{"id":"2d138909-cc84-42a7-b07b-1cc25ab2f21e","arxiv_id":"2505.11482","paper_version":3,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":1,"one_line_summary":"A score-based metric computed on corrupted measurements is claimed to equal the KL divergence between training and test image distributions, enabling unsupervised shift detection and adaptation in inverse problems.","lead":"This paper proposes a metric that estimates how different two image datasets are, using only corrupted measurements, such as partial MRI or inpainting, and diffusion model score functions, without clean test images. If correct, it would let practitioners detect distribution shifts and adapt models in medical imaging and other inverse problems where ground truth is unavailable.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1 is not established: Eq. (22) conflates conditional expectations, and the stated W=E[P]^{-3/2} is refuted by a Gaussian counterexample.","rationale":"The reader's weakest assumption is exactly the invalid step in Appendix B, Eq. (22), and my stress-test confirms it is load-bearing: without it, the measurement-domain integrand is not the image-domain KL integrand, so Theorem 1 collapses. My independent Gaussian calculation sharpens the problem: even in a case where the posterior-mean factorization happens to hold for diagonal Gaussian data, the stated W=E[P]^{-3/2} is wrong by a mask-rate-dependent factor p^{-2}. This explains why the reported empirical curves track the image-domain KL only under an unstated normalization: with the theorem as written, Figures 1 and 6 would show factors up to 25 at p=0.2. The adaptation experiments may still be a useful heuristic, but they do not rescue the central theoretical claim. I therefore agree with the reader's REJECT verdict and see no reason to change it.","tokens_in":19360,"tokens_out":11015,"duration_ms":114428,"concrete_test":"Write out the Gaussian check exactly: let p=N(0,I_n), q=N(0,2I_n), H=P with P_ii~Bern(p), and n~N(0,σ^2I). Compute both sides of Eq. (9) in closed form for any p∈(0,1] and σ. The ratio of the measurement integrand to the image-domain integrand is p^{-2}; if instead W is set to E[P]^{-1/2}, the ratio becomes 1. This single computation distinguishes the stated theorem from a corrected weighting and settles whether Eq. (9) holds as written.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central equality Eq. (9) rests on Appendix B, Eq. (22): E[y|y_σ] = E_P[P] E[x|x_σ]. This is not a harmless use of the law of total expectation. Conditioning on y_σ = P x_σ and P does not make P x_σ sufficient for x given the full x_σ; in general E[P x | P x_σ, P] != P E[x | x_σ], and the outer average over P cannot recover the missing information. For non-Gaussian or even correlated Gaussian p, the posterior mean of the projected measurement depends on which coordinates were observed, not on the full noisy image. The same faulty step drives the W=E[P]^{-3/2} weighting. A direct Gaussian check falsifies the theorem as written: take p=N(0,I), q=N(0,2I), and P_ii~Bern(p) diagonal. Then the image-domain integrand of Eq. (4) is E[||(Σ_q^{-1}-Σ_p^{-1})(x+n)||^2]; the measurement integrand of Eq. (9) with W=E[P]^{-3/2} equals p^{-2} times that quantity, not the same quantity. Matching would require W=E[P]^{-1/2}, not E[P]^{-3/2}. Consequently the claimed equality is false as stated; the empirical plots in Figures 1 and 3 can track Eq. (4) only under an unstated normalization or a different W. The adaptation procedure may remain a plausible heuristic, but the theoretical central claim is unsupported.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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).","tokens_in":19560,"tokens_out":4457,"duration_ms":42034,"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":[{"comment":"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.","section":"Appendix B, Eq. (22)"},{"comment":"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.","section":"Appendix B, Eq. (23), step 7"},{"comment":"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.","section":"Appendix C, proof of Theorem 2"}],"minor_comments":[{"comment":"The text says \"another on ODD samples\" where \"OOD\" is meant; please fix the typo.","section":"Section 2.2"},{"comment":"The header contains the typo \"probablity\"; it should be \"probability\".","section":"Table 1"},{"comment":"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.","section":"Appendix B and Appendix C"},{"comment":"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.","section":"Section 3.2, paragraph after Theorem 1"}],"recommendation":"reject","confidential_remarks":"The central theorem of the paper is not merely unproved but false as stated, as shown by a simple Gaussian counterexample. The error in Eq. (22) is not a presentation issue; it is the mechanism by which the measurement-domain metric is claimed to equal the image-domain KL divergence. The adaptation experiments may be of practical interest as a heuristic, but the paper's framing as a principled, theoretically justified KL estimator cannot be salvaged without substantial new assumptions or a different derivation. I see no straightforward fix within the scope of the current manuscript, so I recommend rejection. If the authors can identify a restricted setting (e.g., Gaussian densities or invertible measurement operators) where the equality holds, they should state that restriction explicitly and re-evaluate the empirical claims against it."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read it. The headline: Theorem 1 as stated is false. The paper's central theoretical claim doesn't hold up.\n\nWhat's genuinely new: a measurement-domain distribution-shift estimator for inverse problems, using only corrupted measurements and pretrained diffusion scores. That's a real gap in the literature, and the empirical setup (inpainting, fastMRI) is substantive. The adaptation trick—fine-tuning the OOD model on projected InD measurements—is a sensible heuristic that seems to improve reconstruction.\n\nBut the proof of Theorem 1 breaks at Eq. (22). The authors write E[y|y_σ] = E_P[P] E[x|x_σ]. The step E[x|P x_σ, P] = E[x|x_σ] is not valid: the projected noisy image does not determine the full noisy image, and the posterior mean depends on which coordinates were masked. The outer average over P cannot fix that. A direct Gaussian check makes the failure concrete: take p=N(0,I), q=N(0,2I), diagonal Bernoulli masks with probability p. The image-domain integrand of Eq. (4) is n/((1+σ^2)(2+σ^2)^2). The measurement-domain integrand of Eq. (9) with W=E[P]^{-3/2} evaluates to p^{-2} times that, not the same value. To match, W would need to be E[P]^{-1/2}. So the stated theorem is not merely missing an assumption; it is false as written.\n\nThe same flawed step drives Theorem 2, so that extension inherits the problem. The experiments don't rescue the theory: the plots tracking image-domain KL could be produced with a different normalization, and the paper doesn't disclose a rescaling that would make Eq. (9) correct. The adaptation loss Eq. (31) may still be a reasonable heuristic, but the claimed principled justification is gone.\n\nThere is a salvageable paper here. The idea is good and the empirical work is thorough. Fix the conditioning step, correct the weight to E[P]^{-1/2}, and state any additional assumptions on the posterior mean; then re-run the experiments under the corrected metric. If the tracking still holds, this becomes a useful contribution.\n\nRecommendation: don't accept in current form. But send to peer review: the claim is significant, the experiments are real, and the error is precise enough to be fixable. I'd cite it only after the correction.","headline":"Theorem 1 as stated is false; the measurement-domain KL identity needs a corrected weight, but the heuristic and experiments are worth engaging.","tokens_in":20232,"tokens_out":4152,"would_cite":false,"duration_ms":38701,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Under stated assumptions, the KL divergence between two image distributions equals a weighted score discrepancy computed from corrupted measurements alone.","keywords":["distribution shift","out-of-distribution detection","diffusion models","score functions","inverse problems","KL divergence","unsupervised adaptation","measurement-domain metric"],"falsifier":"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.","tokens_in":19016,"feed_emoji":"📏","tokens_out":13814,"duration_ms":127989,"temperature":0.7,"pith_summary":"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.","feed_headline":"No clean images needed: measure shift from corrupted measurements","feed_subtitle":"Two diffusion models and masked measurements reproduce the true KL divergence, no ground truth needed.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the image-domain score-based KL divergence formula in Eq. (4), which Theorem 1 extends to the measurement domain.","marker":"[42]"},{"why":"Discusses the score discrepancy as a measure of distribution shift and motivates using learned diffusion denoisers to estimate it.","marker":"[43]"},{"why":"Provides the SVD-based corrupted-data measurement model and the assumptions of shared right singular vectors and collectively full-rank projections.","marker":"[45]"},{"why":"Establishes score-based generative modeling through stochastic differential equations, giving the score functions that the metric evaluates.","marker":"[11]"},{"why":"Gives Tweedie's formula relating the posterior mean to the score of the noise-corrupted density, a key step in converting denoisers into scores.","marker":"[38]"},{"why":"Provides the same Tweedie's formula identity used in Eq. (2) and throughout the proof of Theorem 1.","marker":"[39]"},{"why":"Introduces the SVD decoupling of linear measurement operators in the context of diffusion restoration models, used to define the projection matrix P.","marker":"[44]"},{"why":"Supplies the diffusion posterior sampling algorithm used to evaluate whether measurement-based adaptation improves reconstruction quality.","marker":"[17]"},{"why":"Provides the diffusion model training framework and noise schedule used to obtain the score functions and denoisers in the experiments.","marker":"[15]"}],"fun_headline_variants":["Shift detection without clean images: only corrupted measurements","Measure distribution shift from corrupted measurements alone","Diffusion scores reveal shift from noisy measurements","No clean images needed: KL from corrupted data via diffusion","Unsupervised score metric detects shift in inverse problems"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Shift detection without clean images: only corrupted measurements","Measure distribution shift from corrupted measurements alone","Diffusion scores reveal shift from noisy measurements","No clean images needed: KL from corrupted data via diffusion","Unsupervised score metric detects shift in inverse problems"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000974,"raw_usage":{"total_tokens":4118,"prompt_tokens":903,"completion_tokens":3215,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":519,"completion_tokens_details":{"reasoning_tokens":3144}},"tokens_in":519,"tokens_out":3215,"duration_ms":25633,"temperature":1.0,"reasoning_tokens":3144,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T20:54:54.536990+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Maximum Likelihood Training of Score-based Diffusion Models","cited_arxiv_id":null,"evidence_quote":"Supplies the image-domain score-based KL divergence formula in Eq. (4), which Theorem 1 extends to the measurement domain."},{"cited_title":"Generalization in Diffusion Models Arises from Geometry-adaptive Harmonic Representations","cited_arxiv_id":null,"evidence_quote":"Discusses the score discrepancy as a measure of distribution shift and motivates using learned diffusion denoisers to estimate it."},{"cited_title":"GSURE-Based Diffusion Model Training with Corrupted Data,","cited_arxiv_id":null,"evidence_quote":"Provides the SVD-based corrupted-data measurement model and the assumptions of shared right singular vectors and collectively full-rank projections."},{"cited_title":"Score-based generative mod- eling through stochastic differential equations","cited_arxiv_id":null,"evidence_quote":"Establishes score-based generative modeling through stochastic differential equations, giving the score functions that the metric evaluates."},{"cited_title":"An Empirical Bayes Approach to Statistics","cited_arxiv_id":null,"evidence_quote":"Gives Tweedie's formula relating the posterior mean to the score of the noise-corrupted density, a key step in converting denoisers into scores."},{"cited_title":"An Empirical Bayes Estimator of the Mean of a Normal Population,","cited_arxiv_id":null,"evidence_quote":"Provides the same Tweedie's formula identity used in Eq. (2) and throughout the proof of Theorem 1."},{"cited_title":"Denoising diffusion restoration models","cited_arxiv_id":null,"evidence_quote":"Introduces the SVD decoupling of linear measurement operators in the context of diffusion restoration models, used to define the projection matrix P."},{"cited_title":"Diffusion Posterior Sampling for General Noisy Inverse Problems","cited_arxiv_id":null,"evidence_quote":"Supplies the diffusion posterior sampling algorithm used to evaluate whether measurement-based adaptation improves reconstruction quality."},{"cited_title":"Elucidating the Design Space of Diffusion-Based Generative Models","cited_arxiv_id":null,"evidence_quote":"Provides the diffusion model training framework and noise schedule used to obtain the score functions and denoisers in the experiments."}],"review_version":1}