-sampling accuracy is CGDM better than PiGDM better than DPS at essentially every diffusion time, across the tested bicubic and motion blur kernels.
DPS's severe covariance underestimation explains its known instability: with $\alpha_{\mathrm{DPS}}=1$ the gradient is divided by an artificially small variance, so reducing it (here to 0.2) is needed for stability, and even then a final-time bias remains.
PiGDM approaches the exact posterior near $t=0$ because its covariance model converges to the true one there, while CGDM remains the lowest-error choice throughout the diffusion path.
PiGDM is the computationally efficient fallback when the full Gaussian covariance cannot be inverted quickly, whereas CGDM's exact Gaussian computations are significantly more expensive.
For super-resolution with Gaussian microtextures, preliminary bias computations show the same ranking, but the full method requires approximations because the subsampling operator breaks simultaneous diagonalizability.

Reading between the lines

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

If the Gaussian ranking reflects behavior beyond this toy regime, it suggests that covariance mismatch rather than score-estimation error is a primary source of posterior bias in conditional diffusion sampling; this is a hypothesis the paper does not test on natural images.
A practical diagnostic for real-world use would be to estimate the noisy-likelihood covariance empirically and compare its spectrum with the constant $\sigma^2 I$ assumption of DPS; datasets where the ratio is large should show the largest DPS bias.
Because the exact covariance requires $\Sigma$, CGDM is practical mainly when a structured prior covariance is known, such as texture models; extending it to learned priors would require approximating $\Sigma$, which is not analyzed.
The final-time alignment of DPS and PiGDM with the true posterior hints that bias corrections matter most at intermediate diffusion steps, suggesting schedule-dependent tuning rather than uniform hyperparameter scaling.

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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: this is a genuinely useful paper for anyone who wants ground-truth checks of conditional diffusion samplers in a tractable regime. The CGDM covariance (Eq. 34) is the real new object, and the exact W2 decomposition is a clean way to separate bias from covariance error. But Section 3.3 has typos that have to be fixed before the numerical claims are reproducible, and I would not take the printed Eq. (49) at face value.

What it does well: the setup is honest — Gaussian prior with known covariance, exact score. Within that scope, the derivation of p(v|xt) and the comparison of DPS/PiGDM/CGDM in Table 1 is correct and informative. The W2 decomposition into kernel and orthogonal complement is a nice idea; it isolates the part of the error that depends on the algorithm. The ADSN microtexture testbed is a good choice, and the observation that PiGDM is close to exact at low noise times is useful.

Soft spots: the recurrence b_t in Eq. (49) is wrong as printed. From their own Algorithm 3 and Eq. (46), the constant term should be β_t Σ_t^{-1} μ + β_t ΣΣ_t^{-1} A^T (C^{algo})^{-1}(v − Aμ + α_t AΣΣ_t^{-1} μ). The printed version has spurious √α_t and √α_{t−1} factors that would suppress the observation term, especially at early backward times. The W2 curves cannot have been generated from that formula. This is fixable, but it is load-bearing for the evaluation. The Appendix A.6 Woodbury formula has a sign typo (minus where plus is used two lines later), and the "orthogonal basis" in Proposition 1's proof has bv2·bv3 = bt1 bt2 ≠ 0. Both are minor, but they add to the impression that the manuscript needs a careful proofreading pass. Also, the DPS comparison is for the idealized version without the official gradient normalization and clamping; Remark 2 says so, which is fair, but it should be prominent so readers don't overgeneralize to practical DPS.

Who it's for: people working on diffusion-based inverse problems, especially those who care about posterior fidelity rather than just FID. It deserves a serious referee; I'd send it out with major-revision expectations, mainly to sort out the recursions and get code or exact parameter lists released.

Referee Report

3 major / 7 minor

Summary. This paper analyzes conditional DDPM-based samplers for linear inverse problems under the assumption that the prior p0 is Gaussian N(µ, Σ). The authors derive the exact noisy-likelihood covariance C_v|t = (1−α_t) AΣΣ_t^{-1}A^T + σ²I, introduce CGDM as the algorithm using this exact covariance, and derive closed-form recursions for the backward means and covariances of DPS, ΠGDM, and CGDM. They then compute exact 2-Wasserstein distances between the samplers' output distributions and the true conditional forward distribution, both in low-dimensional Gaussian examples and in deblurring experiments on Gaussian microtextures. The central claim is that, under the Gaussian assumption, CGDM achieves the smallest exact W2 error, ΠGDM is intermediate and computationally cheaper, and DPS severely underestimates the likelihood covariance and becomes unstable unless its α_DPS parameter is reduced. The paper is clearly structured and explicitly acknowledges the restrictiveness of its Gaussian assumption.

Significance. If the stated results hold, the paper provides a valuable controlled testbed for conditional diffusion sampling: closed-form expressions enable exact W2 evaluation, and the comparison exposes structural reasons for DPS's and ΠGDM's posterior bias. The strengths are the explicit derivations in Section 3, the exact formulas for noisy likelihoods, and the careful decomposition of the W2 error on ker Σ and its orthogonal complement. The limitations are the Gaussian prior assumption and the requirement that the blur kernel be identical across channels, both of which the authors acknowledge (Assumption 1 and Remark 4). However, the numerical support is compromised by an inconsistency in the backward-mean recursion (Eq. (49)), so the current version is not reproducible as printed.

major comments (3)

[Section 3.3, Eq. (49)] The affine backward recursion printed in Eq. (49) does not follow from Algorithm 3 and Eq. (46). Substituting the exact Gaussian score and the likelihood gradient into Algorithm 3 gives b_t = β_t Σ_t^{-1} μ + β_t ΣΣ_t^{-1} A^T (C^{algo}_{v|t})^{-1} (v − Aμ + α_t AΣΣ_t^{-1} μ), not the printed expression with the extra factors β_t√α_t and β_t√α_{t−1} on the two terms. The matrix A_t in Eq. (48) is consistent with the corrected b_t, so the extra factors are not absorbed elsewhere. Since Algorithms 4 and 5, and therefore all reported W2 curves, depend on Eqs. (47)–(49), the numerical results in Figures 8–10 are not reproducible from the printed formulas. The same issue propagates into the ker Σ analysis in Section 4.3 (Eqs. (62)–(66)). Please correct Eq. (49) and re-derive the affected recursions, or provide the code used for the experiments.
[Section 4.3, DPS hyperparameter] The DPS curves in Figures 8–10 are computed with 'the lowest stable value' of α_DPS, but no numerical values or a precise definition of stability are given. Because the DPS likelihood gradient scales linearly with α_DPS (Eq. (23)), the DPS contribution to the reported W2 distances and the relative ranking of DPS versus the other algorithms are functions of this free parameter. Please report the values used for each figure and include a sensitivity analysis over α_DPS.
[Appendix B.2, Proposition 1 proof] The proof of Proposition 1 claims that the vectors bv2 and bv3 in Eq. (B86) form an orthogonal basis, but bv2^T bv3 = bt1 bt2, which is not zero in general. The vectors do lie in the kernel of bΣ(ξ) and are linearly independent in the generic case, so the statement of Proposition 1 may survive after orthogonalization, but the proof as printed is invalid. Since Proposition 1 underpins the simultaneous diagonalization used for the efficient W2 evaluation in Section 4, the proof must be corrected.

minor comments (7)

[Eq. (14)] Eq. (14) writes E(Ax0 + σ²n | xt), but the forward model in Eq. (6) is v = Ax0 + σn; the mean is unaffected because E(n)=0, but the formula should be corrected.
[Appendix A.3, Eq. (A32)] In the application of Lemma 1, M is written as AΣA + σ²I; it should be AΣA^T + σ²I. The general statement of Lemma 1 in Eq. (A7) is correct, so the typo is local.
[Eq. (51)] The two Gaussian distributions are both written as N(µ1, Σ1); the second should be N(µ2, Σ2).
[Appendix A.6, Eq. (A69)] Eq. (A69) has a minus sign before α_t AΣ²Σ_t^{-1}A^T inside the inverse, whereas Eqs. (A70)–(A74), which are the ones used in the paper, require a plus sign. The plus sign is the one consistent with the Woodbury derivation. Please correct Eq. (A69) to avoid ambiguity.
[Figures 2–5] The legend label 'GDM' should be 'ΠGDM' for consistency with Table 1 and the main text.
[Section 3.4] The text says that CGDM 'aligns perfectly' with the true theoretical distribution while the 2-Wasserstein distance is non-zero; please rephrase to avoid the apparent contradiction.
[General] There are several typographical errors, including 'theoeretical', 'fisrt', 'biaised', and 'distrbution'; a careful copyedit is needed.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: CGDM's exact-likelihood covariance is definitional and self-citations are minor, but the Wasserstein ranking is computed, not fitted.

full rationale

Walking the claimed derivation chain: Assumption 1 fixes p0 = N(µ,Σ) and provides the exact score (Eq. 29). Equation (32) then derives p_t(v|x_t) = N(A x̂0(x_t), (1−α_t)AΣΣ_t^{-1}A^T + σ²I), and Table 1/Eq. (34) define CGDM by fixing its likelihood covariance to exactly this expression. The later identity Eq. (45) — that CGDM's noisy posterior covariance equals the forward covariance of p_t(x_t|v) — is an algebraic rewriting of that defining choice, stated transparently by the authors ('which is the exact expression under the Gaussian assumption'). This is a definitional consequence, not a hidden prediction; more importantly, no constant is fitted to the W2 benchmark. The numerical ranking is obtained by iterating the Gaussian backward recurrences (Eqs. 47–50, Algorithms 4–5) and comparing the resulting distributions with the independently closed-formed forward distribution (Eq. 31), so the CGDM < ΠGDM < DPS ordering is a computed mathematical consequence. The self-citations [16] and [25] supply ADSN score-evaluation machinery and a kriging lemma, but the formulas needed here are rederived in Appendices A and B, so the citations are not load-bearing. Tuning αDPS to its 'lowest stable value' adjusts a baseline algorithm rather than fitting the CGDM result, and if anything favors DPS. The paper also explicitly delimits its scope (Remark 4; Section 5), and the reviewer-flagged issues with Eq. (49)'s √α factors, the Appendix A.6 sign, and the Proposition 1 basis orthogonality are internal correctness/reproducibility concerns rather than circularity: none exhibits a claimed result that is equivalent by definition to its own inputs. Score 2 reflects only the minor self-citation and the partly definitional character of the CGDM likelihood claim.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The central claim rests on the Gaussian prior assumption and the availability of the exact score; these are stated assumptions, not free parameters. The only fitted constant is the DPS stability hyperparameter α_DPS. No new physical entities, particles, forces, or dimensions are introduced; CGDM is a named algorithm, not an entity.

free parameters (1)

alpha_DPS (DPS stability hyperparameter) = 0.2 in toy examples; 'lowest stable value' in deblurring experiments (exact values not reported)
Introduced by Chung et al. [8]; the paper tunes it to the lowest value that keeps DPS stable, which directly affects the W2 ranking. It is not predicted from the Gaussian model.

assumptions (6)

domain assumption The data distribution p0 is Gaussian N(µ, Σ) with known µ and Σ (Assumption 1, Section 3).
CGDM's exact formulas and the W2 computation rely on closed-form Gaussian conditioning; the paper explicitly restricts to this case.
domain assumption The unconditional score ∇ log p_t is exactly known (Section 3.2).
Used to implement DPS, PiGDM, and CGDM without score estimation error; the paper assumes a perfect score.
domain assumption The inverse problem is linear: v = Ax0 + σn with known A and σ, n ~ N(0,I) (Eq. 6).
All likelihood models rely on this observation model.
domain assumption DDPM forward and backward dynamics with diagonal backward noise β_t I (Eqs. 1, 5, Remark 1).
The sampler analyzed is the discrete DDPM with σ_t = β_t; the paper states results extend to other schedules but all numbers are computed for this one.
domain assumption For ADSN microtextures, the covariance is the autocorrelation of a texton (Eq. 54) and diagonalizable in the Fourier basis; deblurring applies the same kernel to each channel so that A, Σ, and Σ_t share eigenvectors (Proposition 1).
Enables exact spectral W2 computation; the paper proves this for the described deblurring setup and notes failure for per-channel different kernels.
domain assumption 2-Wasserstein distance between Gaussians is the appropriate fidelity metric (Eq. 51).
The evaluation metric is a modeling choice, not an empirical benchmark; the paper motivates it as an exact alternative to FID.

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Cite this review

Pith. "Pith review of Exact Evaluation of the Accuracy of Diffusion Models for Inverse Problems with Gaussian Data Distributions." pith.science (2026). https://pith.science/paper/RRP36XEJ

@misc{pith2026250707008,
  author       = {Pith},
  title        = {Pith review of: Exact Evaluation of the Accuracy of Diffusion Models for Inverse Problems with Gaussian Data Distributions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RRP36XEJ}},
  note         = {Machine review of arXiv:2507.07008}
}

abstract

Used as priors for Bayesian inverse problems, diffusion models have recently attracted considerable attention in the literature. Their flexibility and high variance enable them to generate multiple solutions for a given task, such as inpainting, super-resolution, and deblurring. However, there is still a lack of understanding about how accurately these conditional diffusion algorithms perform conditional sampling. In this article, we investigate the errors induced by these models when applied to a Gaussian data distribution for which the score function is exactly known. Within this constrained context, we are able to precisely analyze the discrepancy between the theoretical resolution of inverse problems via conditional sampling and the practical distributions generated by conditional diffusion models. This is done by characterizing all the involved iterative Gaussian processes and by computing the exact Wasserstein distance between the distributions of the diffusion model samplers and the ideal conditional distribution associated with the inverse problem. Our findings allow for the comparison of two major algorithms from the literature, Deep Posterior Sampling (DPS) and Pseudo-inverse Guided Diffusion Models ($\Pi$GDM), and the introduction of the new paradigm Conditional Gaussian Diffusion Models (CGDM) that is shown to be more accurate for Gaussian data distributions.

Discussion (0). Continue with ORCID to comment.

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Reviewed August 6, 2026 · model on record in the stance chip above.

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