Generalization of Diffusion Models Arises with a Balanced Representation Space
Authors: Zekai Zhang
Xiao Li
Xiang Li
Lianghe Shi
Meng Wu
Molei Tao
Qing Qu
Venue: ICLR 2026 Poster (2026)
Keywords: diffusion models
representation learning
generalization
memorization
denoising autoencoders
Primary area: generative models
TL;DR: Learning good representations is central to novel and meaningful generation.

Abstract: Diffusion models excel at generating high-quality, diverse samples, yet they risk memorizing training data when overfit to the training objective. We analyze the distinctions between memorization and generalization in diffusion models through the lens of representation learning. By investigating a two-layer ReLU denoising autoencoder (DAE), we prove that: *(i)* memorization corresponds to the model storing raw training dataset in the learned weights for encoding and decoding, yielding localized, spiky representations; whereas *(ii)* generalization arises when the model captures local data statistics, producing balanced representations. Furthermore, we validate our theoretical findings on real-world unconditional and text-to-image diffusion models, demonstrating that the same representation structures emerge in deep generative models with significant practical implications. Building on these insights, we propose a representation-based method for detecting memorization and a training-free editing technique that allows precise control via representation steering. Together, our results highlight that *learning good representations is central to novel and meaningful generative modelling*. Code is available at https://github.com/la0ka1/diffusion-gen-from-rep.

Published as a conference paper at ICLR 2026
GENERALIZATION OF DIFFUSION MODELS ARISES WITH A BALANCED REPRESENTATION SPACE
Zekai Zhang* Xiao Li" University of Michigan
Xiang Li
Lianghe Shi
Meng Wu
Molei Tao
Qing Qu
Georgia Institute of Technology
University of Michigan
ABSTRACT
Diffusion models excel at generating high-quality, diverse samples, yet they risk memorizing training data when overfit the training objective. We analyze the distinctions between memorization and generalization in diffusion models through the lens of representation learning. By investigating two-layer ReLU denoising autoencoder (DAE), we prove that: (i) memorization corresponds to the model storing raw training dataset in the learned weights for encoding and decoding, yielding localized, spiky representations; whereas (ii) generalization arises when the model captures local data statisties. producing balanced repre- sentations. Furthermore, we validate our theoretical findings on real-world un- conditional and text-to-image diffusion models, demonstrating that the same rep- resentation structures emerge in deep generative models with significant practi- cal implications. Building on these insights, we propose representation-bun based method for detecting memorization and training-free editing technique that allows precise control via representation steering. Together, our results high- light that learning good representations is central to novel and meaningful gen- erative modelling. Code is available at httpp:: / Sigithh.b... com/ 1a0ka1/
diffus EFFFussio-aaa gen- rep.
INTRODUCTION
Diffusion models ct 2020: Lou all 2024) have rapidly emerged as the dominant class of generative models, powering state-of-the-art systems such Stable Diffusion (Rombach all 2022), Flux (Labs et all R(25), and Veo (Google, 2025).. By iteratively denoising random noise, they achieve unprecedented scalability, controllability, and fidelity. However, their empirical suc- cess raises a fundamental question: in principle, the standard training objective (e.g., denoising score matching) admits closed-fome solution that merely memorizes training examples (Yierall 2023): in practice, however, woaldworld models consistently produce novel and diverse outputs (Zhang et all 2024: Kadkhodane et al 20222)). This distinct mismatch between thcoretical expectation and ob- served behavior poses eritical gap in understanding of diffusion model generalization, with di- rect implications for privacy, interpretability, and trustworthy deployment (Somepalli et all ((23))).
Addressing this question has drawn increasing attention in the machine learning community (Zang atall RIZE: Lretal 4024N: Wang all RIRES Kadkhodaie at 202E: Guret 4025 Bonnaure
ctall 20255 Zhang et all, 20255; Bertrand el all 0025% Zhang et al, 21267)) yet existing explanations
remain far from satisfactory. Early works based on random feature models etll 2023: George
atal (025) provide useful insights but necessarily oversimplify model architectures. Analyses of
linear models on Gaussian mixtures (Li 20246 Waan et all, 20244, Wang (0255) shed light
on generalization but cannot capture memorization. Another line of research explores inductive biases by constructing handcrafted closed-form solutions from empirical data approximate U-Net performance (Kamb and Ganguli, 2025; Niedoba et al, 2125: Lukoranoy ef all MIZA: Eloros et all 2025), attributing success to principles such as locality and equivariance. While these advances are valuable, the findings remain fragmented and phenomenological, and more unified account of how diffusion models both memorize and generalize still lacking (see Appendix A for a more detailed discussion of related work).
"Equally contributing, Email: zzekai teekai@umicie edu edu 1 "Correspondig author. Email: qi ngqu@um edu

---

Published as a conference paper at ICLR 2026
Memorized Sample Delection
Representation Steering
Image Editing
Noruocled Geucration Splky Representations
Representation
Spile Reperacatation ia Not Stocrakle
3.3
you Partan
"eatureexttaattion peolf
Lenel Srerco
Generalized Generation Bahanced Repecantaaions
Movy
Balanced Reprodentations to Steemble
cok Partiig Sigh
Generation
Figure 1: Diffusion models generalize while learning benign internal representations. Activa- tions from intermediate network layers form a representation space, within which distinct patterns emerge: memorized samples produce spiky representations that make them detectable, whereas novel generations yield balanced, information-ric representations that support controllable genera- tion via representation steering.
To address these challenges, we develop a unified mathematical framework based on a theoretical analysis of a nonlinear two-layer ReLU denoising autoencoder (DAE) (Vincent. 20T This frame- work not only unifies the characterization of memorization and generalization, but also bridges distri- bution learning with representation learning, offering profound practical implications. Specifically: (i) Memorization. We prove that when empirical samples locally sparse, the network weights memorize and store individual training examples, leading to overfitting and hence memorization. (ti) Generalization. Conversely, when the empirical data are locally abundant, the weights effectively capture local data statistics, enabling the model to generate novel in-distribution samples.
Crucially, our work provides a unique eepresentation-centie perspective on generalization (Tian, (025), highlighting the pivotal role of bottleneck activations in DAE networks. This view is moti- vated by recent empirical evidence on the duality between distribution learning and representation learning in diffusion models (Li 2025E; Xiang et all, 2025% Tinaz all, 2025);: they inher- ently learn informative features for downstream tasks (Kwon et all 2023: Chen et all 2(225)). and representation alignment regularization has been shown to accelerate training (Yu all 20Z5). Our theory makes this connection explicit: memorized samples are encoded as spiky activations concen- trated on few neurons, whereas generalized samples yield balanced representations that reflect the underlying distribution. These contrasting modes of representation learning manifest in distinct gen- cration behaviors in terms of memorization or generalization, which we comprehensively validated across range of models, including EDM (Karras et all (0222). Diffusion Transformers (DiT) (Pee bles and Xic, 2023), and Stable Diffusion v1.4 (Rombach et all 2022) (SD1.4).
Moreover, our findings show that the representation space is not byproduct but a crucial and control- lable factor for generation. Specifically, we demonstrate two practical implications: (i) Memoriza- tion detection. Leveraging the spikiness of representations identified by our theory as a signature of memorization, we develop theory-driven detector that achieves highly accurate and efficient per- formance in a prompt-free manner. (ii) Model steering. We propose an effective steering method based on additions in the representation space and reveal distinct behaviors between memoriza- tion and generalization: memorized samples are difficult to steer, whereas generalized samples are highly steerable owing their balanced, semantically rich representations. Together, these appli- cations illustrate the far-reaching implications of our representation-ccnttie analysis for the privacy, interpretability, and controllability of diffusion models.
Summary of contributions. Our main contributions are as follows:
Unified framework in a nonlinear ReLU setting. We analyze the optimal solutions of a two- layer nonlinear ReLU DAE under different empirical data sizes, providing a unified characteriza- tion of memorization and generalization that goes beyond prior random-featur or linear model analyses.
A representaatoon-cnntii understanding of generalization. We establish a rigorous connection between representation structures and generalization, identifying distinct patterns that separate memorization from generalization and validating these insights across diverse model settings.
-heory-inspired tools for memorization detection and model steering, Building sis, we propose simple yet effective methods for memorization detection and representation steering, revealing distinet behaviors of generalized versus memorized samples.
analy-
2

---

Published as a conference paper at ICLR 2026
PROBLEM SETUP
In this section, first introduce the basics of diffusion models, and then describe problem setup for theoretical studies in Section E1
2.1 DENOISING PERSPECTIVE OF DIFFUSION MODELS
Basics of diffusion models. Diffusion models comprise two processes: (i) forward noising process and (ii) a reverse denoising/sampling process. The forward process progressively corrupts a clean sample 20 via 22 + a&E with (0, T), while the reverse process (e.g., DDIM (Song et all
(04E) removes noise to generate data:
-t-1
(or 01-1) or Po
(1)
where V log (x))) is the score function of the marginal distribution of the noisy sample 24 at time To estimate (ete); we use a denoising autoencoder (DAE) fo(a;) (Karras et all, 2022: Li and He. 2(255) that predicts Zi from Lt. so that
V logg: = (x: fa((z)))/o? = (x, fo(x,))/of.
where (y) == E| oiE = y: x Pat the ground-trute -tuth denoiser via Tweedie's formula (Etron I0. Thus, the ideal (population) objective to learn the DAE is
ew.(O.) [| [e + m26,)) - a=l?]
Generalization of diffusion models. In practice, we only have finitely many empirical samples X [a: with ~ Pęt- Accordingly, we work with the empirical distribution Pemp - 8(æ: az)), and Equation (2) reduces to its empirical counterpart. Minimizing this empirical
loss leads to the nonparametric empirical denwiser femp ell 2025), which maps noisy input towards the nearest training samples:
femp (y) = B æ x
V Pampl
(3)
E Ny;
Sampling with Semp can provably reproduce training samples (Zhang el all 2024; Baptista et all 2025). In practice, however, this empirical loss is minimized by taking the gradient descent over parameterized neural network, which does not always overfit; instead, it can approximate the popu- lation denoiser fgt (Niedoba et all 2025). In this paper, we aim understand when a parameterized network overfits (learns femp) versus generalizes (learns fgt).
2.2 OUR THEORETICAL FRAMEWORK
Data assumptions. We assume K-component mixture of Gaussians (MoG) for the data distribu- tion:
K
K
Pgt Σ
ph N Es),
Σ
Pk
1,
which is standard approximation to data manifolds used in recent theoretical studies (Wang et all 2024 Zhang et all, 2024 Li cl all 20256: Cu and Zdcborova, 2023: Gatmiry et all 2028 Birof
atall 2024 Kamkari et all 2024; Buchanan all 2025; Li al (025c.).
Model parameterization and training loss. Following Vincent (201)); Chen all (2023) Zend
a (2033); Cui et all (2025), we parameterize the DAE by two-layer ReLU network:
fw. w, (a)) Wzh((w) W₂ [WTT]::.
with Wi, W2 E Rdxy, []- denoting ReLU and h(()) stands for the representation. Training and
sampling can be viewed as operating with collection of DAEs across multiple noise levels. Fol-
lowing prior work (Lietal all 00248 Zeno all 2025: Zhang and Pilanci, 2024; Han et 2025).
we begin with a fixed noise level a. The fz-regularized training objective is
Lx (W2, Wi) W.
min
E2- [| (z, + +Ee) xz mm:12
1 + E=
(6)
3

---

Published as a conference paper at ICLR 2026
Decreasing Noise Level
Nearest
Decreasing Noise Level
Nearest
Traning imaga
traising Image
Sampling with Mem/Gen. ReLU DAES. Left: sampling with a set memorized ReLU DAE produces duplications of training images. Right: sampling with generalized DAEs produces novel images. Details for training and sampling are provided Appendix [a] and single- step denoising results are shown in Appendix B2.2
Figure 2:
Figure illustrates training and sampling across multiple noise levels under this setting; we revisit the effect of different noise levels after Corollary K
We adopt (5) as a minimal, tractable model to analyze memorization and generalization. Recent works (Lukoianov al, 2025: Lretall 2024B; Wang and Vastol, 2022)) imply that real diffusion models exhibit approximate piecewise linearity; our ReLU model shares this structure and can be viewed local approximation of such networks. We verify this connection via SVD analysis of denoiser Jacobians Kadkhodate et all, 2024:; Achilli et all 2024) for EDM, SD1.4, 1,,, and ReLU DAE in Appendix B.3 around generalized samples, the Jacobian reflects local data statistics in Cor. BK. whereas around memorized samples it becomes noticeably low-rank and is dominated by the corresponding data vector, consistent with Cor. 3.2
MAIN THEOREMS
Building the setup in Section D this section presents our main theoretical results for two- layer nonlinear DAE with the ReLU activation, complemented by experiments on state-of-the-an diffusion models. By characterizing the optimal solutions of the training loss, establish:
Three Learning Regimes Training Diffusion Models
Memorization Regime (Section + l)
In over-parameterized models trained on locally
sparse data, memorization arises when network weights store individual training samples, leading to overfitting and producing distinctively spiky representations.
Generalization Regime (Section (: In contrast, when the model is under- parameterized and the data are locally abundant, the weights capture underlying data statistics, enabling novel sample generation and yielding balanced, semantically rich rep- resentations.
Hybrid Regime (Section 3.3) Imbalanced real-world data leads to hybrid regime where models generalize on abundant clusters while memorizing scarce ones. Consequently, the representations can help identify input's 's region and detect its memorized samples.
To substantiate the above results, we first establish general theorem characterizing the local mini- mizers of the training loss (E) for the DAE networks. This theorem then specializes to individually address the memorization and generalization regimes. To simplify the nonlinear DAE problem and obtain more interpretable characterization, we adopt the following separability notion. It is de- signed to match bias-free linear layers (as our ReLU DAE), where cluster structure is naturally captured by within-cluste concentration and angular separation of cluster means; the definition can be extended to standard hyperplane separability by allowing affine (biased) layers.
Definition 3.1 ((a, 3) Separabiliyy of Training Data). Suppose the training dataset X can be par- titioned into M clusters X [XI,..., X,), where X [ER,1..... Lk.ny C RRd has mean
vi 11p 1==1 Tk.j. We say the dataset is (a, --sepaaable if
for all k,j [ 2xl12
[kg Tt)
S a,
for all + t:
$ B.
F 112
The parameters a and 3 are not required to universal constants. Intuitively, tight within-cluster concentration together with well-separated means yields inter-cluster margin 7 that quantifies
4

---

Published as a conference paper at ICLR 2026
negative alignment between samples from different clusters; y depends only a, 3, and the norms of the training data (the explicit expression is given in Appendix Dz Under this separability condition, we show that local minimizers of the DAE admit block-wiss structure.
Theorem 3.1 (Block-wise Structure of Local Minimizers in the DAE Loss). Suppose the training data X i....... X is (a, 5) --ssparrbble according to Definition оا with 3 < Consider minimizing the training loss (G) for DAE trained with fixed noise level a 2 0 and weight decay 2 Then there exists a local minimizer with a block-wise structure, where the weights satisfy
W = Wi where
= (Wx. Wx.
Wxs) + R(o,7))
Here, Wx, Rdxxpe EV
ER=1PE p denotes the block decoppoccoooiiiioon of W. R(o, 7) small residual term whose Frobenius norm is bounded by || R(o, 7) 1/3 S (6 for universal constants C, e > O and margin 7 O determined by (), Each block Wx, (1 s k M) is
constructed from the Gram matrix X U of the k-th data cluster follows:
Wx. = =((
(I+ (A
I I (
2
O
(8)
where (i) U RdxPx is the submatrix of Us containing its top pke eigenvectors, (ii) A (pn)
IP xPh contains the corresponding Ps eigenvalues, and (iii) O₁ E 1 Xph is an orthogonal matrix accounting for rotational symmetry. This holds under the condition nd < ming Amin (A (Pk) which ensures that the matrix square roots in (8) are well-defined.
k
Remarks. The proof is deferred to Appendix D. The local minimizer ( consists of a block-wise main term plus a residual Ro, 7), which vanishes as a becomes small relative to the separation mar- gin y. This is consistent with the low-noise regimes that crucial for diffusion-modee model sampling and representation learning (Niedoba this block-wise structure even for relatively large (Figure B) The (a,, seeasarbryyyy assumption serves mainly to simplify the proof; similar conclusions hold more generally (see Appendix B l). Fi- nally, the optimal solution is not tied a specific block order, since fwa,W, Wi is invariant arbitrary column permutations of the weight matrices (W1, W2).
2025: Pavlova and Wei, 2025). Empirically, we observe
For the remainder of this section, we specialize the result to the memorization (Section and generalization (Section (22) regimes by varying the training-set size. For clarity, we omit the residual term R(a, 7) and focus on the block-wise leading component of the optimal solution.
3.1
CASE 1: MEMORIZATION WITH OVERPARAMETERRIZATII
First, we consider the overparameterized setting where the model parameters are larger than the number of training samples 2 n. In this "sample sparse" regime, each training sample can be treated individual cluster that is sufficiently separated from each other, where a1 = 0 and $1 can be set to max;, (z,, xz). Based on this setup, Theorem 3. can be reduced to the following.
Corollary 3.2 (Memorization in Overparameterized DAES). Under the problem setup of Theo- rem F. consider training data X = [#1...., xx,] C TRd that is (0, (..31--spprabbee (with 31 < 0). Furthermore, let the two-layer nonlinear DAE Ww (x) be overparameterized with p 2 n hidden units. If we further assume the weight decay in (l) satisfies nλ min;e(n) || then there exists a local minimizer of the DAE loss
(6) with the following memorizing block-wise structure:
12
n2
W = W = (F131
W moms
2
Moreover, when X + 0, this solution attains an empirical loss that is independent of the ambient dimension d:
n
Cx Wi)
o2
n
Remarks. The proof is deferred to Corollary D and our result implies the following:
5

---

Published as a conference paper at ICLR 2026
p=100
p=100
d-1024
EF1024
DO
Canier022
Aue 1:
Neoe LCE
Figure 3: Verification of Corollary B and Corollary E.L We visualize the learned encoder matrix Wi of a ReLU DAE trained with noise level 0.2. When trained on CelebA face images, the model stores training samples in its columns, matching Corollary BW2 When trained on 10,000 images, the model generalizes and captures data statistics. consistent with Corollary B. Empirically, the same behavior holds for larger noise, up to d = 5; additional results are in Appendix B.]
Learning the optimal solution with sparse columns. The structured solution with (p n2) trail- ing zero columns in (9) is one among many local minimizers, as dense alternatives can arise by splitting sparse columns. However, empirical evidence and theory (Xie eteeetta all 2025) suggest that standard optimizers such as Adam (Kingma and Ba, 2015) bias training toward -soooth solu- tions of the DAE loss (ef. Corollary D As a result, the solutions observed in practice often align with the sparse structure we construct (Figure B))
Sampling reproduce training samples (memorization). this regime, the learned DAE closely approximates the empirical denoiser femp in Eq. (E), achieving low empirical loss and conse- quently reproducing the training samples under sampling (as shown in Figure 2) This occurs be- cause the DAE's projection and reconstruction over the sparse columns of the weights during the reverse sampling effectively act as power method, recovering memorized training data (Weitzner atall 20ZE).. Quantitatively, by plugging Corollary 3.2 into the overall denoising score matching loss, we find that the KL divergence between the sampled and empirical distributions is bounded by z maxisisn |a confirming strong memorization.
signature of memorization. As consequence of Corollary BM2 for any training sample æ;, its learned representation within the DAE exhibits a distinctive sparse form:
Spiky representations
Amem (2;; + oe) = [ mom (z, + + R (O...., r; (æ;; + oε), 0, 0,...
This sparsity arises because at; is negatively correlated with other samples stored in the learned weight matrix W Wmem- yielding a nearly one-hot feature vector within the representation space (Figure M) Such spikiness could serve as a robust signature of memorization (Hakem cLal 2025: Gan ct all 2025), which we empirically demonstrate on both synthetic (Figure HD and real-world (Figure 5) settings. Building on this insight, we introduce a simple yet effective memorization detection method that achieves strong results, as detailed later in Section I.I. Additionally, analo- gous correlations between sharp, localized activations and the recall of concrete stored knowledge have been empirically observed in Large Language Models (LLMs) (Sun et 2024)), suggesting our findings could also offer a potential explanation for these phenomena in LLMs.
3.2
CASE 2: GENERALIZATION WITH UNDEEPARAMETEERIZZATON
On the other hand, suppose we have sufficiently many i..dd. samples {œg, )e from each Gaussian mode E [K] of the MoG distribution (.)) Then the empirical mean and Gram matrix of each cluster k concentrate around their expectations:
1
1 X St :
TKe
Pk:
Tili + Σt.
(10)
T1k
T1k
If the component means incoherent (i.e., (Ma, / p4x ||x < B2 for k * E) and the within-mode variance is small (i.e., | 1/2 l/ < a2). then with high probability the clus- ters [X satisfy the separability conditions in Definition B.ll with (a, 3) = (a2, 82). In this scenario, we demonstrate below, the optimal weights of the DAE network will learn the local

---

Published as a conference paper at ICLR 2026
Distribution in Represeniallors Side (CelebA)
Distriutionn Representation Sids (MoG)
Deneay
Demelly
Mem.
Gen
Mem.
0
0.8
0.7
0.3
10
1.1
1.3
1.4
18
Mem, Std-O 9576
Acsveliin Value
10
Mem, 3id00
Actuaton
4
40 60
105
10
25
30
50
Latent Dimension (p 100)
Latent Dimension (0= 50)
Figure 4: Mem./Gen. representations in ReLU DAES. Top: Memorized generalized samples can be separated by the standard deviation (Std) of their representations: memorized models produce spiky. high-Std features, whereas generalized models do not. Bottom: Representation of single training data. The memorized model exhibits large outlier activations (high Std); the generalized model yields more balanced representation (lower Std), consistent with theory. All models use 0.2. Left: CelebA. Right: MoG. See Appendix C for details.
data statistics (specifically, the means and variances of the MoG) from these well-separated, non- degenerate clusters of training data enable generalization.
Corollary 3.3 (Generalization in Underparameterizeed DAEs), Under the problem setup of Theo- rem 3 we assume the training data satisfy the separability condition in Definition 7 If the DAE
network in (S) is under-parameterte
with = Ek= < n,, then there exists local minimizer of the DAE training loss (66) such that
W = W (Wx, Wx. Wxx)= W gon;
where each block Wx x, E Rd captures the principal components of the empirical Gram matrix X₃ XX in (S), with Wx, WT, concentrating to the rank-pı: optimal denoiser for N Ex):
A
Wx. W X
((S=-N)( Pk
-oo
rank-pe approx
where S& is introduced (IT) and pk is the ratio of the k-th mode of MoG. Moreover when - the expectation of the test loss (which captures generalization error) can be bounded by
Ex~n, Wi)l
S
K
Σ
eig, (Sk) 0
+ Eeig,
eigg (St) + CkPk
(Sk) + 02)2
JSPa
3>pE
where Ck > O depends only on a and spectral properties of Sk. Here, (Sg) denotes the j-th eigenvalue of S₁. which independent of d.
Remarks. The proof is deferred to Appendix D. and our result implies the following:
Sampling yields novel in-distribution samples (generalization). When the model is under- parameterized, results show that the local optimal solution learned from the training data achieves bounded population loss on the MoG distribution by effectively acting as an optimal lo- cal denoiser for each mode. Consequently, sampling (Lietall all 20248: 20255) from the trained DAE produces in-distribution images that are distinct from the training samples, as illustrated in Figure A
Moreover, the population loss depends on the spectrum S₁ (equivalently. Ez). When Σg has an approximately low-rank structure (De Bortoli, 2022; Cole and Lut 2024; Huang et al], 2024).. the loss is small and decays rapidly with the number of samples per mode rig- This provides a princi- pled explanation for the reproducibility of diffusion models across disjoint training subsets (Zhang
et all 20247 Kadkhodaie eLall 212EF).
Balanced representations as a signature of generalization. Unlike the spiky representations in Corollary 3.21 the underparameterized solution spreads the energy of æ; + ac, with
For simplicity, we take separability assumption; given sufficient samples, it can be verified under extra conditions the means and covariances of MoG using standard measure concentration (Vershynin) 201S).
7

---

Published as a conference paper at ICLR 2026
Distribution of Representialion Sids (ImageNee)
Distribution of Representations Sids (LAION)
g
Demaity
Gen.
Mem.
Gen.
Mem.
0'5
Mem. Std-O 6834
Mem. Sta-1 2217
Activation Valun
Activafoon Volue
Gen, Sid-0.4951
Gen, Sld-0.4331,
2000
4002
250
1000
Latont Dimensian (p 4096)
Latent Dimanslan (p 1280)
Figure 5: Mem./Gen. representations in real-world models. memorized samples have spiky representations while generalized samples have more balanced ones. The layout follows Figure and the results consistent with it. Representations
extracted at timestep 50 (a, R 0.17).
Left: DiT-L/4 pretrained on an ImageNet subset. Right: Stable Diffusion v1.4 pretrained on LAION (Schubmann et all 2022). Results for EDM pretrained CIFARIO and additional details are in Appendix
Vppg Ex), across the Pk coordinates of the active block (see Figure BD The representation behaves like low-dimensional projection for a Gaussian mode (Tipping and Bishop, 1999):
hgon (oe+ W gen (xx++ (0, (.....,0, 0, W W (xx ..... W Xe.pp (ax+tee), o,...,0) 0).
Intuitively, generalized samples activate multiple neurons rather than a single spiky unit; the result- ing projections encode information about the underlying distribution, helping to explain empirical findings on semantic directions (Kwon et 2023) that are useful for editing, which we further explore in Section
Concluding Corollary 3 and Corollary 6. we see the learned structure remains stable across timesteps, with a primarily acting as regularization parameter. Varying only slightly perturbs the solution, which helps explain the empirical success of diffusion models that employ a single neural network for denoising across multiple noise levels (Sun et ala 2025).
3.3 CASE 3: HYBRID OF MEMORIZATION AND GENERALIZATION WITH IMBALANCED DATA
Large-scale diffusion datasets often contain duplicates due to imperfect curation heterogeneous aggregation (Carlini all 2023);; such samples are more easily memorized (Somepalli et all 202M) (See Appendix 4 for more discussions). We model this by allowing duplicated (rank-1) I) clusters alongside well-sampled, nondegenerate clusters, so the DAE can admit local minimizers that mix memorization and generalization blocks:
Corollary 3.4 (DAE memorizes duplicates and generalizes on well-sampled modes). Let X [X Xx] satisfy Definition 5.7. where for { = I,.... m, X = (æe...., xt) is rank 1, and Xm+1,..... Xx contain distinct empirical samples from the remaining Gaussian modes. Suppose a ReLU DAE is trained with weight decay X 2 and input noise o > 0. Then there exists a local minimizer of the form
W W (riæ)
Tm3m
Wx
Wxx),
where the first m columns memorize the duplicated clusters (as in Cor B.Z), and the remaining blocks Wx. implement generalization on the nondegenerate clusters (as in Cor
This corollary interpolates Cases and 2: duplicated training samples memorized, while the model still generalized for the other modes. We verify this in Figure f and defer proof to Ap- pendix D.G.
4 IMPLICATIONS FOR MEMORIZATION DETECTION AND CONTENT STEERING
In this section, we demonstrate that our theoretical insights from Section B yield profound practical implications for model privacy and interpretability. Leveraging the identified dual relationship be- tween representation structures and generalization ability, we present the following two applications:
8

---

Published as a conference paper at ICLR 2026
Ges8im-0.33
003m-022
cos--8 01
0oa-4..00
0os--2.00 00
000-0
003--0.00
Figure 6: Verification of Corollary Fd
The model learns both memorizing and generalizing
columns when data duplication present.
Method
Prompt Free?
LAION
ImageNet
CIFARIO
AUC 1
TPR
Time
AUC
TPR
Time 4 AUC TPR
Time J
ar ((2K)
0.498
0.020
3.724
N/A
N/A
(Wen
0.986
0.961
0.134
N/A
N/A
(Hinersdorr
0.957
0.500
0.009
N/A
N/A
Ross 0025)
0,956
0.915
0.545
0.971
0.528
0.031
0.713
0.013
0.071
Ours
0.987
0.961
0.067
0.995
0.912
0.015
0.998
0.984
0.020
Table 1: Memorization detection results. We report AUROC, true positive rate (TPR) at 1 1% false positive rate, and runtime (s). Evaluated on three dataset-model pairs: LAION-SDIA4 ImageNet- DiT, and 1--ARO--DM. Sample sizes: 500 memorized and 500 generalized for LAION and Ima- geNet; 100 each for CIFAR1O. (t higher better; + lower is better). See Appendix + for details.
Representation-basse memorization detection (Section E.I)). Leveraging the spikiness of data representations, we introduce prompt-free classification method that accurately distinguishes between generalized and memorized samples produced by diffusion models. We demonstrate that our approach achieves strong performance with high efficiency and extensibility.
Representation-space steering for image editing (Section A.2). We introduce training-free editing method that steers generated samples within the representation space. Crucially, we find that generalized samples substantially more steerable, whereas memorized samples exhibit minimal editing effects due to the spikiness of their representations.
4.1 REPRESENTATION B BASED MEMORIZATION DETECTION
Building theoretical insights, investigate whether memorization be detected directly from internal representations. Prior work has largely focused on how certain prompts trigger mem- orization and often relies those for detection (Wen all 2022 Ieon et all, 2025; Ren all 2024). Representative approaches include: (i) Density-based:: detecting samples that are generated disproportionately frequently under prompt (Carlini et all 2023);; and (ii) form-based: comparing conditional vs. unconditional scores (Wen et all 2024) and (iii) Attention-bal basood: locating anomaly the cross-attention induced by memorized prompts (Hintersdorf all, 2024: Chen all 20255).. A notable exception is (iv) landscape-bused method of (Ross et all 2025), which evaluates mem- orization using local score-function geometry around a generated sample. Their method makes detection prompt-free, but is still based on output space.
In contrast, we introduce the first detection method that is both representation-base based and prompt- free. The core intuition is that spiky representations arise when sample has been internally stored by the model, whereas generalized samples yield balanced activations. Therefore, our analysis yields simple yet effective diagnostic: the standard deviation of intermediate features serves proxy for spikiness. High variance indicates memorization; low variance corresponds to generalization. We benchmark this detector against existing baselines on pre-trained diffusion models. As reported in Table M our method achieves the highest accuracy and efficiency, thereby demonstrating the strong informativeness of representation-spate statistics. Pseudocode and further implementation details are provided in Appendix (
4.2 REPRESENTATION-SPACE STEERING FOR INTERPRETABLE IMAGE EDITING
As shown Corollary G representations of generalized samples are governed by data statistics, capturing local semanties and acting dow-dimensional projections of Gaussian modes. This in- sight implies an interpretable steering mechanism: we can inject information about a target mode (e.g., specific concept style) by adding its average representation, thereby smoothly guiding

---

Published as a conference paper at ICLR 2026
Stearing Strnggh
Sleatin Stanggh
florefockng
phocagycpey
<h Golffich-jp Atmrstiptttioo
Sheorigg Bhenggh
Shoring Ehrnggh
"A lighenastiiion fondffocing jarinat of rechiii plategapht"
"Ovk Menina Talkz Star Ausguile BanAfflckk <Din By Mighstoo
(a) +Old (Gen.)
(b) +Old (Mem.)
Stwwiig Shreggh
Swerrg Shangh
"An image of Non, righ quality dk"
EEnna Wazzon play Belt Dinney's <i-Socuty and she becstrip"
Swweing Steeggh
Steerng Shenghh
"Enmo Wetzen"
"Livig in the Light with Ann Graham Locz"
(c) +Oil-painting Style (Gen.)
(d) Oill-painting Style (Mem.)
Figure 7: Image editing via representation steering. We perform image editing on Stable Dif- fusion v1.4 using (II). Generalized samples exhibit smooth and progressive style transfer as the editing strength increases, whereas memorized samples display brittle and threshold-like transfer effects.
generation toward it. Specifically, our proposed steering function is defined as:
fateered (xt,
t, (x,,
(x,, + av), where
Σ ho #ES
(11)
Here, S denotes samples from the target concept/style, he and go represent the encoder and decoder components of the network, respectively, and E IR controls the steering strength, is the text prompt and denotes the desired concept/style prompt.
We evaluate this method on Stable Diffusion v1.4 using both memorized and generalized samples (Figure M) As predicted by our theory, generalized samples exhibit smooth and monotonic edits as a varies, indicative of well-behaved local geometry in their representation space. In contrast, memorized samples display brittle, threshold-like responses, making fine-grained control difficult because of their spiky representations.
5 CONCLUSION
summary, our study establishes that the representation space of diffusion models not a sec- ondary artifact of training but critical factor in how these models operate. Its structure provides principled separation between memorization and generalization: spiky, samplc-specific codes signal memorization, while balanced, low-dimensional representations often imply strong generalization. This perspective allows us not only to detect memorization directly from internal model representa- tions but also leverage representations for practical tasks, such controllable editing via steering. While prior works have used intermediate activations for downstream applications, our framework highlights their important role in shaping diffusion behavior itself. By making these structures explicit, we bridge the theoretical findings on simplified models with the empirical properties of real-world deep nonlinear models, offering a unified view that connects perception and generation and opens pathways toward more interpretable and trustworthy generative models.
10

---

Published as a conference paper at ICLR 2026
ACKNOWLEDGMENTS
We acknowledge funding support from NSF CAREER CCF-2143904, NSF CCF-2212066, NSF CCF-2212326, NSF IIS 2402950, ONR X00014251233 DARPA HR0011578254, and Google Re- search Scholar Award. We also thank Shuo Xie (TTIC; for fruitful discussions on optimization bias), SooMin Kwon, Hujjie Zhang, Alec Xu, Prof. Laura Balzano (UMich; for help with problem formu- lation), Prof. Zhihui Zhu (OSU; for guidance on presentations), Zaicun Li (UMD; for mathematical insights), and Zhenyun Zhu (UMich; for scientific discussions).
CODES OF ETHIC AND REPRODUCIBILITY
Ethics Statement. This study utilizes only theoretical frameworks and controlled experiments on publicly available datasets and research models; no human subjects, sensitive data, or person involved. We acknowledge dual-use risks: memorization detection
ally identifiable information could be misused to extract or target memorized content, and representation steering could be abused to generate harmful outputs. Our intent is the opposite: to improve auditability and safety by (i) de- tecting and characterizing memorization, and (ii) observing (and then disabling) harmful steering directions at the post-training stage. All experiments were conducted under controlled conditions using standard public memorization benchmarks, adhering dataset licenses and established prac- tices as in previous memorization-detecee literature. Code and artifacts, if released, will include safeguards and usage guidelines to discourage misuse.
Reproducibility Statement. All assumptions underlying our theoretical results are explicitly stated in the main paper. Proofs provided in Appendix D with additional verification in Ap- pendix B.ll Empirical details are provided in Appendix ID, and code is included with the submission to support reproducibility.
REFERENCES
Beatrice Achilli, Enrico Ventura, Gianluigi Silvestri, Bao Pham, Gabriel Raya, Dmitry Krotov, Carlo Lucibello, and Luca Ambrogioni. Losing dimensions: Geometric memorization in generative diffusion, arXiv preprint, 2024.
Luca Ambrogioni. In search of dispersed memories: Generative diffusion models are associative memory networks. Entropy, 2024.
Jie An, De Wang, Pengsheng Guo, Jiebo Luo, and Alexander Schwing. On inductive biases that enable generalization of diffusion transformers. NeurIPS, 2025.
Ricardo Baptista, Agnimitra Dasgupta, Nikola Kovachki, Assad Oberai, and Andrew M Stuart. Memorization and regularization in generative diffusion models. arXiv preprint, 2025.
Dmitry Baranchuk, Andrey Voynov, Ivan Rubachev, Valentin Khrulkov, and Artem Babenko. Label- efficient semantic segmentation with diffusion models. ICLR, 2022.
Yoshua Bengio, Li Yao, Guillaume Alain, and Pascal Vincent. Generalized denoising auto-encoders as generative models. NeurIPS, 2013.
Quentin Bertrand, Anne Gagneux, Mathurin Massias, and Rémi Emonet. On the closed-form of flow matching: Generalization does not arise from target stochasticity. NeurIPS, 2025.
Rajendra Bhatia. Matrix analysis, volume 169. Springer Science & Business Media, 2013.
Giulio Biroli, Tony Bonnaire, Valentin De Bortoli, and Mare Mézard. Dynamical regimes diffu- sion models. Nature Communications, 2024.
Tony Bonnaire, Raphaël Urfin, Giulio Biroli, and Mare Mézard. Why diffusion models don't mem- orize: The role of implicit dynamical regularization in training. NeurlPS, 2025.
Sam Buchanan, Druv Pai, Yi Ma, and Valentin De Bortoli. On the edge of memorization in diffusion models. NewrIPS, 2025.
11

---

Published as a conference paper at ICLR 2026
Nicolas Carlini, Jamie Hayes, Milad Nasr, Matthew Jagielski, Vikash Sehwag, Florian Tramer, Borja Balle. Daphne Ippolito, and Eric Wallace. Extracting training data from diffusion mod- cls. USENIX Security, 2023.
Chen Chen, Daochang Liu, and Chang Xu. Towards memorization free diffusion models. CVPR. 2024a.
Chen Chen, Daochang Liu, Mubarak Shah, and Chang Xu. Exploring local memorization in diffu- sion models via bright ending attention. ICLR, 2025a.
Minshuo Chen, Kaixuan Huang, Tuo Zhao, and Mengdi Wang. Score approximation, estimation and distribution recovery of diffusion models on low-dimensional data. ICML, 2023.
Siyi Chen, Huijie Zhang, Minzhe Guo, Yifu Lu, Peng Wang, and Qing Qu. Exploring low- dimensional subspace in diffusion models for controllable image editing. NeurIPS, 2024b.
Xinlei Chen, Zhuang Liu, Saining Xie, and Kaiming He. Deconstructing denoising di diffusion models for sele--upe learning. ICLR, 2025b.
Zhengdao Chen. On the interpolation effect of smoothing. arXiv preprint, 2025.
Frank Cole and Yulong Lu. Score-based generative models break the curse of dimensionality in learning family of sub-gaussian probability distributions. ICLR, 2024.
Hugo Cui and Lenka Zdeborová. High-dimensional asymptotics of denoising autoencoders.
NeurlPS, 2023.
Hugo Cui, Cengiz Pehlevan, and Yue M Lu. A precise asymptotic analysis of learning diffusion models: theory and insights. arXiv preprint, 2025.
Valentin De Bortoli. Convergence of denoising diffusion models under the manifold hypothesis. TMLR, 2022.
Bradley Efron. Tweedies formula and selection bias. Journal the American Statistical Association, 2011.
Andreas Floros, Seyed-Mohsen Moosavi-Dezfooli, and Pier Luigi Dragotti. On the anisotropy of score-based generative models. arXiv preprint, 2025.
Chaofan Gan, Zicheng Zhao, Yuanpeng Tu, Xi Chen, Ziran Qin, Tieyuan Chen, Mehrtash Harandi, and Weiyao Lin. Massive activations are the key to local detail synthesis in diffusion transformers. arXiv preprint, 2025.
Rohit Gandikota, Joanna Materzyúska, Tingrui Zhou, Antonio Torralba, and David Bau. Concept sliders: Lora adaptors for precise control in diffusion models. ECCV, 2024.
Khashayar Gatmiry, Jonathan A. Kelner, and Holden Lee. Learning mixtures of gaussians using diffusion models. COLT, 2025.
Anand Jerry George, Rodrigo Veiga, and Nicolas Macris. Denoising score matching with random features: Insights on diffusion models from precise learning curves. arXiv preprint, 2025.
Google. Veo 3:
Google's most capable video generation model.
Technical report,
Google, 2025. URL
com/ dewpmineemmn..m. perepmiindmeeiia/eeeee media/veo/
Teo-3-Tech-
paf.
Xiangming Gu, Chao Du, Tianyu Pang, Chongxuan Li, Min Lin, and Ye Wang. On memorization in diffusion models. TMLR, 2025.
Shahin Hakemi, Naveed Akhtar, Ghulam Mubashar Hassan, and Ajmal Mian. Deeper diffusion models amplify bias. arXiv preprint, 2025.
Andi Han, Wei Huang, Yuan Cao, and Difan Zou. On the feature learning in diffusion models. ICLR, 2025.
12

---

Published as a conference paper at ICLR 2026
Amir Hertz, Ron Mokady, Jay Tenenbaum, Kfir Aberman, Yael Pritch, and Daniel Cohen-or. Prompt-to-promp -rommppp image editing with cross-attention control. In ICLR, 2023.
Dominik Hintersdorf, Lukas Struppek, Kristian Kersting, Adam Dziedzic, and Franziska Boenisch. Finding nemo: Localizing neurons responsible memorization diffusion models. NeurIPS, 2024.
Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models. NeurIPS, 2020.
Zhihan Huang, Yuting Wei, and Yuxin Chen. Denoising diffusion probabiliitic models are optimally adaptive to unknown low dimensionality. arXiv preprint, 2024.
Dongjae Jeon, Dueun Kim, and Albert No. Understanding and mitigating memorization in genera- tive models via sharpness of probability landscapes. ICML, 2025.
Zahra Kadkhodaie, Florentin Guth, Eero P. Simoncelli, and Stéphane Mallat. Generalization in diffusion models arises from geometry-adaptive harmonic representations. ICLR, 2024a.
Zahra Kadkhodaie, Stéphane Mallat, and Eero P Simoncelli. seature-guided score diffusion for sampling conditional densities. arXiv preprint, 2024b.
Mason Kamb and Surya Ganguli. An analytic theory of creativity in convolutional diffusion models. ICML, 2025.
Hamid Kamkari, Brendan Ross, Rasa Hosseinzadeh, Jesse Cresswell, and Gabriel Loaiza-Ganent A geometric view of data complexity: Efficient local intrinsic dimension estimation with diffusion models. NeurIPS, 2024.
Tero Karras, Miika Aittala, Timo Aila, and Samuli Laine. Elucidating the design space of diffusion- based generative models. NeurlPS, 2022.
Diederik Kingma and Jimmy Ba. Adam: A method for stochastic optimization. ICLR, 2015.
Daniel Kunin, Jonathan Bloom, Aleksandrina Goeva, and Cotton Seed. Loss landscapes of regular- ized linear autoencoders. ICML,, 2019.
Mingi Kwon, Jacseok Jeong, and Youngjung Uh. Diffusion models already have A semantic latent space. ICIRR, 2023.
Black Forest Labs, Stephen Batifol, Andreas Blattmann, Frederic Boesel, Saksham Consul, Cyril Diagne, Tim Dockhorn, Jack English, Zion English. Patrick Esser, Sumith Kulal, Kyle Laccy, Yam Levi, Cheng Li, Dominik Lorenz, Jonas Müller, Dustin Podell, Robin Rombach, Harry Saini, Axel Sauer, and Luke Smith. Flux.1 kontext: Flow matching for in-context image generation and editing in latent space. arXiv preprint, 2025.
Gen Li, Yuting Wei, Yuejie Chi, and Yuxin Chen. A sharp convergence theory for the probability flow odes of diffusion models. arXiv preprint, 2024a.
Gen Li, Yuting Wei, Yuxin Chen, and Yuejie Chi. Towards non-asymptotic convergence for diffusion-based generative models. ICLR, 2025a.
Puheng Li, Zhong Li, Huishuai Zhang, and Jiang Bian. On the generalization properties of diffusion models. NeurIPS, 2023.
Tianhong Li and Kaiming He. Back to basics: Let denoising generative models denoise. arXiv preprint, 2025.
Xiang Li, Yixiang Dai, and Qing Qu. Understanding generalizability of diffusion models requires rethinking the hidden gaussian structure. NeurIPS, 2024b.
Xiang Li. Rongrong Wang, and Qing Qu. Towards understanding the mechanisms of classifier free guidance. NeurlPS, 2025b.
13

---

Published as a conference paper at ICLR 2026
Xiao Li, Zekai Zhang. Xiang Li, Siyi Chen, Zhihui Zhu, Peng Wang, and Qing Qu. Understanding representation dynamics of diffusion models via low-dimensional modeling. NeurIPS, 2025c.
Jiadong Liang, Zhihan Huang, and Yuxin Chen. Low-dimensional adaptation of diffusion models:
Convergence total variation. COLT, 2025.
Aaron Lou, Chenlin Meng, and Stefano Ermon. Discrete diffusion modeling by estimating the ratios of the data distribution. ICML, 2024.
Artem Lukoianov, Chenyang Yuan, Justin Solomon, and Vincent Sitzmann. Locality in image diffu- sion models emerges from data statistics, NeurlPS, 2025.
Yang Lyu, Tan Minh Nguyen, Yuchun Qian, and Xin T Tong. Resolving memorization in empirical diffusion model for manifold data in high-dimensional spaces. arXiv preprint, 2025.
Matthew Niedoba, Dylan Green, Saeid Naderiparizi, Vasileios Lioutas, Jonathan Wilder Lavington, Xiaoxuan Liang, Yunpeng Liu, Ke Zhang, Setarch Dabiri, Adam Scibior, et al. Nearest neighbour score estimators for diffusion generative models. ICML, 2024.
Matthew Niedoba, Berend Zwartsenberg, Kevin Patrick Murphy, and Frank Wood. Towards mech- anistic explanation of diffusion model generalization. ICML, 2025.
Elizabeth Pavlova and Xue-Xin Wei. Diffusion models under low-noise regime. arXiv preprint, 2025.
William Peebles and Saining Xie. Scalable diffusion models with transformers. ICCV, 2023.
Bao Pham, Gabriel Raya, Matteo Negri, Mohammed J Zaki, Luca Ambrogioni, and Dmitry Krotov. Memorization to generalization: Emergence of diffusion models from associative memory. arXiv preprint, 2025.
Adityanarayanan Radhakrishnan, Mikhail Belkin, and Caroline Uhler. Overparameterized neural networks implement associative memory. PNASS, 2020.
Jie Ren, Yaxin Li, Shenglai Zeng, Han Xu, Lingjuan Lyu, Yue Xing, and Jiliang Tang. Unveiling and mitigating memorization text-to-image diffusion models through cross attention. ECCV, 2024.
Robin Rombach, Andreas Blattmann, Dominik Lorenz, Patrick Esser, and Björn Ommer. High- resolution image synthesis with latent diffusion models. CVPR, 2022.
Brendan Leigh Ross, Hamidreza Kamkari, Tongzi Wu, Rasa Hosseinzadeh, Zhaoyan Liu, George Stein, Jesse C. Cresswell, and Gabriel Loaiza-Ganent Ganem, A geometric framework for understanding memorization in generative models. ICLR, 2025.
Christopher Scarvelis, Haitz Sáez de Ocáriz Borde, and Justin Solomon. Closed-form diffusion models. arXiv preprint, 2023.
Christoph Schuhmann, Romain Beaumont, Richard Vencu, Cade Gordon, Ross Wightman, Mehdi Cherti, Theo Coombes, Aarush Katta, Clayton Mullis, Mitchell Wortsman, al. Laion-5b: An open large-scale dataset for training next generation image-text models. NeurIPS, 2022.
Lianghe Shi, Meng Wu, Huijie Zhang, Zekai Zhang, Molei Tao, and Qing Qu. A closer look at model collapse: From gemeraliiatioontt--mmmooiii.
perspective. NeurIPS, 2025.
Jaskirat Singh, Xingjian Leng, Zongze Wu, Liang Zheng, Richard Zhang, Eli Shechtman, and Sain- ing Xie. What matters for representation alignment: Global information spatial structure? arXiv preprint, 2025.
Gowthami Somepalli, Vasu Singla, Micah Goldblum, Jonas Geiping, and Tom Goldstein. Diffusion art or digital forgery? investigating data replication diffusion models. CVPR, 2023a.
Gowthami Somepalli, Vasu Singla, Micah Goldblum, Jonas Geiping, and Tom Goldstein. Under- standing and mitigating copying in diffusion models. NewrIPS, 2023b.
14

---

Published as a conference paper at ICLR 2026
Jiaming Song, Chenlin Meng, and Stefano Ermon. Denoising diffusion implicit models. ICLR, 2021a.
Yang Song, Jascha Sohl-Dickstein, Diederik P. Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. Score-based generative modeling through stochastic differential equations. ICLR, 2021b.
Mingjie Sun, Xinlei Chen, J Zico Kolter, and Zhuang Liu. Massive activations in large language models. COLM, 2024.
Qiao Sun, Zhicheng Jiang, Hanhong Zhao, and Kaiming He. Is noise conditioning necessary for denoising generative models? ICML, 2025.
Yuandong Tian. Provable scaling laws of feature emergence from learning dynamics of grokking. arXiv preprint, 2025.
Berk Tinaz, Zalan Fabian, and Mahdi Soltanolkotabi. Emergence and evolution of interpretable concepts in diffusion models. NeurIPS, 2025.
Michael E Tipping and Christopher M Bishop. Probabilistic principal component analysis. Journal of the Royal Statistical Society Series B: Statistical Methodology. 1999.
Roman Vershynin. High--iimnnsioall Probability: An Introduction with Applications in Data Sci- ence. Cambridge University Press, 2018.
Pascal Vincent. A connection between score matching and denoising autoencoders. Neural compu- tation, 2011.
Pascal Vincent, Hugo Larochelle, Isabelle Lajoie, Yoshua Bengio, Pierre-Antoine Manzagol, and Léon Bottou. Stacked denoising autoencoders: Learning useful representations in deep network with a local denoising criterion. JMLR, 2010.
Binxu Wang. An analytical theory of power law spectral bias in the learning dynamics of diffusion models. NeurIPS, 2025.
Binxu Wang and John J Vastola. The unreasonable effectiveness of gaussian score approximation for diffusion models and its applications. 7MLR, 2024.
Chenyu Wang, Cai Zhou, Sharut Gupta, Zongyu Lin, Stefanie Jegelka, Stephen Bates, and Tommi Jaakkola. Learning diffusion models with flexible representation guidance. NeurIPS, 2025a.
Peng Wang, Huijie Zhang, Zekai Zhang, Siyi Chen, Yi Ma, and Qing Qu. Diffusion models learn low-dimensional distributions via subspace clustering. arXiv preprint, 2024a.
Qingsong Wang. Zhengchao Wan, Mikhail Belkin, and Yusu Wang. Seeds of structure: Patch pca reveals universal compositional cues in diffusion models. NeurIPS, 2025b.
Yuqing Wang, Ye He, and Molei Tao. Evaluating the design space of diffusion-based generative models. NeurIPS, 2024b.
Ryan Webster, Julien Rabin, Loic Simon, and Frederic Jurie. On the de-duplication of laion-2b. arXiv preprint, 2023.
Dana Weitzner, Mauricio Delbracio. Peyman Milanfar, and Raja Giryes. On the relation between linear diffusion and power iteration. arXiv preprint, 2024.
Yuxin Wen, Yuchen Liu, Chen Chen, and Lingjuan Lyu. Detecting, explaining, and mitigating memorization diffusion models. ICLR, 2024.
Yu-Han Wu, Pierre Marion, Gérard Biau, and Claire Boyer. Taking a big step: Large learning rates in denoising score matching prevent memorization. COLT,, 2025.
Weilai Xiang, Hongyu Yang. Di Huang, and Yunhong Wang. Denoising diffusion autoencoders are unified seff-supervised learners. ICCV, 2023.
15

---

Published as a conference paper at ICLR 2026
Weilai Xiang, Hongyu Yang, Di Huang, and Yunhong Wang. Ddae++: Enhancing diffusion models towards unified generative and discriminative learning. arXiv preprint, 2025.
Shuo Xie and Zhiyuan Li. Implicit bias of adamw: -nom constrained optimization. ICML, 2024.
Shuo Xie, Mohamad Amin Mohamadi, and Zhiyuan Li. Adam Exploits Eo-gemettyy of Loss Landscape via Coordinate-wise Adaptivity. ICLR, 2025.
Ruofeng Yang, Bo Jiang, Cheng Chen, Baoxiang Wang, Shuai Li, al. Few-shot diffusion models escape the curse of dimensionality. NeurIPS, 2024.
Xingyi Yang and Xinchao Wang. Diffusion model representation leamer. In ICCV, 2023.
Zeqi Ye. Qijie Zhu, Molei Tao, and Minshuo Chen. Provable separations between memorization and generalization in diffusion models. arXiv preprint, 2025.
Mingyang Yi, Jiacheng Sun, and Zhenguo Li. On the generalization of diffusion model. arXiv preprint, 2023.
Sihyun Yu, Sangkyung Kwak, Huiwon Jang. Jongheon Jeong, Jonathan Huang, Jinwoo Shin, and Saining Xie. Representation aligmment for generation: Training diffusion transformers is easier than you think. ICIRR, 2025.
Chen Zeno, Greg Ongie, Yaniv Blumenfeld, Nir Weinberger, and Daniel Soudry. How do minimum- norm shallow denoisers look in function space? NeurIPS, 2023.
Chen Zeno, Hila Manor, Greg Ongie, Nir Weinberger, Tomer Michaeli, and Daniel Soudry. When diffusion models memorize: Inductive biases in probability flow of minimum-norm shallow neu- ral nets. ICML, 2025.
Fangzhao Zhang and Mert Pilanci. Analyzing neural network-basse -bsse generative diffusion models through convex optimization. arXiv preprint, 2024.
Huijie Zhang, Jinfan Zhou, Yifu Lu, Minzhe Guo, Peng Wang, Liyue Shen, and Qing Qu. The emergence of reproducibility and consistency in diffusion models. ICML, 2024.
Huijie Zhang, Zijian Huang, Siyi Chen, Jinfan Zhou, 7ekai Zhang, Peng Wang, and Qing Qu. Under- standing generalization diffusion models via probability flow distance. arXiv preprint, 2025a.
Huijie Zhang, Peng Wang. Siyi Chen, Zekai Zhang, and Qing Qu. Generalization of diffusion models: Principles, theory, and implications. SIAM News, 2025b.
Lvmin Zhang, Anyi Rao, and Maneesh Agrawala. Adding conditional control text-to-image diffusion models. ICCV, 2023.
Wenliang Zhao, Yongming Rao, Zuyan Liu, Benlin Liu, Jie Zhou, and Jiwen Lu. Unleashing text- to-image diffusion models for visual perception. ICCV, 2023.
16

---

Published as a conference paper at ICLR 2026
A ADDITIONAL RELATED WORKS
A.1 ANALYSIS OF LEARNING DIFFUSION MODELS WITH SPECIFIC MODEL
PARAMETERIZATTIOS
There has been large body of work analyzing the learning of diffusion models (Wang et all, 2024h:
Chen all 2023: Liang et all 2025; Yang et all 20244. Recently, more attention has turned to when and how they overfit generalize: Li (2023); Bonnaire et (2025) use mundom-featur assumptions, while Wang et al, (2024a); Buchanan et all (2025) studied empirical denoisers with learnable attractors; Wu all (2025); Chen (2025); Ye etll (2025) investigated smoothing effects
induced by learning rates and weight decay that promote generalization. These works are theoreti- cally rigorous but often lack real-world validation.
A.2 MEMORIZATION AND GENERALIZATION WITH ANALYTICAL DMS
Constraineddreegllriizee models. Recent works characterize how architectural or inductive bi- ases can push empirical scores toward more generalizable solutions. For instance, Scarvelis et all (2023): Lukonanoy et (2025) constructed closed-fome diffusion models from data; Niedoba et all
(2012) 2025);; Kamb and Ganguli (2025);; Wang et all (2025))) imposed locality or translation- equivariance constraints to mimic U-Net behavior; and Kadkhodate all (202247); An all (2(25);
Floros et all (2022) analyzed architectural biases of CNNs and DiTs. Baptısta et all (2025)
empiri-
cally evaluated the impact of various regularization schemes.
Associative Memory (AM) models. Radhakrsshaan et all (2020); Ambrogioni (2122) Pham Caamctal ct (2025) model imperfect training and sampling jointly as an AM recall process, viewing novel image generation as attraction basins and memorization perfect recalls (Biroli at 2024) Lyu et all (027)) However, this perspective can understate the role learned neural networks in enabling generalization.
A.3 STUDIES ON REPRESENTATION LEARNING OF DIFFUSION MODELS
Concurrent work studies co-emerging representation learning (K et all 2023: Han et all 2025:
Yang and Wang (02)) with distribution learning in diffusion models. As recent works (Chen al. 2025H: Xiang et al,, 2025) re-em e--ehassize that the diffusion objective is fundamentally self- supervised autoencoder loss (Vincent 2010; Vinceni, 20T,; Bengio all, 2013), which induces encoder-decoder behavior [Chen.etaa all 2025h) and the model autonomously learns informative fea- tures for downstream tasks (Baranchuk all 20ZZ Xiang arl 2IZ3): Zhao all RIZg). Moreover, supervising the representations can accelerate training (Yu all MOZAS Wang et all 202T;; Singh atall 2025), and different representation behaviors correlate with different degrees of overfitting (225c).
et a
B ADDITIONAL EXPERIMENTS
B.1 FURTHER VERIFICATION OF THFOREM
Verification with MoG data Under Mixture of Gaussians (MoG) setting, we directly ver- ify Theorem 6.T since the separability assumptions can be enforced by construction. We use two-mode MoG in a 1000-dimensional space with symmetric means μ4 = = = 5e1, where e1 (1, 01......00 0), and covariance matrices Σ,, Σg each having exponentially decaying spectra. We sample 5,000 points from each mode, yielding two separated clusters. Training ReLU DAE with p 50 hidden units and a = 0.2, we find that the model effectively learns rank-25 ap- proximation of the Wiener filter for each cluster, as defined in Theorem B.I. as shown in Figure
Robustness to large noise levels We show here that the vanishing remainder in Theorem G is negligible even for large øs. For instance, we train the ReLU DAE under o = 0.2, 1, 5 on CelebA and we find the model still learns the constructed solution shown in Figure g
17

---

Published as a conference paper at ICLR 2026
Eigen Values
Top-25
Eigenspace Alignment (mean cos=0.997)
1.00
1.0
Elgen value
0.95
cosine UUT UW)
0.8
0.6
0.90
0.4
0.85
Wiener target (top 25)
0.2
WW (top 25)
0.80
0.0
0
10
15
20
25
12
16
20
24
Index
Component index
Figure 8: Comparison between the learned ReLU DAE and the constructed solution from Theo- rem 3 under the MoG setting. They agree both eigenvalues and eigenvectors.
Top-norm encoder columns across noise levels
norm 1.02
norm 1,02
norm 1.01
norm 0.96
nerm 0.96
norm 0.07
norm 0.05
nom 0.04
norm
nom
norm 0.98
nzrm 0.92
norm 0.87
norm 0.85
norm 0.84
norm 0.22
norm 0.18
nom 0.18
Figure 9: training with larger a will give us less perfect memorization, but the trend holds
Robustness beyond separability When the separability assumption is relaxed (3 > 0, so training images overlap), the memorized ReLU DAE still learns processed version of the solution in Theo- rem G.T. Empirically, it recovers a denoised/proceesed data matrix, or approximately an orthonormal basis for the data span; see Figure II].
For the generalized ReLU DAE on CelebA, it is already non-separable dataset. And the model continues to (i) generate novel images (Figure D) (ii) capture dataset statistics (Figure R), and (iii) produce balanced representations (Figure ED). Thus, separability mainly simplifies the form of local minimizers; is not required for either memorization or generalization.
Robustness different optimization setups We show that the local minimizer characterized in Cor. 3.2 is robust to different random seeds and optimizers (RMSProp. Adam, Adam ())..)).. In all cases, modern adaptive optimizers converge to a sparse solution that stores individual training samples as columns. We also varied the random seed and found that it essentially only permutes the columns, and omit those results for brevity (Figure (T].
Tying vs. untying the encoder-decoder matrices Our theorem shows that even when the encoder and decoder are parametrized independently. training drives them to a symmetric (tied) solution. confirm this empirically Figure 12 consistent with prior observations (Kunın et all 2019). Accordingly, for Figures [ and @ in the main text we train weight-tied ReLU DAEs.
B.2 DENOISING AND REPRESENTATIONS OF TEST SAMPLES WITH RELU DAE
As in (Kadkhodare.ed et all 20245) ) the ability to denoise unseen test image is equivalent check
for generalization or overfitting, as shown in Figure I.. Memorizing DAE (Corollary (X) per- fectly denoises training image. On test image, it still produces training-data like output (visu-
18

---

Published as a conference paper at ICLR 2026
p-50
Normelized (top-5)
cotumn inder.
Eigenspace Alignment span(W top5) spen(ing)
0.8
0.6
Fide Pidecaao Ses
mec aond 2r1
mas coen Laze
mus ciNeUSk
tan come TIM
LOUNE-200 200
Non- separable Training set
0.2
angle
Figure 10: When separability breaks, ReLU DAE still learns a processed version of the data matrix (approximately an orthonormal basis of the data span).
RAISorop
Nom 0.89
Nom 0.07
Nom 1000
Norn 0:88
0.88 CouBin
Nom 0,20
Namo D.
Adam
Nom 0.9
Nom 0.95
Nom 0.64
Norn 0.04
Norm 0,22
Norn D.6
Norm 0.08
AdanIN
Nom
Kome 1.00
Nom 69
Nom
0:66
Nom 0,36
Narm 0.07
1:00
CusBinn
Figure 11: The local minimizer from Cor. 3M is robust to different random seeds and optimizers.
ally clear but discarding input-specific information, producing high test MSE). Generalizing DAE (Corollary (333) denoises both while preserving input-specific structure.
Moreover, we also visualize the representations of test samples for the memorizing and generaliz ing DAEs in Figure [4 Since the memorizing DAE learns sparse columns, the representation of test image is also sparse: positive activations indicate positive alignment with specific memorized training samples, and the resulting code is highly spiky. For the generalizing DAE, which learns statistics reflecting the underlying data distribution, the representations of test samples are bal- anced as those of training samples.
B.3 CONNECTION BETWEEN RELU DAE AND REAL-WORLD WORLD MODELS
this section, we demonstrate that our ReLU model exhibits piecewise linearity, consistent with observations in real-world models (Lukoranov et all 0025). Consequently, it can be viewed as a localized approximation
these counterparts: large model can implement the mechanisms of Corollary (222 and Corollary in distinct local regions (Ross all, 2025), thereby simultaneously generalizing and memorizing. We verify this via SVD analysis of the Jacobian (Kadkhodare et all 2024a: Achilli et all 2024) for SD1.4 EDM, and ReL.U DAE:
19

---

Published as a conference paper at ICLR 2026
Encoder-Decoder Frobenius error over training
Frob. err
10
Top Copummn by norm
frai weighte
5
Dec
Dices
10000
20000 30000 Epoch
40000 50000
Eno
Ere
Ere
17
Figure 12: An untied ReLU DAE learns (approximately) symmetric encoder-decoder matrices.
Ofyan funt
pa purtoo Hivht
DAR WArF PORN
DAR Joad PORNE
Oighnipand
HIGHT
EDAE goart
SaeLOde yoar
Mew DAE
Dos DAP bes
Man DAE
DeNDAE
(a) Denoising 0.2 with Mem./Gen. DAEs
(b) Denoising 1.0 with Mem./Gen. DAEs
Figure 13: One-step denoising result of train/test samples with ReLU DAE
Around memorized data, the model's Jacobian is extremely low-rank and dominated by that spe- cific data vector. This indicates the model is storing and denoising along the memorized sample, confirming the results Cor. 62 Moreover, the model denoises with near-perfect certainty.
Around generalized samples, the Jacobian matrix reflects the data structures described in corol- lary GK Accordingly, the model produces a smoothed result, having learned a ground-truth de- noiser that incorporates the constraints of the underlying distribution (Niedoha etall (2225).
We visualize these findings in Figures
and
B.4 DUPLICATION OF TRAINING DATA INDUCES MEMORIZATION
Large-scale diffusion datasets often contain duplicates due to imperfect deduplication or aggregation from heterogeneous sources (Carlim et al, 2023; Shi et all 2025). Such duplicates dispropor- tionately memorized by generative models (Somepalli et all 20235: Chen_etal 2024a). Interpolat- ing Corollary EW and Corollary 3. suggests that, when subset is duplicated, the model tends to memorize those duplicated samples while still generalizing on the rest. We observe this behavior empirically in Figure IIA for EDM trained CIFARIO with duplicated subset (and similarly for DiT on ImageNet as in Figure )
C EXTRA TECHNICAL DETAILS
C.1 TRAINING AND SAMPLING SETUP FOR RELU DAES
Optimization. We train with RMSprop. For memorized models, we use learning rate x 10 10-3 weight decay 1 1x002 and run 5 10s gradient steps. For generalized models, use learning rate 1x10 -4 weight decay 1x10 and run 4x x 10" steps. Perturbing these choices (e.g., Adam/ RMSprop, slightly different learning rates weight decays, or tying untying the encoder- decoder) can slightly shift the final solution, but the memorization aeemorization-gneeaaizzzzaiooooi characterization remains clear.
20

---

Published as a conference paper at ICLR 2026
Representations of Mem. DAE
Train
Train (3td-1 815) Test (Std- (88)
Activation
Latent dimension (p 100)
Representations Gen. DAE
Test
Train (Bid-1 487)
Activation
Test (Stpet 471)
40
100
Latent (p 100)
Figure 14: Representations of train and test samples under memorizing vs. generalizing ReLU DAEs.
Sampling, We train set of DAEs with VE noise scheduling (Song et al. 20216 over E [0.02, 2] and run DDIM sampling (Eq. II).
Data. MoG: In 1000, we consider two symmetric modes with means μ4i = - = 5ei (where e1 = (1, O...,... ,, 0)) and covariances Σ, Σg having exponentially decaying spectra. For the memo- rized model, we use 2 samples per mode; for the generalized model we use 10,000 samples (5,000
per mode). CelebA: We use training images (chosen for clear separability) for the memorized model and the first 10,000 for the generalized model.
C.2 MEMORIZATION DETECTION DETAILS
Collecting mem./gen. sets. For LAION-Stable Diffusion, we follow (Wen et all 2024) and use publicly available prompts curated to elicit either memorization or generalization (Webster et all For CIFARI0-EDM and ImageNet-DiT,
compute the SSCD similarity (Zhang all between each generated image and its nearest neighbor in the training set; samples with similarity 0.9 labeled memorized and those with similarity <.0. generalized.
2023).
2024)
Feature extraction. For EDM we extract activations at 8x8 [e4__block3.o no ror ;; for Stable Diffu-
sion v1.4 at up blocks 0. resnets. www.eetts..nnnnnnnanna.mmmmm nonninearity and for DiT-L/4 we use the SiLU activation in block 12 (of 24). We apply global max pooling (spatial for Stable Diffusion v1.4 and EDM; token-wise for DiT) to obtain compact representations, though detection also works even if not. Unless otherwise noted, representations are taken at DDPM timestep 50, corresponding to an equivalent noise level a x 0.17 cL (02().
Algorithm 1: Detection via representation standard deviation (STD)
Input: generated image zp, timestep f, threshold THRES
Output: intermediate representation h, detection flag Umom
2e + ADDFORWARDNOISE(C., t);
h + he (2,, condition = Ø); (ær, 4, Ø)] with and h the decoder/encoder components;
where fa (Zt, 1) go (a
Lmem + (STD(h) > THRES);
return h, Imem:
The detection metric need not be limited to standard deviation; other effective choices include the l ratio (Vershymin 2018), entropy, and max-min of the representations. We found these alterna- tives yield similar separability between memorized and generalized samples.
21

---

Published as a conference paper at ICLR 2026
mam
dancisac menm
genn
gen
Singular va lues
Sincular Vectors
Mem Gen
2.5
Left Singular Vectors (Gien.)
1.4-
0.5
2500 5000
7500 10000 150000000110000
idx
LE: 00
M1.51
J8. 2
ME.31
41:41
(a) SD1.4's Jacobian at t 200
gth
Singular
Lefi Singular Vectors (Mem)
1.75
Mem
Gen
0.75
Left Singuler Vectors (Gen
302
1000
1550
2000
2000
3000
idk
18.41
L0: 11
Lt:
10, 41
(b) EDM's Jacobian at 0.2
Denaact
Singular values
Singuler (Mam.)
Dae
46.11
(Gen.)
20
43
60
(c) ReLU DAE's Jacobian 0.2
Figure 15: Jacobians for SD1.4, EDM, and ReLU DAE at the indicated time/noise settings.
C.3 IMAGE EDITING DETAILS
We use Stable Diffusion v1.4 for our image editing experiments. For each style transfer task, we first generate 100 images in the target concept/style. We then extract feature representations at timestep
= 10 (out of 1000) from the condi- layers upp stp_blocks.om C. resnets.))
tional path
up wwwbblock..o 0. resnets.
up blocks. resnets.2) up blocks. resseets.
uр blocks 1, resnets.11 and [up blocks. C resnets. The resulting tensor has size 100 C. x H W. We compute the mean across the image, height, and width dimensions, yielding a steering vector of size 1 x C x 1. Representation steering is performed by adding this steering vector to the conditional path representation of source image with varying editing
22

---

Published as a conference paper at ICLR 2026
Distriution of Rupresertaainn Sids (CIFAR10)
Similrity Distribttion
Dinoliy
1.
Mom.
1.20
1,25
1.50
1.75
2:25
275
S1d-2.1801
Gen. Stoin 2868
10
Actvvttinn
250
Semiuaiiy
Latent Dimonsion (p 12810)
(a) Bimodal similarity of generated samples to the training set (CIFARI0) under duplication
(b) Mem./Gen. representation statistics for EDM
pretrained on CIFARIO with duplicated subset.
Figure 16: Effect of training-set duplication. Duplicates induce a memorization mode while non- duplicated data continue to support generalization.
Swewing Strengtt
Saewring Stangth
'A high-rssoution facing poruett a man realitic photogrophy
"Donna Tortt's <he Goldfochsijp Scores Film Adoptation
Swewing Staeggh
Suewrigg Staungh
AA highresalution fiont portrott recleac photography
"hhtts Messina Tolks Star Alongskle Ben Affleck <D<Live By Mightrjo
Strength
high-esolutonn facing portest man ... photogrophy
DuVerny
Ponthercip
Stewing Stnengh
Sheering Shaugth
"A highresolution fiont facig portratt of wamon, ..., reolssi photography"
"Aaron Paul to Play Luke Skywaker at Th-The Empire Strkes Back"
(a) +Old (Gen.)
(b) +-Old (Mem.)
Figure 17: Image editing via single-layer representation steering. We follow the setup of Figure M
but extract and apply the steering vector using only layer.
strengths. Sampling is performed with 40 total generation steps, where representation steering is applied during the final 20 steps. All experiments use classifier-free guidance (CFG) scale of 3.5.
C.4 EXPLORATION ON STEERING-BASE BASED IMAGE EDITING
In the main body of the paper, show that simple representation-bate based steering method enables effective image editing. More importantly, the editing outcomes differ systematically between gen- eralized and memorized images. In this subsection, evaluate the robustness of this method.
23

---

Published as a conference paper at ICLR 2026
Stewring Strengh
Swewing Streggh
*A hightessootion intenior photugroph bedroom...."
"A high-resoktionn prodect photograph of stylsh lecther hondbag
(a) +ddar
(b) Red
Figure 18: Image editing on SD 3.5. We follow the setup of Figure D using more recent DiT-based
Stable Diffusion 3.5 model for image editing.
Using fewer layers. As described in Appendix the steering results in Figure are obtained by extracting and applying representation addition across layers of the network. As an ablation, we find that the method does not require such depth: even a single layer sufficient. To illustrate this, we use up blocks. 1 resnets... 1 for both extraction and application, and present the results in Figure IA. The outputs closely match those the main paper, and the distinction between memorized and generalized examples remains evident.
Stable Diffusion 3.5. The representation space in this architecture is more elu- sive, likely due to components such Adaptive LayerNorm. To investigate this, we applied representation steering using layers transformer bocnmerr__looo wol.blloccooo Blockk... norml trransformer_blocks.. norm
transformer cttpcmeer_llcccce 12. norml
transformer blocks. 13. 1t..ckss.3...mml1. and
We generated 200 reference images to extract repre- sentations for each task. Sampling performed with 40 total generation steps, with steering applied during steps 35-30. All experiments utilized CFG scale of 4.5. These results are visual- ized in Figure IT
(transformer www.kl..........o....oo... normi
We note that do not intend to compete with existing outstanding steering methods, several such approaches have already demonstrated impressive empirical success (Hertz et all 2023: Zhang et all 2022: Gandıkota
2024; Kadkhodaae all 20245; Chen all 20246) Rather, our focus is on showing how steering reveals the dual relationship between representation structure and generation behavior. In particular, when the model generalizes, representations form compositional and interpretable spaces, enabling continuous and controllable edits.
D DEFERRED PROOFS
D.1 PROOF OF LEMMA D
Lemma D.1 (Global minimizers of Regularized LAE), Consider the regularized p-neuron LAE objective with W1, W2 dxx;
Σx 22. Wi = W&W X X no W. W.W.T. 1P + X(IWI 12
where X = (DI.... I.n) and S = XX = UAUT, Assume X' < Ap where \1 2 2 Au are the eigenvalues of S. Then every global minimizer has the form
W WT ( 2 A A - NA
(12)
where U(p) contains the top-p eigenvectors, Aip) the corresponding eigenvalues, and O 12pxp is any orthogonal matrix.
Proof. (0) Idea. Set = W W, and replace the separate Frobenius penalties with a nuclear
norm via minw, Wa: Wa W.W.T = W W2 [ A|]- Rotate to the S-basis and pinch to
24

---

Published as a conference paper at ICLR 2026
diagonalize, yielding d decoupled ID convex problems with solutions af (xptno>) + Keep the top p directions (largest X;), then factor A optimally obtain (I2).
(1) Reduction to a convex objective in A -W W For any with rank () S p.
min Wi.Wa: W.W, =A (1 + W2 (p)) |||
Hence
min 4x (W2, Wi) = min Aggexa rank(A)st
(IAX- - X + no² ||ll + 2X 11 =: min F(A),
Wi.Ws
A
where F is convex in A (the rank constraint is nonconvex).
(2) Diagonalization in the S-basis. Let = ÄU with = T Using
AA X l Tr(ASAT) 2T((AS) + Tr(S),
we obtain
F(A) = TH(AAA)) -2 Σ X.ā + no | + 2X | Tr(A).
=E,- E, 0
Zeroing the off--iaaonal entries of weakly decreases the quadratic terms and does not increase the
nuclear norm (pinching (Bhatia 2013) Thus a minimizer can be chosen diagonal in the U-basis:
A = U diag(a1..... a.) U
(3) Scalar decoupling and positivity. With A diagonal above,
F(A)=
1:(1 a()=-a))2 no + 2X 2x|]ss]] + const.
For X; 2 0, negatives are suboptimal (replacing a by |a| decreases the first term), so we minimize 0g 0:
A X aj = A; + nO
(4) Rank-p constraint and form of the minimizer. Enforcing rank(A) (A) S keeps the indices with largest A₂ (equivalently, largest unconstrained a;)) and sets the rest to Writing ot sf on this set,
s; = (1+ no x-1) no²x=1)
=1
and
# (1 (I- XA (p)
1
W = WT = U() diag(s;) 0 = =U(n) (I= Ap))
OT.
with any orthogonal 0 E RPxp, This matches (I2). (All inverses//quare--ooos are taken entrywise on (p)-)
Remark (large X' or degenerate S). some X; < X' (including X; = 0), then the unconstrained coefficients a; = (>==>)) vanish on those indices. In that case, keep the largest indices with X; X' (the rank may drop below p if fewer exist), and the same formulas apply entrywise on the retained cigenvalues; any remaining columns can be set to zero and O arbitrary.
D.2 PROOF OF THEOREM &R
Definition D.1 ((a, Separabbbyy of Training Data). Suppose the training dataset D can be par- titioned into M. clusters X [X.,..., XM), where X [ [Fy. ,
C IRd has mean == Kk.j- We say the dataset is (a,, -)-spababble for some a E (0, 1) and B < O. Tt
Tbll2
(Ø Tt)
< a for all k.j.
S 3 for all k # l. T ll | ll2
25

---

Published as a conference paper at ICLR 2026
Theorem D.2 (Restatement of Theorem (_) Assume (a, sseparabbyl with < 0 and nonde-
generate means ming @xxl2 > Let =k=1 Tik and define
W W = (Wx.
Wxa).
For each k, let X U be the digo-ddecopooiition and let U(!h) collect the top pk
eigenvectors (with eigenvalues A (1))). Assume nd < Amia (A (pz)), that the block solutions below are well-defined (real). Then there exist absolute constants C. c 0 and a margin 7 > (defined
(A
explicitly below, depending only a, B, b, {pu). and the block scalings, and independent of the noise level) such that, for all
(W2, W1) E BB = (1 W I[1 Wi S {6} and all > 0,
can decompose the DAE loss into LAE losses introduced in Lemma D.
M
<().0,)) s c( +
Lx(W2, Wi)
Σ Lx, (W2,()), WI,00) + 6(6,0,7), k=1
(13)
where Lx2 is the LAE objective in Lemma D for cluster k with noise weight ngo² and weight decay X' nd. Moreover, each block is minimized by
W 2.kk) = Wi.00) = Wx. == -U (I + + (pL)
(I- nA (px) 11
for some orthogonal Og Consequently (W2. Wi) is local minimizer.
Furthermore, the constructed minimizer is close to an actual minimizer: On B5 if we fix the ReLU masks be those induced by (W2, Wi) at the center (see Steps (1)(2) below), then the map
1 (W2, W1)
n Exx,
is mo-strongly corvex around (W2, Wf) with
mo 2 co + n2),
for numerical constant co > 0 independent of (8, a, 7). Therefore, any local minimizer (W2, W1) of the full DAE loss inside BB obeys
25(0, 0,7)
2C (0² + n2)
-1/2 ( /0"
1/2
(22 Wi) (WZ, WT) S
ma
(14)
In particular, 6/1 - O and a/7 0 (with nd 0 fixed), the right-hand side is o(1), giving explicit o(1) control on the distance between the constructed solution and the actual local minimizer.
Proof. Let fws.W, (z) W2[ [T,Ta 2]+ and write Wi [Wx..... Wx.>. so the columns are partitioned into M blocks. We also use [v] := [- entrywise.
(0) Idea. We compare the nonlinear DAE fws.w, with its mask-fixed linearized counterpart ALAF W. The intended behavior block & is active on X2 with positive margin, while all other w. blocks are inactive with negative margin. For small (a, (), the ReLU masks are preserved with high probability; concretely.
fw..w (X + oE) = W2 W X1 0E1....
XM + JEM)]
W (X1,
(W2.0)
W2(M))
WTin (XMr + +ow (15)
CLAB JW..W, (X + tee)
26

---

Published as a conference paper at ICLR 2026
With masks fixed to the "correct" ones (block i Xs, others off), the network reduces to linear map
LAAE W, (z) (2,(k) W 1.(k)
E
Le., each cluster is reconstructed by small number of neurons in its corresponding block. Equiv- alently. writing A := (2, W 1.(4) and A == blkdiag( (A,aaa((A..... Al...., AM), we have Sw..w. (X OE) = A(X + oe). With fixed masks, the loss decouples into M LAE problems and becomes solvable.
S (1) Masks and margins the block center (no noise). Write Wx, = =U(( O with Se 18..,............... > 0. Let 8min = mink,r 8k,r and smax = maxa,r 8k,r, and choose O2 so that
0 U (Px) 1 aell2
VDE
Pm.
Since
X X = nh & +
Σ (o (xF(. - Txx),
the within-cluster tightness (a) implies the "residual" term has spectral norm
Σ [xx. (x)( - Txx)
[ke Tx s ny a² a
Therefore H is well aligned with the top eigenspace of X In particular, there exists Spruj (a) E (0, such that
| | æxll2
2 Cproj(a).
||0,1|2
(One explicit choice.) Let u 1 be the top eigenvector of X (so uR,1 E span((((u)) for any Pk 2 1). A Davis--ahan/Wedin--a bound gives
a²
(un,1, Rx)
sin (un,1, 2x) 5
V
1 a
A ||2
I-ax
hence one may take Cproj (a) == v (a²/(1 - )²))² (for < 1).
Hence, for any Xx and any column w of Wi.(k)>
(w) (, a) 2 |
&min
Spooj(a) VPv
Smax))))
For any L * and any unit vector u span(U((e)), (a, B) separrbbbyl yields (u, ax) S a. Therefore, for any column wā,r-
| 2
(win x) < | Smax (3 2a) < - || maax
provided a is sufficiently small compared to | (absorbed into constants below). Define the margin
min | 2| min
Smin
(prci(a)
mmax
Smaz
3
0.
(16)
Then, on X₁, every unit block has pre-activation 2 and every unit of A has pre-activation
<-7
(2) Mask stability with noise E, and loss error. Fix a noise draw E = ".,,)..,,,,), and set
e(e) = fw,.w, X ae)
LLAE Jw.w, (X +oe) (el e,] E Rd
Any right-orthogonal choice of O₂ yields the same objective value; the choice above maximizes the first- order margin and is convenient for the mask analysis.
27

---

Published as a conference paper at ICLR 2026
where, for sample Ti from the i-th cluster, the deviation is
Eig
Σ
W2,10)
W (oy + oe..))
W2.60 W
(a,, + te
(17)
tFi
off-block, should be
on-block, should be
Intuitively, 6i
0 unless some pre-activation crosses the margin. Moreover,
||e(e) P Σ |e
Loss difference. For a fixed noise realization e, define
Le W1) == [w.. (X + +o) x11 + ^(1 A(1 + 1|
and analogously LLAF with fL.... Writing a == fw..w, (X + oE) - X and b = cLAF oæ) X (so a b = e(E)),
J, (X
1
||al |6| 1
ILe CLAB
1a + b, b)
71
n (Ial]rr [[b|l) ||e())]lr
1
$ n
[|]b|] [|{())]le) [|e(s)]le,
where the last line uses ||a||r & |b|| p + ||({)]] Thus remains to bound E e(s) and B|| r.
We further simplify ||b|| by splitting out the noise. Denote A& := W2,0k) W 1.(k) and A :
blkdiag( AM)
LLAE W (X + oε) X| = |A(X + Oε) X S (A I) + a || |||||
Taking expectations and using Cauchy-Schwaad with E[[e|? = dn reduces the problem to bounding {)
Bounding (l) Fix column eip The entries in [W]. (e) + oew are rectified Gaus-
sians. By the margin argument in Step (1), at the center (Wg, Wi) we have W (, I --y1 for + and W 1.(6) T4 - +.1.5
Thus it suffices to control the first and second moments of rectified Gaussian whose mean is separated from 0 by y. Let Z ~ _N with < -y. A standard Gaussian tail bound (Mills ratio/ Chernoff) implies
82 S e 1120))
22
E S 82 -12//02
82
E E
(2/ 7
4
112
72
and the same bounds hold for within-block terms with [Z]_ Z|+
Now, generic off-block pre-activation (an entry in (I7) has the form
Z = w T x + tww
EZ
S -
Var(Z) =
==2l|ul
Let
M
K max W1,r 12,
L3
W 2,0)
2
opp
p p:= D j=1
Pj-
j=1
Using (22(e)ell2 < W| flop |ll|l and the second-moment bound above,
E Ele, < 22 -- L2p.
A
OOn Bs, the pre-activations shift by most 0(8) (absorbed into the 8/7 term in the final bound), so may treat the -y (off-block) 2 +y (on-block) up constants.
28

---

Published as a conference paper at ICLR 2026
Since ||e())]13 Eラ= 1ley1l2. we obtain
o'R²t ( T
E Blee(e) Ee
go
Final plug-in. Let == blkdiag (A, Am), LA = AAlops and BLAE :=
The
preceding bounds imply
CLAP (W2, Cx Wi)| S of -
Cx(W2, W1)
uniformly on B8 for some absolute constants C, > 0. (Here 5/7 accounts for deterministic mask changes across Bs, while the exponential term accounts for noise-induced sign flips.)
(3) Expectation yields the LAE loss. For ε ~ NO 1) and A == W2,(k) W. 1.(k)*
A (a + ae) x= = | - 2212 + a0 |
Summing over æ E XR. averaging by n, and adding weight decay gives
M
CLAF
W2 W1) = E2 | |LAA
1 n.
Ex ( 22,0), WL.0)),
Ex
using X' = nd and Σx W 1 W P for i = 1,2.
(4) Block solutions and distance to strict minimizer. By Lemma D each block is minimized by W2,0) Wi,(k) = Wx,; concatenating blocks yields (Wa, WT), which minimize the leading LAE term in (IL)). The leading term E. E. is quadratic in (W2, W1) on B3 (with masks fixed). Its Hessian (Wa WF) equals the Hessian of the quadratic reconstruction term plus 2nA from weight decay, together with the positive-semidefiii curvature from A By continuity of the Hessian, this yields uniform lower bound
Σ
Cx.)
= mo I neighborhood of (W2, Wi).
mo 2 colo² + nd),
for a numerical > 0 independent of (6, (,) Hence, for any local minimizer (W2, Wi) of the full DAE loss in Bs.
mh0
Wi) (Wi, WIll & (W, Wi) Lx (W2, Wi) < a(0,a,7).
which yields (L4). The right-hand side is o(1) whenever 8/7 -> 0 and a/7 + 0, completing the proof.
L
D.3 PROOF OF COROLLARY LTO.
Corollary D.3 (Restatement of Cor. the separability condition in Definition T hidden units, weight decay X O, and imput noise level a 0. Then, by Theorem (applied to singleton clusters), there exists a local minimizer of the form
Assume the dataset X [x: ...., En, Rdd is satisfy
Consider an overparameterized ReLU DAE with p 2 n
x
nd
W = W = (r131
Tn
W mem:
|
2 x3
(18)
(The trailing (p n) columns are zero; see also Corollary D.4 on -mo-ooothhssss.) Moreover, for X + 0 this solution attains small empirical loss independent of d:
Lx.
memo W ) S
< 0 Vivisisn
x
29

---

Published as a conference paper at ICLR 2026
Proof. (0) Optimal weights align with data, Apply Theorem B.1 with clusters X₁ { of size
ng 1. The block solution from Thm. B. now yields xx Fk
#t with Th V || +o Tæall2
my nn
so the corresponding column of W (and W2) equals r2 with T sk s*/l|ax ll2, giving (I8).
Since p 2 n, we may set the remaining (p n) columns to zero without affecting the network output. Other equivalent parametrizations (e.g.. duplicating columns and rescaling) have larger Ex-
smoothness; our choice the sparsest among these.
(1) Empirical loss bound (case À - 0). By Theorem B.I and Lemma D.I. with singleton clusters and À = 0, the expected denoising loss decouples over samples and, for each i,
0² 12:13
< 12
min [(1 (0l-))211 | o²a²] aE(0,1]
attained at of
0² 1
2 +
Averaging over gives the stated bound, which strietly less than x² and independent the ambient dimension d.
D.4 PROOF OF SMOOTHNESS WITH RESPECT TO THE INFINITY NORM
Corollary D.4 (Sparse solution has the smoothest to local landscape). At the memorized, sparse
solution W₂ W1= Wmem from Corollary B.A the loss decomposes singleton clusters as
L., (22 Wi) | =:i12 o² A211 + 2nd 2nA 1
A W.W, a.,..aa.
With masks frozen (singleton case), the Hessian w.r.t W1 is block diagonal:
W1) blkdiag H(x1) + XI, AI..... (xn) + XI, AI AL)
p-n blocks
where each active block has rank-1 plus diagonal (xx,) ag ar a, > 0 (smooth in a, r, ||ac]12))
Consequently, the Loo Lipschitz constant of the gradient at Weem is
Loc = V., (W mem, W Wmem)) lmem))loo-sss max
[[(x)) + oo }}.
max Isis
Among all equivalent local minima obtained by orthogonal re-mixing within the active span (i.e.. W1 + W1Q Q and W2 + W2Q with block-orthogonal Q that preserves masks and the LAE optimum), the choice Wmem minimizes L and hence yields the smoothest local landscape in Eoo-
Proof. (1) Block structure. Freezing masks singleton clusters forces second-order decoupling across columns, giving the block-diagonal Hessian displayed above. A direct differentiation of the decomposed objective shows each active block equals a; F AI for some a; > 0.
(2) Bound loo via (1, 1)-morm of Hessian. Recall || 000 max =1 | ooo ||M [|=-;; for symmetric blocks this equals the maximum absolute column sum. Since the Hessian block diagonal.
Ww Lx 00-+00 max max ax + AZ 1 }}.
For rank-1 matrix, the (1, operator norm is the max column sum: ||o: ب. Il,-, az ||æ:||ı || Adding AI increases each (absolute) column sum by at most X, hence
||a, I. + AI a oo = ||a x. + AZ S a, ||0,||a || + X,
which yields the claimed expression for Loc.
(3) Minimality under orthogonal re-mixing. Let an equivalent optimum obtained by block- orthogonal mixing that preserves masks. Each active block is conjugated to Q, (a; æ.,.. xi) While eigenvalues are invariant, (1, 1) (hence + oc) norms are sensitive to densification. The memorized alignment keeps the block rank-1 plus diagonal along I;, which minimizes absolute column/row sums; mixing spreads mass across coordinates and (weakly) increases the max colum- n/row sum. Therefore the memorized choice minimizes Las among all such equivalents (Xie and Li. 2024).
30

---

Published as a conference paper at ICLR 2026
D.5 PROOF OF COROLLARY 1
Corollary D.5 (Restatement of Corollary (.S.). Under the setup of Theorem 3.1 assume the training
data satisfy the separability condition in Definition H. If the DAE in (5) is under-parameterized
with p 2k=1 Pk < n, then there exists local minimizer of (6) of the form
W -W (Wx, Wx.
Wxx)= W Wgens
where each block Wx. x, E Rdxxs consists of the leading principal components of X as (8). and Wx. WX. concentrates to the rank-Pk optimal denoiser for Nu Ex):
Wx. Wx, + [S
PE S) + - runk
where Sy is introduced (IT) and Pk is the weight the k-th mixture component. Moreover, when X + O, the expected test loss (generalization error) satisfies
eig, (Sk) 0
Ck Pk x² 1ik
ExxnolCx(W5) WT)]
Pk
(eig, (Sk) 02)2
+
Σ
eig, +
JSPa
52Pk
where Cs > 0 depends on and spectral properties of Sk, and eig, (Sx) denotes the j-th eigenvalue of Sy (independent of d).
Proof. Notation. For a PSD matrix A, define f(A) = (A 2 (AA+o21)) and let fpe (A) be
f(A) truncated to its top Ps eigendirections. Set
6p. == eigp, (Sx) ei > 0,
Teff,k == Tr((St)(([S
All high-probability statements with respect to the draw of Xx; C' 0 denotes universal constant.
(1) Plug in Theorem 3.I. By Theorem 1 in neighborhood of a block- point the DAE loss decouples across clusters, and each block solves a regularized LAE on Xx with effective noise weight nya² and decay nλ. Hence the learned denoiser on cluster k is Dr == Wx. WX.-
(2) Concentration to the population denoiser. For Gaussian clusters, 1 Xx X concentrates around S The LAE solution depends smoothly on its Gram matrix; combining this with Davis- Kahan perturbation yields
ns
| Sn, (Sa)ll
( (
1 X Sa11E n
Moreover, with probability at least et
5 S || / x0)
1
X
Combining the last two displays gives the explicit deviation
Dr (St) < ||Skllop ( 2
Pe (FeH R2
(w.h.p.).
(19)
(3) Population rank-p; DAE risk. Let o' ~ N( Σx) and ~ NOO, be independent. Define
(p)) = E [11 (2x) + =2'1)
Diagonalizing S₁ and using f(A) A(A 021)= gives
eig, 04
(po
Σ
Σ (St). j>pb
(eig, (St) + 02)²
iSPE
(4) Generalization loss on cluster Let Di = Sp. (Sx). Then
E B||| Da(a' +ae) -ar" x 2/12_ L (px) Ty Sx (ix D) < L (px) S| 11 Di112.
31

---

Published as a conference paper at ICLR 2026
Plug (19) into the last inequality to obtain, with probability least 1 e-
) (Teft +
([IIDx(2' + oe) 2'1 P + C' C'I + )
(20)
n
This makes the 1/ng rate and its dependence on o, 6pg and the spectrum of S₁ explicit.
(5) From clusters to the mixture (population) bound. Let P&t linearity of expectation,
Σk Pa N Dz). By
K Σ Ps Elj (x + OE) - =x11].
Ex~per ( Wi)
k=1
Apply (20) to each term and take a union bound over k 1,..., K by choosing = log (gg(K/)) With probability least 1 7).
ppx (Teff + log(K/n))
Ex~pal (W Wi)] I
Σ Pk C (px) + C|S cpp
Jmm
Absorbing Feff. and log( (//()))) into a cluster-dependent constant C. yields exactly the last term in the corollary statement. (If one prefers bound in expectation without failure probability, the same inequality holds with the right-hand side plus an ((n) additive term by integrating the tail; choosing r makes this negligible.)
D.6 PROOF OF COROLLARY 4
Corollary D.6 (Restatement of Corollary Let X for l = 1,.., X; (æz,.... xz) is rank 1, and Xm+1..... X K contain distinct empirical
[x....., Xx] satisfy Definition B. where
samples from the remaining Gaussian modes. Suppose a ReLU DAE trained with weight decay and input noise 0. Then there exists local minimizer of the form
W Wi (riæ1
Tmmm
W WXm+1
W Wxk).
where the first m columns memorize the duplicated clusters (as in Cor, B.2)), and the remaining blocks W x, implement generalization on the nondegenerate clusters (as in Cor. B.3).
Proof. The proof follows by combining Cor. and Cor. k* and using the block-wise structure guaranteed by Thm. B.I.l In particular, Thm. allows us to treat each cluster X₁ independently at local minimizer.
For the first SJ S m clusters, X, is rank and Cor. implies that the corresponding columns of WT and W simply scaled data vectors ryæy. For the remaining clusters Xm+1.... .x. Cor. BR yields the blocks Wx2 that implement generalization on the nondegenerate modes. Stack- ing these columns and blocks gives precisely the stated form of Wi W₁.
This corollary illustrates the local adaptivity of ReLU DAE models: they can memorize duplicated subsets while simultaneously generalizing well-sampled regions of the data distribution.
32


Reviews:
[{"review_id": "MU4BT4MiEz", "reviewer": "Reviewer_Eg8y", "summary": "This paper investigates the mechanisms of memorization and generalization in diffusion models, starting with a theoretical analysis based on a two-layer ReLU denoising autoencoder.\n\nThe claim is that mem. or gen. is determined by the learned representations: (1) memorization corresponds to learning spiky, sample-specific representations and weights that store the training data sparsely, when model parameters $p$ are larger than data size $n$ ($p\\geq n$), and (2) generalization arises when the model learns balanced representations that capture local data statistics, when it is under-parameterized ($p \\ll n$).\n\nBased on the analysis, the authors introduce two practical applications: a highly efficient and effective method for detecting memorized content by measuring the spikiness of representations, and a training-free image editing technique based on representation steering, which demonstrates that generalized samples are more editable than memorized ones. The paper also validates these methods on some real-world models, including EDM, DiT and SD1.4.", "questions": "Please refer to the weaknesses regarding the image editing approach and experiments.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- The paper presents a valuable contribution by proposing a more fundamental, representation-centric framework to explain the behaviors of memorization and generalization in diffusion models.\n- The quality of the theoretical analysis is high, and the findings are shown to be consistent with observations in real-world models, as validated through two practical applications.\n- The memorization detection method demonstrates high performance and broad applicability across different models and datasets.\n- The observation that generalized samples are more steerable, while memorized samples exhibit brittle editing behavior, provides a novel and interesting insight.", "weaknesses": "- The theoretical framework is built upon a two-layer DAE. While empirical results suggest the conclusions hold more broadly, I do not think it is very clear why the findings based on the linear projection dimension $p$ vs. data size $n$ ($p\\geq n$ or $p \\ll n$) should transfer to deep, multi-layered UNet / DiT.\n- The image editing experiments are limited in scope and somewhat unclear in their details.\n  - The representation steering method is only demonstrated on Stable Diffusion 1.4, which is somewhat old in this area. Its effectiveness and the observed behavior have not been tested on more recent models like DiT / MM-DiT based text-to-image models.\n  - The paper lacks a justification for how the \"encoder\" $g_\\theta$ and \"decoder\" $h_\\theta$ are determined (Line 461, Lines 990-992). The appendix specifies that features are extracted from 6 distinct layers in up_blocks.0 and up_blocks.1, but how these specific layers are selected is not provided. Additionally, I assume they form a collection of features with the size $100\\times C\\times H\\times W \\times 6$. Are these features averaged, or is the steering performed on 6 layers in parallel?\n- The image editing approach requires generating 100 reference images to compute a mean representation for the target concept, which makes it inapplicable to image-guided editing scenarios using a few (<10) provided reference images. Furthermore, will the reference image generation procedure affect the editing quality? For example, how the editing process would be affected if the generated reference samples were themselves memorized or of low quality.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Sh7TjqvyuR", "reviewer": "Reviewer_NqY1", "summary": "This paper aims to understand memorisation and generalisation of diffusion models from the representation perspective. Specifically, the authors built a theoretical framework in a two-layer ReLU diffusion model, which is more manageable in the theoretical sense. Firstly, the authors analysed the optimal solution of the above simplified diffusion model, which is over-parameterized. Through the analysis, they found that spiky representations happened, which are a a signal of memorisation. Secondly, the authors analysed the scenario of under-parameterisation, and showed that balanced representation as a signature of generalisation. Finally, the authors demonstrate the theoretical analysis can result in real-world impact, such as better memorisation detection metric and a steering approach for image editing.", "questions": "See the weakness.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "This paper benefits from both theoretical analysis and further empirical impact. Especially, through the theoretical analysis, the authors showed that data representations (spikiness or balance) could be signals for memorisation/generalisation. Such signals could be used for memorisation detection, and have high accuracy than previous metrics and meanwhile is prompt-free.", "weaknesses": "I have the following concerns or questions, which may need authors' clarifications. Also please correct me if  I am wrong.\n\n1. In case 1 of over-parameterisation, the authors found that with the optimal solution, the representations of a **single training sample** exhibits spikiness (in line 269). I am wondering if we input a different sample (not in training data) to such a neural network, whether the learned representations become a zero vector. \n\n2. In case 2 of under-parameterisation, the authors found that with the optimal solutions, the representations of a **single training sample** exhibits balance (in line 369), hence smaller std. I am also wondering what would happen if we input a different sample (not in training data, or a generalised sample). \n\n3. It seems that the authors consider two neural networks, one could memorise all samples, one could generalise new samples. However, for a trained diffusion model, it could both memorise and generalise. Why is the spikiness in representations an indicator for memorisation? \n\n4. The authors mainly discuss about optimal solutions for diffusion model. However, [1,2,3] show that as long as the number of training samples is finite, whether diffusion model is over-parameterised or not, there exists a theoretical optimum which could always generate memorised training data. Can you clarify the connection between such a theoretical optimum and the optimal solution shown in this paper? Is this because the model family of two-layer ReLU network cannot represent the theoretical optimum?\n\nReferences:\\\n[1] Yi et al. On the generalization of diffusion model. 2023.\\\n[2] Gu et al. On memorization in diffusion models. 2023.\\\n[3] Kamb et al. An analytic theory of creativity in convolutional diffusion models. ICML 2025.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "i8FxEA4D8j", "reviewer": "Reviewer_jzL9", "summary": "The authors present a study on diffusion model generalization. They use a two-layer Relu network in a denoising autoencoder framework. In this framework, and assuming a type of cluster-separability, the authors present a theorem relates the optimal weights to the data clusters. They then proceed to derive properties for the low and high data limits, demonstrating they relate to memoization and generalization, and explain how these two domains have different weight properties. The authors then demonstrate these properties exist in pretrained diffusion models and how they can explain limitations of \"steering\" where the data is sparse.", "questions": "- (see also weaknesses)\n- Do the authors use sigma in the denoiser network? It is common practice for sigma to be part of the denoising representation. Would adding versus omitting it change any results?\n- Can the authors confirm that in the low data case, the network's output matches the optimal empirical denoiser? It seems like this would be the case, but I did not find that back in the paper. It would be useful to the reader to make a statement about that. \n- Related to the last question: is the reason that the under-parametrized networks do not output the optimal empirical denoiser that the network does not have enough capacity to learn it?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "Overall I very much enjoyed reading this paper, here are some specific points of strength:\n- The paper is clearly organized, and concepts are well explained and presented.\n- I very much like the paper is centered around rigorous theory, but makes it approachable in presentation, and demonstrates the strengths and downstream applications their theory through practical experiments with real datasets and models.\n- This is a paper that makes a strong contribution toward understanding diffusion model generalization fundamentals, which is still quite poorly understood. The authors not only deliver great insight for people working on understanding this generaliation, but also make it approachable for practitioners working on downstream tasks", "weaknesses": "- There are a few things about definition 3.1 that seem odd to me. What 3.1 says is that the cluster means must be separated by at least some angle related to beta. This seems like a very specific type of clustering. It excludes for example clustering in the same direction (imagine a multi-modal gaussian with a sequence of modes in one dimension). Perhaps this is a common assumption I am not aware of. Could the authors defend this choice? Is it necessary for the theorems?\n- It seems to me that depending on beta, the assumption the authors use sets some limitations on the ratio between the number of clusters and the data dimension. When I think about such clusters and data dimension in the image space, I would argue that the number of clusters is much larger than the data dimension. This would limit beta for such datasets, and because of that make theorem 3.1 less applicable since it assumes beta < 1. Can the authors comment?\n- A small comment here is that the \"margin\" gamma (line 184) does not seem to be defined in the text.\n- It seems surprising to me that the \"spikiness\" observation from Cor. 3.2 holds in more complex networks than two-layer Relu networks. While it seems like a perfectly reasonalbe intermediate state for the two-layer net, for more complex networks it seems like that would be much more complex. Beyond empirical results, can the authors give some intuition for why this would hold?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "XKWzQeNWbE", "reviewer": "Reviewer_6mi7", "summary": "In this work, authors study the representation space of diffusion models, showing especially for simplified scenario with 2-layered DAE that memorization can be attributed to the spikiness of activations. In particular, authors show that for overparameterized models examples are encoded in single neurons, while this is not the case for generalized underparametrized scenario.", "questions": "-", "limitations": "", "rating": 2, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 1, "strengths": "- Insightful formulation with ReLU DAE which clearly introduces the main claims of the paper\n- The theoretical contribution of the paper is clear and sound", "weaknesses": "- The paper's central observation—that memorization is linked to \"spiky\" neural activations—appears to be a rediscovery of the main finding from [1]. The authors propose using \"the standard deviation of intermediate features\" as a proxy for this spikiness. This metric is functionally identical to the z-score introduced in [1], which also identifies memorized examples by measuring the number of standard deviations an individual neuron's activation is from the mean.\n- The analysis is restricted to a comparison between two well-understood extremes: an overparameterized model that memorizes and an underparameterized model that generalizes. This setup does not address the more complex and realistic scenario where a single, large-scale model simultaneously memorizes some training examples while generalizing others.\n- In Section 3.1 there is an interesting example of memorization in overparametrized setup, where authors show that small DAE with large enough latent space (bigger than the dataset) can be optimized by memorizing individual samples (following block-wise structure). While this is a viable example, I am not sure that the conclusions are rigorous enough. I agree with comments presented in the first dot, but given the fact that the analysed solution is only the one of many local minimas, we cannot be sure that other minimas do not promote some generalization via neurons entanglement. \n- The paper makes the strong claim that “our findings show that the representation space is not a byproduct but a determining factor for generation.” This implies a causal relationship that the experiments do not support.\n- The conclusions on steering and editing representations are drawn from an insufficient sample size of just 8 qualitative examples. There is no quantitative evaluation to validate these claims.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)