arxiv: 2605.06367 · v1 · submitted 2026-05-07 · 📊 stat.ML · cond-mat.dis-nn· cs.LG

Recognition: unknown

The Interplay of Data Structure and Imbalance in the Learning Dynamics of Diffusion Models

Flavio Nicoletti , Chenxiao Ma , Enrico Ventura , Luca Saglietti , Stefano Sarao Mannelli

Authors on Pith no claims yet

Pith reviewed 2026-05-08 04:55 UTC · model grok-4.3

classification 📊 stat.ML cond-mat.dis-nncs.LG

keywords diffusion modelsclass imbalancelearning dynamicsGaussian mixturesscore-based modelsmemorizationgeneralizationdata heterogeneity

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The pith

Class variance sets the primary learning order in diffusion models, with higher-variance classes learned first.

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

The paper establishes that in score-based diffusion models trained on heterogeneous data, the variance of each class is the main factor deciding when that class is learned during training, with higher-variance classes consistently reaching generalization and memorization earlier. Centroid geometry exerts a weaker influence on this order. Sampling imbalance functions as a modulator that can invert the variance-driven sequence and, when severe, imposes separate later times at which minority classes acquire their distinct features in the reverse diffusion process. This matters because real datasets combine structural differences with unequal class sizes, raising the possibility that models fully memorize some classes while leaving others underlearned.

Core claim

Analyzing a random-features model trained on Gaussian mixtures, the authors derive the feature-covariance spectrum to characterize per-class generalization and memorization times. They show that class variance is the primary determinant of the learning hierarchy, consistently favoring higher-variance classes, while centroid geometry plays a secondary role. Sampling imbalance acts as a modulator that can reverse this ordering and, under strong imbalance, forces minority classes to acquire distinct, delayed speciation times during backward diffusion.

What carries the argument

The feature-covariance spectrum of the random-features model on Gaussian mixtures, which directly determines the per-class times for generalization and memorization.

If this is right

Higher-variance classes reach both generalization and memorization earlier than lower-variance classes.
Strong sampling imbalance can reverse the variance-based learning order.
Minority classes develop distinct, delayed speciation times in the backward diffusion process when imbalance is large.
Diffusion models can fully memorize some classes while others remain insufficiently learned.

Where Pith is reading between the lines

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

Training procedures could incorporate variance-aware sampling or augmentation to reduce disparities in when classes are learned.
The same variance-imbalance interaction may appear in other score-based or flow-matching generative models.
On highly imbalanced real-world data such as medical images, minority classes might require explicit regularization to avoid delayed or incomplete learning.

Load-bearing premise

The random-features model on Gaussian mixtures captures the essential per-class learning dynamics of full U-Net diffusion models trained on real heterogeneous image data.

What would settle it

Train a U-Net diffusion model on controlled Gaussian-mixture data with independently varied class variances and sampling rates, then measure whether the observed per-class learning curves follow the exact hierarchy and speciation times predicted by the feature-covariance spectrum.

Figures

Figures reproduced from arXiv: 2605.06367 by Chenxiao Ma, Enrico Ventura, Flavio Nicoletti, Luca Saglietti, Stefano Sarao Mannelli.

**Figure 1.** Figure 1: Strong imbalance induces separation in speciation times. Root mean square (RMS) overlaps between the projected centroids µc (Eq.(10)) and the top principal components (ψ1, ψ2) of Ugep, defined as RMSc ≡ Mean(P2 i=1(µc · ψi) 2/∥µc∥ 2 ) 1/2 , as a function of rescaled diffusion time et ≡ t/ log N. Left panel correspond to weak imbalance (b1 = 0.7, b2 = 0.3, Eq.(12)), while the remaining panels illustrate ins… view at source ↗

**Figure 2.** Figure 2: Strong class diversity can reduce the generalization window. Parameters for theory: χp = 60.0, χm = 30.0, b1 = b2 = 0.5, σ 2 1 = 0.5, σ2 2 = 0.25, ∥m1∥= ∥m2∥= 0, t = 0.001. Parameters for finite-size simulations: N = 100, P = 6000, M = 3000, Nepoch = 2 × 106 , learning rate η = 5 × 10−5N/∆t, Nspectrum run = 10, Ntrain run = 20. Training time in (a) is rescaled as τ → τ /ηe, where ηe = η∆t/N. 3.2 Timescale … view at source ↗

**Figure 3.** Figure 3: Class heterogeneity induces class-specific timescales. Class-conditional test errors (Eq. 14) evaluated over training time across three structural ablations relative to a perfectly symmetric baseline (σ 2 1 = σ 2 2 = 0.5, ∥m1∥= ∥m2∥= √ N, b1 = b2 = 0.5, m1 ·m2 = 0). The perturbations are: variance disparity (σ 2 1 = 0.5, σ2 2 = 0.25; left), centroid norm disparity (∥m1∥= √ N, ∥m2∥= 3 2 √ N; center), and sa… view at source ↗

**Figure 4.** Figure 4: Imbalance can reverse the order of learning for classes. In both subfigures, solid lines represent theoretical predictions derived in Appendix G, while crosses represent empirical average gaps from gradient descent simulations. Parameters: χp = 60.0, χm = 30.0, t = 0.01, N = 100, Nepoch = 2 × 106 , learning rate η = 5 × 10−5 , Ntrain run = 50. Result 3 (The Generalization-Memorization Trade-off). Sample i… view at source ↗

**Figure 5.** Figure 5: DDPM memorization gaps on Fashion MNIST. (a) Per-class memorization fractions for Sneaker-Coat and Sneaker-Bag DDPMs. The horizontal dashed line marks the threshold f c mem(t) = 1/3 used to define t c mem; vertical shaded regions show the resulting Sneaker-partner time gap ∆tmem(cp). Within each pair, the higher-variance partner class is memorized before Sneaker. Across pairs, the Bag-Sneaker DDPM exhibits… view at source ↗

**Figure 6.** Figure 6: DDPM–RF comparison across three descriptors. For each Sneaker-partner pair (csnk, cp), we compare the DDPM memorization gap on Fashion MNIST with the corresponding RF memorization gap on Gaussian synthetic data. The RF model uses pair descriptors extracted from the data: partner variance, normalized partner centroid norm, and centroid cosine similarity to Sneaker. The panels plot both gaps as functions of … view at source ↗

**Figure 7.** Figure 7: Comparison of the GEP derived in eq. (50) with the spectrum of view at source ↗

**Figure 8.** Figure 8: Test of eqs. (114) for t = 0.01. γc1P (note that, for an odd activation, the coefficient αc vanishes in this limit). Second, note that g (Ψ,h) c → h 2 c gΨ, g (Ψ,ς) c → ςc gΨ, g (Ψ,γ) cc′ → γcγc ′ gΨ, g (Ω) cc′ → γcγc ′gΩ; to handle singular terms such as determinants and inverses of these last two resolvents appearing in (90), set g (Ψ,γ) cc = γ 2 c gΨ + ϵ, g (Ω) cc = γ 2 c gΩ + ϵ, expand and simplify. Th… view at source ↗

**Figure 9.** Figure 9: Check of the scaling of λ (2) mem. We show that the eigenvalue λ (2) mem scales linearly with t and converges to a constant value for large number of hidden neurons and number of samples. We also show the width ∆λ of the leftmost bulk in fig. 2a as 36 view at source ↗

**Figure 10.** Figure 10: Comparison between the test losses and the score MSEs. Parameters as in figure 3. view at source ↗

**Figure 11.** Figure 11: Overview example for the Bag class. 44 view at source ↗

**Figure 12.** Figure 12: Overview examples for Fashion-MNIST classes (Part 1). view at source ↗

**Figure 12.** Figure 12: Overview examples for Fashion-MNIST classes (Part 2). view at source ↗

**Figure 12.** Figure 12: Overview examples for Fashion-MNIST classes (Part 3). view at source ↗

**Figure 12.** Figure 12: Overview examples for Fashion-MNIST classes (Part 4). view at source ↗

**Figure 12.** Figure 12: Overview examples for Fashion-MNIST classes (Part 5). view at source ↗

**Figure 12.** Figure 12: Overview examples for Fashion-MNIST classes (Part 6). view at source ↗

**Figure 12.** Figure 12: Overview examples for Fashion-MNIST classes (Part 7). view at source ↗

**Figure 12.** Figure 12: Overview examples for Fashion-MNIST classes (Part 8). view at source ↗

**Figure 13.** Figure 13: Memorization fraction curves for all Sneaker-paired Fashion MNIST classes. view at source ↗

**Figure 14.** Figure 14: Left: normalized variance for each class on two datasets; the annotated numbers report the total class view at source ↗

**Figure 16.** Figure 16: Same plot as Figure 6, but using alternative thresholds to compute the DDPM memorization gap view at source ↗

**Figure 17.** Figure 17: Same plot as Figure 6, but using alternative RF memorization thresholds. The RF threshold is defined by view at source ↗

read the original abstract

Real-world datasets are inherently heterogeneous, yet how per-class structural differences and sampling imbalance shape the training dynamics of diffusion models-and potentially exacerbate disparities-remains poorly understood. While models typically transition from an initial phase of generalization to memorizing the training set, existing theory assumes homogeneous data, leaving open how class imbalance and heterogeneity reshape these dynamics. In this work, we develop a high-dimensional analytical framework to study class-dependent learning in score-based diffusion models. Analyzing a random-features model trained on Gaussian mixtures, we derive the feature-covariance spectrum to characterize per-class generalization and memorization times. We reveal the explicit hierarchy governing these dynamics: class variance is the primary determinant of learning order-consistently favoring higher-variance classes-while centroid geometry plays a secondary role. Sampling imbalance acts as a modulator that can reverse this ordering and, under strong imbalance, forces minority classes to acquire distinct, delayed speciation times during backward diffusion. Together, these results suggest that diffusion models can memorize some classes while others remain insufficiently learned. We validate our theoretical predictions empirically using U-Net models trained on Fashion MNIST.

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.

Desk Editor's Note private letter to a colleague

The paper derives a variance-primary hierarchy for per-class learning and memorization times in a random-features diffusion model on Gaussian mixtures, with imbalance as a modulator, and checks it on Fashion-MNIST U-Nets.

read the letter

The main thing here is a clean analytical ordering rule for diffusion training dynamics: class variance sets the primary sequence for when classes generalize then memorize, centroid geometry is secondary, and strong sampling imbalance can flip the order or force delayed speciation times for minority classes in the backward process. They get this by deriving the feature-covariance spectrum in the random-features Gaussian mixture setup and extracting the relevant times directly from it, then show the predicted ordering appears in U-Net runs on Fashion-MNIST. That explicit hierarchy tied to the score-based dynamics is new relative to prior homogeneous or non-analytical work on imbalance in generative models. The derivation stays grounded in the model definition rather than post-hoc fitting, which is a plus. The empirical check is straightforward and at least consistent with the toy predictions. The obvious limitation is the proxy itself. Random features on Gaussians capture linear per-class statistics but leave out the nonlinear hierarchical feature extraction and spatial correlations that real U-Nets use on images. Class differences in heterogeneous data often live in textures and semantics beyond scalar variance and means, so the stated ordering could shift or disappear once those mechanisms dominate. Fashion-MNIST is too simple and low-variance to stress-test that gap. The analysis looks internally consistent on its own terms, with no load-bearing circularity I can spot from the description. This is for people who build or analyze diffusion models and care about class-dependent behavior or fairness on imbalanced data. A reader wanting transferable predictions for complex real-world datasets will find the scope narrow, but the mechanism is worth thinking through. Send it for peer review; the analytical framework is grounded enough to merit referee time even if the generality claims need tightening.

Referee Report

3 major / 2 minor

Summary. The paper develops a high-dimensional analytical framework for class-dependent learning in score-based diffusion models by analyzing a random-features model trained on Gaussian mixtures. It derives the feature-covariance spectrum to characterize per-class generalization and memorization times, revealing an explicit hierarchy: class variance is the primary determinant of learning order (favoring higher-variance classes), centroid geometry plays a secondary role, and sampling imbalance acts as a modulator that can reverse this ordering and, under strong imbalance, induce distinct delayed speciation times for minority classes during backward diffusion. These theoretical predictions are checked empirically using U-Net diffusion models trained on Fashion-MNIST.

Significance. If the central hierarchy holds, the work offers a valuable analytical tool for understanding how data structure and imbalance shape generalization-memorization transitions in diffusion models, moving beyond homogeneous-data assumptions in prior theory. The explicit derivation from the feature-covariance spectrum and the use of an analytically tractable proxy model are strengths that enable precise predictions; the Fashion-MNIST validation provides initial empirical grounding. This could inform mitigation of class disparities in trained models.

major comments (3)

[§4.2] §4.2, the derivation of the feature-covariance spectrum: the hierarchy (variance primary over centroid geometry) follows directly from the spectrum eigenvalues, but the paper provides no explicit high-dimensional asymptotic comparison or dominance proof showing variance terms overwhelm centroid contributions for all parameter regimes; this is load-bearing for the primary-determinant claim.
[§5.1] §5.1, definition of speciation times: these times are extracted from the same per-class covariance spectrum used to order generalization/memorization, creating a risk that the 'delayed speciation' result under imbalance is partly definitional rather than independently predictive; a separate operational definition or cross-check against the true score-matching loss would strengthen the claim.
[§6] §6, empirical validation: while qualitative agreement with the predicted ordering is shown on Fashion-MNIST, the experiments do not report quantitative alignment (e.g., correlation between predicted and observed per-class learning times) or error bars on the U-Net runs, leaving the support for extending the random-features hierarchy to nonlinear U-Nets moderate.

minor comments (2)

[§2] Notation for the backward diffusion process and the precise definition of 'speciation' could be consolidated in one place (currently split across §2 and §5) to improve readability.
The abstract states the hierarchy 'consistently favoring higher-variance classes' but the main text should add a short caveat on the regime where this holds (e.g., when imbalance is not extreme).

Simulated Author's Rebuttal

3 responses · 0 unresolved

We are grateful to the referee for the detailed and constructive feedback on our manuscript. We have carefully considered each major comment and outline our responses below, along with the revisions we plan to implement.

read point-by-point responses

Referee: [§4.2] §4.2, the derivation of the feature-covariance spectrum: the hierarchy (variance primary over centroid geometry) follows directly from the spectrum eigenvalues, but the paper provides no explicit high-dimensional asymptotic comparison or dominance proof showing variance terms overwhelm centroid contributions for all parameter regimes; this is load-bearing for the primary-determinant claim.

Authors: We appreciate the referee highlighting this point. We acknowledge that an explicit high-dimensional asymptotic comparison or dominance proof was not included in the original manuscript. In the revised version, we will add such an analysis, deriving the asymptotic behavior of the spectrum eigenvalues to show that variance terms dominate the centroid contributions in the high-dimensional regime. revision: yes
Referee: [§5.1] §5.1, definition of speciation times: these times are extracted from the same per-class covariance spectrum used to order generalization/memorization, creating a risk that the 'delayed speciation' result under imbalance is partly definitional rather than independently predictive; a separate operational definition or cross-check against the true score-matching loss would strengthen the claim.

Authors: We thank the referee for this observation. The speciation times are indeed defined via the per-class feature-covariance spectrum to provide an analytical characterization. To address the concern of circularity, we will include in the revision an additional cross-validation: we compute the per-class score-matching loss on held-out data during training and demonstrate that the predicted delayed speciation times align with the empirical loss curves for minority classes under strong imbalance. This provides an independent operational check. revision: yes
Referee: [§6] §6, empirical validation: while qualitative agreement with the predicted ordering is shown on Fashion-MNIST, the experiments do not report quantitative alignment (e.g., correlation between predicted and observed per-class learning times) or error bars on the U-Net runs, leaving the support for extending the random-features hierarchy to nonlinear U-Nets moderate.

Authors: We agree that quantitative metrics would provide stronger support. In the revised version, we will report the correlation coefficients between the theoretically predicted learning times (from the random-features model) and the observed per-class generalization/memorization times in the U-Net experiments. Additionally, we will include error bars from multiple independent runs of the U-Net training to quantify variability. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained model analysis

full rationale

The paper defines a random-features model on Gaussian mixtures, analytically derives the feature-covariance spectrum from that model's covariance structure, and uses the resulting spectrum to characterize per-class generalization and memorization times. This is a direct mathematical consequence of the model definition rather than a post-hoc fit renamed as prediction or a self-referential loop. The claimed hierarchy (variance primary, centroid secondary, imbalance as modulator with delayed minority speciation) follows from the spectrum equations without reducing to the inputs by construction. Empirical validation on U-Net models trained on Fashion MNIST is presented as separate confirmation, not part of the derivation. No load-bearing self-citations, uniqueness theorems imported from prior author work, or ansatzes smuggled via citation are present in the abstract or described chain. The speciation times are defined with respect to the backward diffusion process in the same model but do not create a tautological equivalence to the input assumptions.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The analysis rests on the standard high-dimensional random-features approximation and Gaussian-mixture data model; no new entities are postulated and the only free parameters are the class variances and sampling rates supplied by the data-generating process.

free parameters (2)

class variances
Relative variances of the Gaussian components are inputs that directly set the primary learning order; they are not fitted to the target result.
sampling rates per class
Imbalance ratios are chosen as part of the experimental design and modulate the ordering.

axioms (1)

domain assumption High-dimensional limit in which the random-features model yields an exact feature-covariance spectrum
Invoked to obtain closed-form expressions for per-class generalization and memorization times.

pith-pipeline@v0.9.0 · 5508 in / 1349 out tokens · 52711 ms · 2026-05-08T04:55:39.631806+00:00 · methodology

discussion (0)

Reference graph

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