{"id":"c90d4c4b-2a8a-4b9f-b0cd-a27bdd8cbf4c","arxiv_id":"2506.07841","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Diffusion models trained on disjoint data converge at high noise but diverge near the data manifold, and they fail to denoise very small perturbations accurately.","lead":"Diffusion models trained on separate halves of the same dataset agree when the input noise is large, but they pull apart when the noise is small. This limits their usefulness for denoising small perturbations and reveals that current models do not learn the fine local shape of the data distribution.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Missing same-data control: low-noise divergence may be due to training stochasticity rather than disjoint training data.","rationale":"The reader's weakest assumption is exactly the concern I identify: the paper lacks a same-data, different-seed control, so the divergence attributed to disjoint training data could instead arise from training stochasticity or architecture sensitivity. This is the most load-bearing issue because the abstract's causal wording ('models trained on disjoint data diverge') depends entirely on the disjoint-vs-same comparison. The missing control is internally relevant, not a matter of consensus, and it is testable with a straightforward experiment. If the same-data control also shows low-noise divergence, the conclusion must be softened to 'independently trained models diverge locally' rather than 'disjoint data causes divergence'. Since the reader's CONDITIONAL verdict already reflects this missing control and asks for it to be addressed, my read does not change the verdict; it strengthens the rationale for the condition.","tokens_in":12426,"tokens_out":3043,"duration_ms":36276,"concrete_test":"Train two UNet denoisers with identical architecture, optimizer, and hyperparameters on the exact same 100k CelebA subset, using two different random seeds. Run the Section 4.1 local probe on the same 200 inputs and compute the cosine similarity of denoising directions at sigma = 0.001. If the same-data pair shows a similar drop (around 0.55), the divergence is not caused by disjoint training data and the central causal claim fails; if the same-data pair stays high (for example, above 0.8) while the disjoint pair drops to about 0.55, the data-split attribution is supported. Repeat over several seeds to obtain error bars and significance estimates.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim, stated in the abstract and Section 4.1, is that models trained on disjoint data diverge near the data manifold even when their high-noise outputs converge. The evidence compares Model A and Model B trained on disjoint CelebA subsets, but the experiments never train two models on the same subset with different random seeds. Under low noise, denoising directions are highly sensitive to local details, so the observed divergence could stem from optimization stochasticity, random initialization, or architecture sensitivity rather than from the disjointness of the training sets. This is not a minor omission: it is the only comparison that separates 'disjoint data' from 'independently trained model'. Without a same-data baseline, the cosine similarity of approximately 0.55 at sigma = 0.001 establishes that independently trained models disagree locally, but it does not establish that disjoint training data causes this disagreement. Appendix B1 states that 'the only source of variation is the model's learned vector field', implicitly assuming training randomness contributes nothing; that assumption is untested.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper investigates the low-noise behavior of diffusion models. It trains UNet denoisers, noise-conditional score networks (NCSN), and sliced-score-matching (SSM) models on disjoint CelebA subsets of varying sizes, as well as on analytic Gaussian mixture benchmarks. The main empirical claim is that models trained on disjoint data produce similar outputs in the high-noise regime but diverge near the data manifold: at sigma = 0.001 the cosine similarity of denoising directions between two 100k-subset models is reported to be about 0.55, versus near-perfect agreement at sigma = 1.0. The paper also reports that denoising performance degrades for very small perturbations, that small models exhibit discrete-attractor re-denoising dynamics, and that all tested models learn curved or anisotropic score geometry poorly. It introduces a re-denoising probe and uses analytic Gaussian-mixture scores as ground truth.","tokens_in":12597,"tokens_out":5416,"duration_ms":69126,"significance":"If the central claim is supported, the paper identifies a practically relevant limitation of diffusion denoisers: the learned local score field is not uniquely determined even with large training sets, which has implications for robustness, adversarial purification, and fine-grained generation control. The use of analytic Gaussian-mixture scores as ground truth is a strength, as is the proposed re-denoising attractor test. The paper also ships a clear experimental framework and promises public code. However, the causal interpretation of the main result currently rests on a missing control, and the quantitative evidence lacks multi-run uncertainty estimates. The core phenomenon is worth publishing only after these gaps are addressed.","major_comments":[{"comment":"The central claim—that models trained on disjoint data diverge near the data manifold—is not established, because the experiments compare only two models trained on disjoint subsets and never train two models on the same subset with different random seeds. The observed divergence at sigma = 0.001 could equally arise from random initialization, batch ordering, or other optimization stochasticity. The statement in Appendix B1 that 'the only source of variation is the model's learned vector field' assumes this source away. Please add a same-data control with at least two seeds and report the same L2 and cosine metrics; if same-data pairs show comparable divergence, the causal wording in the abstract, §4.1, and Discussion must be revised.","section":"§4.1, Fig. 1; Appendix B1"},{"comment":"The quantitative support for the main result lacks error bars, confidence intervals, or significance tests. The text states that metrics are averaged over 200 image samples, but with one trained pair per condition this captures only image-level variability, not model-training variability. Please report multiple training seeds and show the spread of the L2 and cosine metrics; this is necessary to confirm that the gap between sigma = 1.0 and sigma = 0.001 (e.g., cosine similarity falling to about 0.55) is robust rather than an artifact of one run.","section":"§4.1, Figs. 1b,c and Appendix B2"}],"minor_comments":[{"comment":"The sentence 'quantification of the differences of the two below reveal divergence at low-noise' is grammatically awkward and should be revised.","section":"§4.1"},{"comment":"There is a typo: 'celebA susbet' should read 'CelebA subset'.","section":"§4.2"},{"comment":"The phrase 'soL2 distances remain highest' is missing a space and should read 'so L2 distances remain highest'.","section":"Appendix B3"},{"comment":"The caption repeats 'cosine similarity' many times and is hard to read; consider a cleaner layout with concise labels.","section":"Figure 7 caption"},{"comment":"The NCSN objective is written without explicitly stating the distribution of sigma over which the expectation is taken; please specify the noise schedule or refer to [20] for the exact weighting.","section":"§3.2"}],"recommendation":"major_revision","confidential_remarks":"The main concern is the missing same-data control, which is fixable and central to the paper's causal claim. I would not recommend acceptance in the current form. If the authors add the control, report multi-seed statistics, and revise the wording accordingly, the paper would be a solid empirical contribution to the study of low-noise diffusion behavior."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper does something genuinely useful: it looks at the low-noise regime that Kadkhodaie et al. skipped, and it builds a local probe—initializing two models from near-identical corrupted inputs and watching trajectories diverge—plus a re-denoising test for attractor behavior. Extending the analysis to NCSN and SSM objectives and to analytic Gaussian mixtures is a real step beyond the original harmonic-representation story. The synthetic benchmarks with known scores give a clean way to see that all architectures cut corners on curved or anisotropic structure. These tools will likely be reused.\n\nThe biggest problem is the missing same-data control. The abstract and Section 4.1 say models trained on disjoint data diverge at low noise, but there is no pair of models trained on identical data with different seeds. Without that, the divergence could come from optimization stochasticity, random initialization, or architecture sensitivity rather than the data split. Appendix B1 says \"the only source of variation is the model's learned vector field,\" but that is an assumption, not an experiment. This is not a minor omission—it is the difference between \"independent models disagree locally\" and \"disjoint data causes local disagreement.\" The authors should either add the control or soften the causal wording.\n\nAlso, the key figures lack error bars or significance tests. With n=200 image samples, the 0.55 cosine similarity at sigma=0.001 is a point estimate; a confidence interval would help. The discrete-attractor claim for small-dataset models is supported by boxplots, but again there is no replication across seeds.\n\nMinor: the leap from 2D synthetic results to \"unlikely that current diffusion models can accurately learn the local structure of the CelebA dataset\" is extrapolation. The high-dimensional synthetic results help, but they are still not natural images. That phrasing should be tempered.\n\nOverall, the central observation—low-noise divergence between independently trained models—is likely correct and interesting. The causal attribution to disjointness is not yet established. This is fixable with one control experiment and some uncertainty quantification. I would send it to review, with those requests. It is a solid empirical paper that deserves referee time.","headline":"Useful low-noise probe and attractor test, but the causal claim about disjoint training data needs a same-data control before it is supported.","tokens_in":13110,"tokens_out":1857,"would_cite":true,"duration_ms":23063,"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":"Diffusion models trained on disjoint data agree at high noise but diverge near the data manifold, where small-perturbation denoising also fails.","keywords":["diffusion models","low-noise regime","score field consistency","memorization versus generalization","local probe","denoising accuracy","attractor dynamics","Gaussian mixture benchmarks"],"falsifier":"Train Model A and Model B on the identical 100k-image subset using only different random seeds, and run the local probe at $\\sigma = 0.001$. If the same-data pair also shows a cosine similarity of denoising directions near 0.55, the claim that disjoint training data causes the divergence is refuted; if the same-data pair stays well above the disjoint-pair value, the data-disjointness explanation survives.","tokens_in":1739,"feed_emoji":"📉","tokens_out":1742,"duration_ms":88631,"temperature":0.7,"pith_summary":"This paper claims that the agreement between diffusion models trained on different data does not survive at low noise. Two denoisers trained on disjoint 100k-image subsets of CelebA produce nearly identical outputs when started from high noise ($\\sigma = 1.0$), but the cosine similarity of their denoising directions falls to about 0.55 at $\\sigma = 0.001$, in the neighborhood of the data manifold. The same local probe shows that denoising very small perturbations is unreliable: improvements stop around $\\sigma = 0.05$ and outputs drift away from the clean image at lower levels, even for a 100k-image model. A small model trained on ten images returns to the same training images under repeated perturbation, while large models do not. On synthetic Gaussian mixtures with known score functions, all tested architectures learn trajectories that cut straight toward density centers instead of following the true manifold geometry, a bias the paper argues would only worsen for high-dimensional natural images. The paper matters because low-noise conditions are exactly the regime of adversarial purification, medical image denoising, and fine-grained generation control.","feed_headline":"Denoisers agree at high noise, split near the data","feed_subtitle":"Two 100k-image denoisers agree at high noise but drop to 0.55 cosine similarity at the smallest noise tested.","key_machinery":"The load-bearing device is the 'local probe': instead of initializing denoisers from pure random noise, the procedure starts both models from identical images corrupted by Gaussian noise with decreasing variance $\\sigma \\in \\{0.001, 0.01, 0.05, 0.1, 0.2, 1.0\\}$ and compares their denoising trajectories through the L2 distance between outputs and the cosine similarity of predicted denoising directions. A companion 're-denoising' procedure adds noise, denoises, and repeats, testing whether trajectories return to the same samples (discrete attractors) or wander continuously. For the synthetic benchmarks, analytic score functions of Gaussian mixtures provide a ground truth against which the learned score fields are measured in L2 error. Together these tools convert 'local consistency' from a qualitative idea into quantitative measurements across noise scales.","core_discovery":"The paper's central discovery is that the generalization documented in the high-noise regime does not extend to the low-noise regime. Using a 'local probe' that initializes denoisers near clean images, the authors show that two models trained on disjoint subsets of 100k images converge at high noise but diverge near the manifold: the L2 distance between their trajectories grows over denoising steps at $\\sigma = 0.001$, and the cosine similarity of their denoising directions drops to roughly 0.55, versus near-perfect agreement at $\\sigma = 1.0$. The same instability limits denoising: at $\\sigma = 0.05$ and below the denoised output moves away from the clean image rather than toward it. A second finding is that a model trained on ten images returns to the same training images under repeated perturbation, indicating discrete-attractor dynamics, whereas a 100k-image model does not, behaving more like a continuous attractor. On analytic Gaussian mixtures where the true score is known, all three model objectives (UNet reconstruction, NCSN, and sliced score matching) learn trajectories that head directly to density centers and miss curved or strongly anisotropic structure, showing a systematic bias in how diffusion models learn local geometry.","pith_inferences":["The data-disjointness attribution is not yet isolated: a same-data, different-seed control is missing, so optimization stochasticity or architecture sensitivity remains a viable alternative explanation for low-noise divergence.","A direct test would train two models on the same subset with different random seeds and repeat the local probe at $\\sigma = 0.001$; a comparable cosine drop would move the explanation from data content to inherent low-noise instability.","If the straight-line-to-center bias persists in latent-space and video diffusion models, near-manifold edits and inverse problems built on those models inherit the same curvature error; the paper explicitly lists latent-space models as open territory.","A practical noise-floor corollary: in applications that rely on near-manifold denoising, such as image editing, MRI reconstruction, and purification, the learned score field's local inconsistency means improvements below roughly $\\sigma = 0.05$ should not be expected to track the true clean image."],"forward_implications":["The memorization-versus-generalization dichotomy does not settle local behavior: even a 100k-image model that generalizes globally is an inconsistent denoiser within 0.001-scale perturbations of the manifold.","Diffusion-based adversarial purification has a floor: at $\\sigma = 0.05$ and below, denoising drifts away from the clean image, so small adversarial perturbations near that scale cannot be stripped reliably.","Small training sets convert the denoiser into a lookup of discrete attractors, with repeated perturbation returning to the same training image, while large sets interpolate continuously between training images.","Explicit score-matching objectives (NCSN, sliced score matching) track low-density tails of anisotropic mixtures better than reconstruction-based UNets and slightly reduce low-noise divergence, but do not remove it.","Trajectories that cut straight to density centers on the spiral mixture indicate a systematic bias in learned score fields that would only be stronger for high-dimensional natural images."],"supporting_citations":[{"why":"Supplies the disjoint-subset training setup, the UNet denoiser architecture, and the high-noise convergence baseline that this paper replicates and extends into the low-noise regime.","marker":"[11]"},{"why":"Defines NCSN and sliced score matching, supplying the two explicit score-matching objectives whose low-noise behavior is compared against the reconstruction UNet.","marker":"[20]"},{"why":"Introduces diffusion-based adversarial purification, the motivating application whose small-perturbation scale the paper tests against.","marker":"[17]"},{"why":"Fixes the score-based SDE formulation and reverse denoising dynamics on which the trajectory comparisons are built.","marker":"[7]"},{"why":"Provides the discrete-attractor theory of associative memory that motivates the re-denoising test for memorization in small models.","marker":"[41]"},{"why":"Documents that diffusion models can reproduce training data verbatim, grounding the memorization-regime framing that the low-noise results connect to.","marker":"[9]"}],"fun_headline_variants":["Low-noise diffusion splits models near the data","Diffusion generalization fails at low noise","Denoisers diverge where data lives","At low noise, diffusion models part ways","Small noise exposes diffusion's memorization trap"],"cache_read_input_tokens":15360,"weakest_assumption_plain":"The argument rests on the assumption that the observed low-noise divergence is caused by the two models seeing different training data; because the experiments never train two models on the same data with different random seeds, the divergence could instead be an artifact of optimization stochasticity or architecture sensitivity rather than data content.","fun_headline_variants_meta":{"raw":{"variants":["Low-noise diffusion splits models near the data","Diffusion generalization fails at low noise","Denoisers diverge where data lives","At low noise, diffusion models part ways","Small noise exposes diffusion's memorization trap"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000193,"raw_usage":{"total_tokens":1356,"prompt_tokens":959,"completion_tokens":397,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":575,"completion_tokens_details":{"reasoning_tokens":332}},"tokens_in":575,"tokens_out":397,"duration_ms":4967,"temperature":1.0,"reasoning_tokens":332,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T05:23:54.642338+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Train Model A and Model B on the identical 100k-image subset using only different random seeds, and run the local probe at $\\sigma = 0.001$. If the same-data pair also shows a cosine similarity of denoising directions near 0.55, the claim that disjoint training data causes the divergence is refuted; if the same-data pair stays well above the disjoint-pair value, the data-disjointness explanation survives.","supporting_citations":[],"review_version":1}