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arxiv: 2603.13419 · v2 · pith:XTYU6EUZnew · submitted 2026-03-12 · 💻 cs.LG

Diffusion Models Memorize in Training -- and Generalize in Inference

Tim Kaiser , Markus Kollmann This is my paper

Pith reviewed 2026-05-21 10:48 UTC · model grok-4.3

classification 💻 cs.LG
keywords diffusion modelsmemorizationgeneralizationdenoising objectiveoverfittingflow fieldsampling trajectories
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The pith

Diffusion models overfit the denoising objective but generalize in inference because model error shifts sampling trajectories away from training data.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper establishes that diffusion models progressively overfit their denoising training objective, creating a clear performance gap between training and validation samples that peaks at intermediate noise levels. Despite this memorization during training, the models still produce diverse outputs at inference time. Through analysis of a fully analytic error-prone toy model, the authors trace how the optimal flow field would sharply localize around training points, yet model error smooths this into a generalizing field. The training gap fails to produce overfitting at inference because sampling trajectories remain distant from the distribution of noisy training samples.

Core claim

The flow field generalizes through model error, which moves sampling trajectories outside the domain of noisy training samples and thereby naturally prevents overfitting, even as the model fully memorizes the training data in the denoising objective.

What carries the argument

The denoising flow field, which localizes sharply around training points in its optimal form but is smoothed by model error into a generalizing version.

If this is right

  • The generalization gap between training and validation performance is largest at intermediate noise levels.
  • Model error suppresses exact recall of individual training points and produces a smooth flow field instead.
  • The training generalization gap does not carry over to inference time.
  • Generated samples show no strong similarity to training samples despite the objective-level overfitting.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same error-driven smoothing might stabilize other generative models whose training objectives differ from their inference paths.
  • Controlled amounts of model error could be deliberately introduced during training to improve generalization without changing the architecture.
  • Quantifying trajectory distances in large-scale diffusion models would test how far this separation holds in practice.

Load-bearing premise

The intermediate states of sampling trajectories are sufficiently far from the distribution of noisy training samples the model is trained on.

What would settle it

Measuring the distance of intermediate sampling states to the nearest noisy training samples and checking whether this distance correlates with increased similarity between generated and training samples.

Figures

Figures reproduced from arXiv: 2603.13419 by Markus Kollmann, Tim Kaiser.

Figure 1
Figure 1. Figure 1: Generalization gap and overfit￾ting for ED(σ) at σ ≈ 1.67. We de￾fine overfitting as a decrease in valida￾tion performance while training perfor￾mance improves. we find that state-of-the-art denoiser models exhibit a significant relative gen￾eralization gap between training and vali￾dation data ( [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The relative generalization gap (Eval − Etrain)/Etrain increases with images seen during training (colorbar in millions) and with model size in relation to dataset size. The black line indicates the beginning of overfitting (validation error starts in￾creasing, [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Flow field lines and predictions of the optimal target predictor y ∗ (x, σ) at different noise levels σ = 28, 2.8, 0.63. Color indicates the magnitude of the prediction error y ∗ (x, σ)−x. Field geometry is (a) global, predictions tend towards the superposi￾tion of all training points, (b) moderately localized around training points, predictions approximate the data manifold, and (c) highly localized aroun… view at source ↗
Figure 4
Figure 4. Figure 4: Relative generalization gap (Eval − Etrain)/Etrain in our 2D toy model for different settings of the model error parameter δ (col￾orbar). Analyzing the behavior of yδ at different regimes of the ratio σ/δ explains why the peak emerges: At high noise (σ ≫ δ), yδ behaves like the posterior mean predictor y ∗ , but noisy samples x are far away from the data man￾ifold. Predictions tend towards the superposi￾ti… view at source ↗
Figure 5
Figure 5. Figure 5: Filled contours showing the relative generalization gap (E − Etrain)/Etrain at intermediate noise (σ ≈ 1.1) between the training points and each point in space, respectively. The black lines enclose the area where this gap is 0.5 or less. As the model error δ decreases, this region shrinks until it no longer contains the validation points, indicating that the generalization gap to the validation set is lar… view at source ↗
Figure 6
Figure 6. Figure 6: Relative generalization gaps (Mval−Mtrain)/Mtrain vs training time and model capacity for various metrics on ImageNet-64. Approximately linearly scaling for one￾step reconstruction metrics, using (a) L2-norm to compare paired samples, (b) Fréchet Distance to compare distributions of paired samples. (c) Relative generalization gaps using tFD (FDD) as a metric, comparing the output of denoising trajectories … view at source ↗
Figure 7
Figure 7. Figure 7: Metric results for Mtrain and Mval for various metrics on ImageNet-64. (a),(b) For reconstruction-based metrics, the training performance improves with training time and model size, whereas the validation performance eventually degrades, resulting in classical overfit behavior. (c) With FDD, overfitting is absent, because training per￾formance degrades as well [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: (a) The relative generalization gap (Mval − Mtrain)/Mtrain in pixel space increases and shifts to the right when the number of classes (colorbar) for a class￾conditioned model increases. (b) In feature space, this trend is mostly mitigated, and (c) stays absent with Fréchet Distance-based metrics. σ fixed near the peak of the gap [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Relative generalization gaps (Mval − Mtrain)/Mtrain for Autoguidance and CFG with EDM2-S on ImageNet-64 with varying guidance weights. w = 0 corresponds to no guidance. Reconstruction-based metrics fix σ near the peak in the generalization gap. a moderate yet steady increase in the generalization gap as the guidance weight increases, given that the primary and auxiliary models differ only in their model er… view at source ↗
Figure 10
Figure 10. Figure 10: (a)-(b) Relative generalization gap (Mval − Mtrain)/Mtrain and (c)-(d) training and validation results Mtrain/val, with EDM2-M on ImageNet-64 with tFD and truncated inference. The indices in the legend show at which step inference was stopped, with 16, 17, 18, 19, 20 corresponding to intermediate noise levels σ ≈ 2.2, 1.6, 1.2, 0.8, 0.6, comparable to the peak in the relative generalization gap for the re… view at source ↗
Figure 11
Figure 11. Figure 11: (a) Decreasing the receptive field mitigates the relative generalization gap (Eval − Etrain)/Etrain. Colorbar shows the sliding window size. 64 corresponds to regular inference. (b) During inference, the model shifts from long-range correlations to short-range correlations. The correlation score is averaged over pixels and images. (c) Edge pixels tend to have longer-range dependencies. σ ≈ 1.2. RGB shows … view at source ↗
Figure 12
Figure 12. Figure 12: Edge pixels tend to have longer-range dependencies. Shown is the pixel-wise correlation score, averaged over 128 images, computed with EDM2-S on ImageNet-64 at σ = 1. CS_{ij} = \frac {\frac {1}{N}\sum _{k, l} d(i,j,k, l)\frac {\partial D(\x )_{ij}}{\partial \y _{kl}}}{\frac {1}{N} \sum _{k, l} \frac {\partial D(\x )_{ij}}{\partial \y _{kl}}} , \quad \x = \y + \sigma \bm \eta with d(i, j, k, l) = [PITH_FU… view at source ↗
Figure 13
Figure 13. Figure 13: Relative generalization gap (Eval − Etrain)/Etrain with EDM2 on ImageNet￾64 for various model sizes. Black lines indicate σ values used in [PITH_FULL_IMAGE:figures/full_fig_p021_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Relative generalization gap (Mval − Mtrain)/Mtrain with EDM2-M on ImageNet-64 for various metrics. Black lines indicate σ values used in [PITH_FULL_IMAGE:figures/full_fig_p021_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Relative generalization gap (Mval−Mtrain)/Mtrain with EDM2 on ImageNet￾64 for various metrics and model sizes. For reconstruction-based metrics, σ is fixed at the peak of the generalization gap, see [PITH_FULL_IMAGE:figures/full_fig_p022_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Training and validation results Mtrain/val with EDM2 on ImageNet-64 for various metrics and model sizes. For reconstruction-based metrics, σ is fixed at the peak of the generalization gap, see [PITH_FULL_IMAGE:figures/full_fig_p023_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Relative generalization gap (Eval − Etrain)/Etrain with EDM2 on ImageNet￾512 for various model sizes. Black lines indicate σ values used in [PITH_FULL_IMAGE:figures/full_fig_p024_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Relative generalization gap (Mval − Mtrain)/Mtrain with EDM2-M on ImageNet-512 for various metrics. Black lines indicate σ values used in [PITH_FULL_IMAGE:figures/full_fig_p025_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: (a)-(c) Relative generalization gap (Mval−Mtrain)/Mtrain and (d)-(f) train￾ing and validation results Mtrain/val, with EDM2 on ImageNet-512 for various metrics and model sizes. ForrL2pix, σ is fixed at the peak of the generalization gap, see [PITH_FULL_IMAGE:figures/full_fig_p025_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Relative generalization gap (Mval − Mtrain)/Mtrain with EDM on CIFAR-10 for various metrics. Black lines indicate σ values used in [PITH_FULL_IMAGE:figures/full_fig_p026_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Relative generalization gap (Mval−Mtrain)/Mtrain with EDM on CIFAR-100 for various metrics. Black lines indicate σ values used in [PITH_FULL_IMAGE:figures/full_fig_p026_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Relative generalization gap (Mval − Mtrain)/Mtrain with EDM on CIFAR￾10/100 for various metrics and model sizes. For reconstruction-based metrics, σ is fixed at the peak of the generalization gap, see [PITH_FULL_IMAGE:figures/full_fig_p027_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Training and validation results Mtrain/val with EDM on CIFAR-10/100 for various metrics and model sizes. For reconstruction-based metrics, σ is fixed at the peak of the generalization gap, see [PITH_FULL_IMAGE:figures/full_fig_p028_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: Flow field lines and predictions of the optimal target predictor y ∗ (x, σ) at different noise levels σ = 28, 2.8, 0.63. Color indicates the magnitude of the prediction error y ∗ (x, σ)−x. Field geometry is (a) global, predictions tend towards the superposi￾tion of all training points. (b) moderately localized around training points, predictions approximate the data manifold (c) highly localized around ea… view at source ↗
Figure 25
Figure 25. Figure 25: Relative generalization gap (Eval − Etrain)/Etrain in our 2D toy model with random data and (a) different settings of the model error parameter δ, (b) different number of training points, and (c) different settings for the guidance weight w. w = 0 corresponds to no guidance. (a) δ = 2.0 (b) δ = 1.6 (c) δ = 1.2 [PITH_FULL_IMAGE:figures/full_fig_p030_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: Filled contours showing the relative generalization gap (E − Etrain)/Etrain at intermediate noise (σ ≈ 1.1) between the training points and each point in space, respectively. The black lines enclose the area where this gap is 0.5 or less. As the model error δ decreases, this region shrinks until it no longer contains the validation points, indicating that the generalization gap to the validation set is la… view at source ↗
Figure 27
Figure 27. Figure 27: (a)-(b) Relative generalization gap (Mval − Mtrain)/Mtrain for various met￾rics and (c)-(d) training and validation results Mtrain/val with FDD/FID, with EDM on CIFAR-10. We retrained for 100Mimg (half of the default) with different numbers of classes. Classes were computed algorithmically using k-means clustering in DINOv2 feature space. For reconstruction-based metrics, σ is fixed at the peak of the gen… view at source ↗
Figure 28
Figure 28. Figure 28: Relative generalization gaps (Mval −Mtrain)/Mtrain with Autoguidance with EDM2-S on ImageNet-64. Reconstruction-based metrics fix σ at the peak of the gen￾eralization gap (see [PITH_FULL_IMAGE:figures/full_fig_p032_28.png] view at source ↗
Figure 29
Figure 29. Figure 29: Relative generalization gaps (Mval−Mtrain)/Mtrain with Classifier-Free Guid￾ance with EDM2-S on ImageNet-64. Reconstruction-based metrics fix σ at the peak of the generalization gap (see [PITH_FULL_IMAGE:figures/full_fig_p032_29.png] view at source ↗
Figure 30
Figure 30. Figure 30: Relative generalization gaps (Mval −Mtrain)/Mtrain with Autoguidance with EDM2-S on ImageNet-64. Black lines indicate σ values used in [PITH_FULL_IMAGE:figures/full_fig_p033_30.png] view at source ↗
Figure 31
Figure 31. Figure 31: Relative generalization gaps (Mval−Mtrain)/Mtrain with Classifier-Free Guid￾ance with EDM2-S on ImageNet-64. Black lines indicate σ values used in [PITH_FULL_IMAGE:figures/full_fig_p033_31.png] view at source ↗
read the original abstract

Diffusion models generalize well in practice. However, an optimal diffusion model fully memorizes the training data and therefore fails to generalize, raising the question of what induces generalization in a real diffusion model. We show that, despite generalizing at the sample level, diffusion models progressively overfit the denoising training objective and thereby create a generalization gap between the performance on validation and training samples. This gap is most pronounced at intermediate noise levels. Using a fully analytic error-prone toy model, we trace the factors affecting the generalization gap. We find that the optimal denoising flow field localizes sharply around training points, but the model error suppresses the exact recall of training points, yielding a smooth, generalizing flow field. Finally, we find that the generalization gap observed in training does not translate to inference, which would result in a strong similarity between generated samples and training samples. This is because the intermediate states of sampling trajectories are sufficiently far from the distribution of noisy training samples the model is trained on. Together, these findings reveal a novel picture of how diffusion models generalize: the flow field generalizes through model error, which moves sampling trajectories outside the domain of noisy training samples and thereby naturally prevents overfitting.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims that diffusion models progressively overfit the denoising training objective despite generalizing at the sample level, creating a generalization gap most pronounced at intermediate noise levels. Using a fully analytic error-prone toy model, it traces how the optimal denoising flow field localizes sharply around training points but model error suppresses exact recall to yield a smooth generalizing flow field. The authors conclude that the training generalization gap does not translate to inference because intermediate states of sampling trajectories remain sufficiently far from the distribution of noisy training samples, revealing that the flow field generalizes through model error which naturally prevents overfitting.

Significance. If the result holds, the work offers a mechanistic account of generalization in diffusion models that credits model error for smoothing the flow field and separating inference trajectories from memorized noisy data. The fully analytic toy model is a clear strength, providing reproducible derivations and a falsifiable picture that could inform training practices and architectural choices in the field.

major comments (2)
  1. [Toy Model Analysis] The construction of the analytic error-prone toy model, its specific error model, and the mapping from toy error structure to high-capacity neural denoisers are not detailed sufficiently. This leaves derivation gaps in verifying the localization of the optimal flow field and the independent smoothing effect of error, which are load-bearing for the central claim that model error induces generalization.
  2. [Inference and Generalization] The inference claim that sampling trajectories remain outside the support of noisy training samples rests on an untested distance assumption extrapolated from the toy model. Without quantitative checks in neural-network regimes, this premise is insufficient to establish that the observed training/validation gap at intermediate noise levels does not produce overfitting at inference time.
minor comments (2)
  1. The abstract would benefit from one or two quantitative statements (e.g., measured gap sizes or distance statistics from the toy model) to make the claims more concrete.
  2. [Toy Model Analysis] Notation for the error term and flow-field localization in the toy-model derivations could be introduced more explicitly to aid readers.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for their positive assessment of the work's significance and for highlighting the value of the fully analytic toy model. We address each major comment below with point-by-point responses, indicating where revisions will be made to strengthen clarity and support for the claims.

read point-by-point responses

  1. Referee: [Toy Model Analysis] The construction of the analytic error-prone toy model, its specific error model, and the mapping from toy error structure to high-capacity neural denoisers are not detailed sufficiently. This leaves derivation gaps in verifying the localization of the optimal flow field and the independent smoothing effect of error, which are load-bearing for the central claim that model error induces generalization.

    Authors: We agree that the toy model section would benefit from greater explicitness to facilitate independent verification. In the revised manuscript we will expand the relevant section and add a dedicated appendix containing: (i) the complete step-by-step derivation of the optimal denoising flow field and its localization around training points, (ii) the precise mathematical definition of the error model (including how additive perturbations to the score are introduced and scaled with noise level), and (iii) an explicit discussion mapping the toy error structure onto high-capacity neural denoisers by linking finite optimization and capacity constraints to analogous smoothing behavior. These additions will close the noted derivation gaps while preserving the analytic character of the model. revision: yes

  2. Referee: [Inference and Generalization] The inference claim that sampling trajectories remain outside the support of noisy training samples rests on an untested distance assumption extrapolated from the toy model. Without quantitative checks in neural-network regimes, this premise is insufficient to establish that the observed training/validation gap at intermediate noise levels does not produce overfitting at inference time.

    Authors: The distance claim follows directly from the analytic smoothing of the flow field derived in the toy model: once model error is present, the resulting vector field steers trajectories away from the localized support of noisy training points at intermediate noise levels. We acknowledge that the current manuscript does not supply quantitative distance measurements or trajectory analyses performed with actual neural-network denoisers. In revision we will add a discussion paragraph outlining how the assumption could be tested empirically (e.g., via latent-space distance statistics or controlled low-dimensional network experiments) and will note this as an important direction for follow-up work. We maintain, however, that the toy-model derivation already supplies a mechanistic, falsifiable account consistent with the observed training/validation gap; the absence of large-scale numerical checks does not invalidate the analytic insight but does limit the strength of the extrapolation. revision: partial

standing simulated objections not resolved
  • Quantitative validation of the sampling-trajectory distance assumption in high-capacity neural-network regimes

Circularity Check

0 steps flagged

No significant circularity; toy-model derivations are independent of inference claim

full rationale

The paper's core chain relies on a fully analytic error-prone toy model to derive localization of the optimal denoising flow field around training points and the smoothing effect of model error, both obtained independently via explicit equations rather than by redefining inputs or fitting. The subsequent claim that inference trajectories remain outside the support of noisy training samples follows directly from this error-induced smoothing in the toy setting, without reducing to a self-citation, a fitted parameter renamed as prediction, or an ansatz smuggled through prior work. No load-bearing step equates to its own inputs by construction, and the analysis remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the representativeness of the analytic toy model for real diffusion networks and on the unverified premise that sampling trajectories remain distant from noisy training samples; no free parameters or invented entities are explicitly introduced in the abstract.

axioms (2)
  • domain assumption An analytic error-prone toy model captures the essential localization and smoothing behavior of the denoising flow field in real diffusion models.
    Invoked to trace factors affecting the generalization gap between training and validation samples.
  • domain assumption Intermediate states along sampling trajectories lie sufficiently far from the distribution of noisy training samples.
    Used to conclude that the training-time generalization gap does not produce overfitting at inference.

pith-pipeline@v0.9.0 · 5732 in / 1466 out tokens · 59094 ms · 2026-05-21T10:48:38.804789+00:00 · methodology

discussion (0)

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