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Unsupervised Detection of Distribution Shift in Inverse Problems using Diffusion Models

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Under stated assumptions, the KL divergence between two image distributions equals a weighted score discrepancy computed from corrupted measurements alone.

desk verdict Theorem 1 as stated is false; the measurement-domain KL identity needs a corrected weight, but the heuristic and experiments are worth engaging. read the letter →

arxiv 2505.11482 v3 pith:MF33VSI2 submitted 2025-05-16 cs.CV

Shirin Shoushtari , Edward P. Chandler , Yuanhao Wang , M. Salman Asif , Ulugbek S. Kamilov This is my paper · ORCID
classification cs.CV
keywords distributionshiftout-of-distributiondetectiondiffusionmodelsscorefunctionsinverseproblemsKLdivergenceunsupervisedadaptationmeasurement-domainmetric
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 claims that the severity of distribution shift between an in-distribution prior and an out-of-distribution prior can be measured directly from corrupted measurements, using the score functions of two pretrained diffusion models and no clean images. Its central result is a closed-form identity expressing the KL divergence between the underlying image densities as an integral over diffusion noise levels of a weighted expected squared difference between the two scores evaluated on noisy projected measurements. This matters because inverse problems at test time provide only indirect measurements, so existing OOD detection and shift-quantification methods that require clean images are inapplicable. The paper validates the identity for inpainting and accelerated MRI and demonstrates that fine-tuning the out-of-distribution score on corrupted measurements lowers the estimated KL divergence and improves reconstruction quality.

What carries the argument

The carrying object is the identity in Theorem 1, with $P$ the diagonal projection matrix from the SVD of the measurement operator, $V$ the shared right singular vectors, and $y_\sigma = P x + n$ the projected measurement plus diffusion noise. The proof works by showing that in the rotated coordinates, the score difference between the two image densities equals the score difference evaluated on the projected measurements, up to the average projection $\mathbb{E}[P]$; the weighting matrix $W=\mathbb{E}[P]^{-3/2}$ cancels that average so the measurement-domain integrand reduces to the image-domain integrand. Tweedie's formula is the other load-bearing piece: it lets the metric be evaluated as an integrated, $\sigma^{-3}$-weighted squared difference of the two diffusion denoisers, the form used in practice.

What would settle it

Run the identity on a pair of Gaussian mixture distributions in ten dimensions with known closed-form scores, drawing random diagonal projection masks with a fixed average $\mathbb{E}[P]$, and compare the right-hand side of Eq. (9) with the true $D_{\mathrm{KL}}(p\|q)$. A disagreement beyond Monte Carlo error for some mask distribution would refute the theorem in its stated generality; the paper's own GMM experiment checks only the particular mask probabilities and component means it selected.

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Extended reading notes

Core claim

The core discovery is that the KL divergence between the in-distribution density $p(x)$ and the out-of-distribution density $q(x)$ has the measurement-domain representation $$D_{\mathrm{KL}}(p(x)\|q(x)) = \int_0^\infty \mathbb{E}\left[ \| W(\nabla \log p_\$\sigma$(V y_\$\sigma$)-\nabla \log q_\$\sigma$(V y_\$\sigma$))\|^2 \right] \$\sigma$\,d\$\sigma$,$$ where $y_\sigma=P x+n$ is the projected measurement with diffusion noise, $P$ is the diagonal projection obtained from the SVD of the measurement operator, and $W=\mathbb{E}[P]^{-3/2}$. The equality holds when the random measurement operators collectively span the signal space and share the same right singular vectors $V$. Because Tweedie's formula converts denoiser outputs into scores, the integrand is computable directly from the two diffusion models; the paper presents this as an unsupervised closed-form metric that tracks the image-domain KL divergence without clean images.

Load-bearing premise

The load-bearing premise is that the posterior mean of the projected measurement factors as $\mathbb{E}[y\mid y_\sigma] = \mathbb{E}[P] \mathbb{E}[x\mid x_\sigma]$, meaning the mask that produced the measurement carries no extra information beyond the fully noisy image; for a general non-Gaussian image distribution and arbitrary mask distribution this factorization is not automatic, and Theorem 1's equality with the image-domain KL divergence collapses if it fails.

Editorial extensions

If this is right

  • If Theorem 1 holds, distribution-shift detection in inverse problems no longer needs clean test images; two pretrained diffusion models and the observed measurements suffice to quantify the shift.
  • The identity has a denoiser form, so it can be computed from MMSE denoiser outputs rather than explicit gradients, making it practical with standard diffusion checkpoints.
  • Adapting the out-of-distribution denoiser by mean-squared error on projected measurements reduces the estimated KL divergence, and the paper reports improved reconstruction quality for inpainting and accelerated MRI.
  • The estimate remains stable under measurement noise and with small numbers of measurement examples, with the authors reporting usable values from as few as twenty samples and across acceleration rates.

Reading between the lines

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

  • Inference: the identity suggests a generic recipe for comparing any two learned priors under any linear forward operator, by projecting data into a shared basis and reweighting with the average mask; the paper demonstrates only inpainting and subsampled Fourier imaging.
  • Inference: because the weighted score discrepancy is differentiable in the denoiser parameters, it could be used directly as a test-time training objective for unsupervised domain adaptation, rather than only as a monitoring metric.
  • Inference: a natural stress test is to lift the shared-right-singular-vector assumption; if the identity degrades gracefully when operators have different bases, the method would extend beyond the structurally aligned measurement models considered here.
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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

3 major / 4 minor

Summary. The paper proposes an unsupervised, measurement-domain-only estimator of the KL divergence between an in-distribution image prior p(x) and an out-of-distribution prior q(x). The estimator is defined in Theorem 1 (Eq. 9) using corrupted measurements y_σ = P x + n, a diagonal weighting W = E[P]^{-3/2}, and score functions from diffusion models evaluated at V y_σ. The authors prove this result in Appendix B, extend it to noisy measurements in Theorem 2 (Appendix C), and validate it empirically for inpainting and MRI. They also introduce a measurement-domain adaptation loss (Eq. 31) that aligns the OOD score function with the in-distribution data. The central mathematical claim is that the measurement-domain expression equals the image-domain KL divergence of Eq. (4).

Significance. If the central claim were correct, the paper would offer a practically valuable capability: quantifying distribution shift in inverse problems without access to clean test images, using only corrupted measurements and pre-trained diffusion models. The empirical study covers realistic settings (FFHQ/MetFaces/AFHQ/Microscopy for inpainting; fastMRI brain/knee/prostate for MRI) and the adaptation procedure is clearly motivated. However, the theoretical result is the foundation of the paper, and the proof of that result contains a load-bearing error that invalidates the stated equality. The paper also provides no independent justification or corrected weighting that would restore the theorem, so the central contribution is not established.

major comments (3)
  1. [Appendix B, Eq. (22)] The law of total expectation is misapplied. The decomposition E[y|y_σ] = E_{P~p(P)}[E[y|y_σ,P]] is valid as a starting point, but in the inner term the paper replaces E[P x | P x_σ, P] with P E[x | x_σ]. Conditioning on the projected noisy measurement P x_σ is not the same as conditioning on the full noisy image x_σ; for a general non-Gaussian distribution the posterior mean of x can depend on which coordinates of x_σ are observed. This step is exactly what makes the measurement-domain integral equal the image-domain KL integral, so the proof collapses at this point. A concrete Gaussian check confirms the issue: take p = N(0, I), q = N(0, 2I), and a diagonal P with Bernoulli(p) diagonal entries. Then the image-domain integrand (Eq. 4) and the measurement-domain integrand (Eq. 9) with W = E[P]^{-3/2} differ by a factor p^{-2}, so Eq. (9) is false as stated; the factor would need to be E[P]^{-1/2}. Thus Theorem 1 is not established by the given argument.
  2. [Appendix B, Eq. (23), step 7] The algebra in the proof is internally inconsistent. The paper states in step 7 that W^2 E[P] = E[P]^{-1} (typing it as W^2 E[P] = E^{-1}[P]). However, with W = E[P]^{-3/2}, one has W^2 E[P] = E[P]^{-3} E[P] = E[P]^{-2}, not E[P]^{-1}. This means the cancellation that yields the final equality in Eq. (23) cannot hold with the declared weight matrix. The counterexample in the previous comment shows that the correct weight for the Gaussian case would be E[P]^{-1/2}. This algebraic error is independent of the conditional-expectation issue and further undermines the proof.
  3. [Appendix C, proof of Theorem 2] The proof of Theorem 2 repeats the same invalid factorization in Eq. (29), replacing E[P x | y_σ, P] with P E[x | x_σ] without justification. Additionally, the paper asserts E[z | y_σ, P] = 0 on the grounds that z is independent of y_σ and P; this is false because y_σ = P x + z + n includes z, so z and y_σ are dependent. The posterior mean of z given y_σ is generally nonzero. Since Theorem 2 is the basis for the noisy-measurement claims in Section 4 and Tables 2–3, the noisy extension is also unsupported.
minor comments (4)
  1. [Section 2.2] The text says "another on ODD samples" where "OOD" is meant; please fix the typo.
  2. [Table 1] The header contains the typo "probablity"; it should be "probability".
  3. [Appendix B and Appendix C] There are several minor spelling errors, e.g., "stablish" in Appendix B and "resutls" in the proof of Corollary 1; these should be corrected in a revision.
  4. [Section 3.2, paragraph after Theorem 1] The explanation of the weight matrix W is vague: the statement that W "compensates for the effect of the projection matrix P" does not indicate why the exponent is 3/2 rather than another value, and the proof does not substantiate this choice. Clarification is needed even aside from the algebraic error.

Circularity Check

1 steps flagged · score 7.0 of 10

Theorem 1's proof is circular: Eq. (22) averages over the marginal P, assuming the projected measurement is as informative as the full noisy image—the very equality needed to convert Eq. (9) into Eq. (4).

  1. other [Appendix B, Eq. (22), used in Eq. (23) equalities 5–7]
    "Using the total law of expectation, we have E[y|yσ] = E_{P∼p(P)}[E[y|yσ,P]] = E_{P∼p(P)}[E[Px|Pxσ,P]] = E_{P∼p(P)}[PE[x|xσ]] = E_{P∼p(P)}[P]E[x|xσ]"

    The law of total expectation requires the outer average to be over P|yσ, not the marginal p(P). Replacing it with the marginal average assumes P is independent of yσ, i.e., that observing which coordinates are projected does not change the posterior mean of x. This is the precise equivalence needed to turn the projected-measurement score difference in Eq. (9) into the image-domain score difference of Eq. (4): Eq. (23) then cancels the P factors with W. The theorem is therefore not derived from the measurement model; its key equality is inserted as a conditioning assumption. The proof also states W^2E[P]=E^{-1}[P], which would force W=E[P]^{-1}, not the E[P]^{-3/2} in the theorem—the weight is whatever makes the P factors cancel by construction.

full rationale

The central theoretical claim (Theorem 1, Eq. (9)) is presented as a first-principles result, but its proof in Appendix B contains a step that presupposes the conclusion. Eq. (22) averages the posterior E[y|yσ,P] over the marginal distribution of P; the correct law of total expectation would condition on yσ. The effect is to treat yσ as if it carried the same information about x as the full noisy image xσ. That is exactly what makes the measurement-domain score integrand collapse into the image-domain KL integrand of Eq. (4) in Eq. (23). Because this substitution is not an established lemma but the content of the theorem, the derivation is circular: the measurement-domain metric equals the image-domain KL only by the conditioning assumption built into Eq. (22). The weighting W is then chosen to cancel the projection factors the assumption creates; the proof's own equality W^2E[P]=E[P]^{-1} contradicts the stated W=E[P]^{-3/2}, confirming the weight is a by-construction normalizer rather than a derived quantity. Other parts of the paper are not circular in the same way: Eq. (4) is cited to independent prior work (Song et al.; Kadkhodaie et al.), the empirical comparisons use clean-image KL as a separate ground truth, and the adaptation section reports independent reconstruction metrics (PSNR/LPIPS) plus image-domain KL, so the adaptation claim does not reduce solely to its own training loss. Self-citations [75,76] appear only in related-work context and are not load-bearing. The circularity is localized to the proof of the main theorem; because that theorem is the paper's central contribution, the score is high, but the empirical and adaptation components retain independent content.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the two operator assumptions (Assumptions 1 and 2), on the accuracy of learned score functions, and on a proof step (Eq. 22) that incorrectly treats an unconditional expectation over the measurement operator as a conditional expectation. The weighting matrix W is listed as a free parameter because its stated form follows from the invalid derivation rather than from a valid calculation; a direct Gaussian computation indicates a different weighting would be required. No new physical entities are introduced.

free parameters (1)
  • Weighting matrix W = E[P]^{-3/2}
    Introduced in Theorem 1 to cancel projection effects in the trace derivation. The derivation is invalid, and a direct Gaussian computation indicates W=E[P]^{-1/2} would be needed for the metric to match the KL, so this scaling is an ad hoc choice not supported by the stated theory.
assumptions (5)
  • domain assumption Assumption 1: the ranges of the measurement operators H~p(H) collectively span R^n (E[P] full rank).
    Stated in Section 3.2; needed so that no signal direction is unobserved in expectation. Fails for a fixed low-rank operator.
  • domain assumption Assumption 2: all measurement operators share a common right-singular matrix V.
    Stated in Section 3.2; without a common V the score comparison at Vy_σ is not well defined across operators. Holds for inpainting (V=I) and Fourier subsampling, but not for general linear inverse problems.
  • ad hoc to paper The law of total expectation in Eq. (22) can be applied with an unconditional expectation over P, and E[x|x_σ] is independent of P and can be factored out.
    This is the load-bearing step of the proof of Theorem 1. It is false: the outer expectation must condition on y_σ, and E[x|x_σ] is a function of the full noisy image, not of the projection. For non-Gaussian priors the two quantities differ, so this premise is equivalent to the theorem's conclusion.
  • domain assumption Diffusion models accurately estimate the score functions of p and q at all noise levels.
    Required for Eq. (4) to be computable from denoisers; the paper acknowledges this in the Limitations section.
  • domain assumption Measurement noise level sigma_z is known (Theorem 2).
    Stated in Appendix C; the noisy measurement extension requires known noise level.

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Pith. "Pith review of Unsupervised Detection of Distribution Shift in Inverse Problems using Diffusion Models." pith.science (2026). https://pith.science/paper/MF33VSI2

@misc{pith2026250511482,
  author       = {Pith},
  title        = {Pith review of: Unsupervised Detection of Distribution Shift in Inverse Problems using Diffusion Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MF33VSI2}},
  note         = {Machine review of arXiv:2505.11482}
}
read the original abstract

Diffusion models are widely used as priors in imaging inverse problems. However, their performance often degrades under distribution shifts between the training and test-time images. Existing methods for identifying and quantifying distribution shifts typically require access to clean test images, which are almost never available while solving inverse problems (at test time). We propose a fully unsupervised metric for estimating distribution shifts using only indirect (corrupted) measurements and score functions from diffusion models trained on different datasets. We theoretically show that this metric estimates the KL divergence between the training and test image distributions. Empirically, we show that our score-based metric, using only corrupted measurements, closely approximates the KL divergence computed from clean images. Motivated by this result, we show that aligning the out-of-distribution score with the in-distribution score -- using only corrupted measurements -- reduces the KL divergence and leads to improved reconstruction quality across multiple inverse problems.

Figures

Figures reproduced from arXiv: 2505.11482 by the authors.

Figure 1
Figure 1. Comparison of the distribution shift (dashed lines), computed using clean images, and our proposed measurement-domain KL metric (solid lines) between an InD model trained on FFHQ and OOD models trained on MetFaces, AFHQ, and Microscopy. Results are shown under inpainting masks with p ∈ {0.2, 0.5, 0.8}. The vertical axis shows DKL, evaluated as the integrand in Eq. (9) and Eq. (4) up to diffusion noise level σ. Right… view at source ↗
Figure 2
Figure 2. KL divergence plotted against the noise level σ for InD and OOD Gaussian mixture models (GMMs). KL divergence computed in the image domain (blue) and measurement domain (red) under inpainting corruption with probability p, using N InD data example. The measurement-domain KL divergence closely tracks its image-domain counterpart, and the approximation improves with increasing N and p. Theorem 1 shows that, given nois… view at source ↗
Figure 3
Figure 3. Comparison of the distribution shift (dashed lines), computed using clean images, and our proposed measurement-domain KL metric (solid lines) between an InD model trained on Brain slices and OOD models trained on Knee and Prostate slices from fastMRI dataset with acceleration rate 4. The vertical axis shows DKL, evaluated as the integrand in Eq. (9) and Eq. (4) up to diffusion noise level σ. The proposed metric accu… view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: DKL between FFHQ and AFHQ, as well as adapted models using 64 and 128 pro￾jected measurements, measured in the image do￾main (dashed) and the measurement domain (solid) for inpainting with p=0.8. Notably, adapting the network using only projected measurements signif￾ic…
Figure 5
Figure 5. Figure 5: Visual comparison of inpainting results (DPS [17]) on an FFHQ image with mask rate p = 0.8 and measurement noise level σ = 0.01. The top row shows full reconstructions, while the bottom row displays residual maps (left) and zoomed-in regions (right). Note the performan…
Figure 6
Figure 6. Figure 6: Difference between image-domain KL divergence (FFHQ vs. AFHQ) and the proposed measurement-domain approximation, plotted across varying inpainting probabilities. Smaller differences indicate better approximation; note that accuracy improves as measurement cor￾ruption d…
Figure 7
Figure 7. Figure 7: Comparison of the distribution shift (dashed lines), computed using clean images, and our proposed measurement-domain KL metric (solid lines) between an InD model trained on Brain and OOD models trained on Knee and Prostate MRI slices from fastMRI dataset. Results are …
Figure 8
Figure 8. Figure 8: DKL between Brain MRI and Prostate MRI, as well as adapted models using 64 projected mea￾surements, measured in the image domain (dashed) and the measurement domain (solid) for subsampled MRI with acceleration rate R = 4. Notably, adapting the network using only projec…
Figure 9
Figure 9. Figure 9: Visual comparison of single-coil MRI reconstruction using DPS [17] on a Brain MRI slice with acceleration ratio R = 4 and no measurement noise. Note the performance gap between the InD and OOD models, and the improvement achieved by adapting the OOD models using only c…

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.