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Feature Selective Model Collapse in Diffusion Models: Total Replacement versus Fixed-Budget Training
Hanna Malet, Gabriel Turinici
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Summary
This paper investigates model collapse in diffusion models under two training protocols: total replacement (last generation) and fixed-budget accumulation. The authors demonstrate that while total replacement leads to rapid and total dataset degradation, fixed-budget training results in partial collapse where 'fragile' features are lost quickly, but 'robust' features are preserved over long horizons. Theoretical analysis using linear-response models and stochastic recursion confirms these empirical findings, providing halving time formulas for feature decay.
Entities (18)
Relation Signals (10)
Total Replacement Protocol → causes → Model Collapse
confidence 95% · replacing real data with synthetic data causes collapse
CIFAR-10 → usedin → Empirical Evaluation
confidence 95% · Experiments on... CIFAR-10... show that replacement protocol degrades dataset rapidly
MNIST → usedin → Empirical Evaluation
confidence 95% · Experiments on... MNIST... show that replacement protocol degrades dataset rapidly
Diffusion Models → uses → DDPM
confidence 95% · We train diffusion models with DDPM training
Diffusion Models → uses → DDIM
confidence 95% · and DDIM sampling
Fixed-Budget Protocol → mitigates → Total Model Collapse
confidence 92% · fixed budget degrades only partially, sparing some features
Total Replacement Protocol → eliminates → Fragile Features
confidence 90% · some features will be fragile and lost within a few generations for both protocols
Fixed-Budget Protocol → preserves → Robust features
confidence 90% · some will be robust and preserved over practically unbounded horizons under the fixed budget protocol
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Abstract
Abstract:Model collapse arises when generative models are trained on synthetic data produced by earlier models. The phenomenon has attracted considerable attention because of its societal and technical implications. However, previous studies have reached seemingly contradictory conclusions: replacing real data with synthetic data causes collapse (Shumailov et al.), yet accumulating real data alongside synthetic data can prevent it. For diffusion models, we study an intermediate regime typical of finite-budget pipelines: all past datasets and the real data are kept, but each new model is trained on a fixed-size sample from this growing pool, so the real fraction vanishes without any data being removed. Experiments on a 2D spiral dataset as well as the image benchmarks (MNIST, Fashion-MNIST, and CIFAR-10) show that replacement protocol degrades dataset rapidly as in the literature, whereas the fixed budget degrades only partially, sparing some features. A linear-response model of the multi-generational parameter dynamics, analyzed by stochastic recursion, confirms that the two protocols differ: some features will be fragile and lost within a few generations for both protocols, while some will be robust and preserved over practically unbounded horizons under the fixed budget protocol.
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- Source: https://arxiv.org/abs/2610.01318v1
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Feature Selective Model Collapse in Diffusion Models: Total Replacement versus Fixed-Budget Training Hanna Malet Affiliation: CEREMADE CNRS Affiliation: Université Paris Dauphine - PSL Affiliation: Paris 75116 FRANCE Affiliation: & Affiliation: Cornell University Affiliation: Ithaca, NY, USA Gabriel Turinici Affiliation: CEREMADE CNRS Affiliation: Université Paris Dauphine - PSL Affiliation: Paris 75116 FRANCE Email: gabriel.turinici@dauphine.fr Affiliation: https://turinici.com Abstract Model collapse arises when generative models are trained on synthetic data produced by earlier models. The phenomenon has attracted considerable attention because of its societal and technical implications. However, previous studies have reached seemingly contradictory conclusions: replacing real data with synthetic data causes collapse (Shumailov et al.), yet accumulating real data alongside synthetic data can prevent it. For diffusion models, we study an intermediate regime typical of finite-budget pipelines: all past datasets and the real data are kept, but each new model is trained on a fixed-size sample from this growing pool, so the real fraction vanishes without any data being removed. Experiments on a 2D spiral dataset as well as the image benchmarks (MNIST, Fashion-MNIST, and CIFAR-10) show that replacement protocol degrades dataset rapidly as in the literature, whereas the fixed budget degrades only partially, sparing some features. A linear-response model of the multi-generational parameter dynamics, analyzed by stochastic recursion, confirms that the two protocols differ: some features will be fragile and lost within a few generations for both protocols, while some will be robust and preserved over practically unbounded horizons under the fixed budget protocol. 1 Introduction Generative models are increasingly trained on data that already contains the output of earlier models. As synthetic content becomes ubiquitous and training pipelines rely on ever larger datasets collected over long periods of time, understanding the long-term consequences of such recursive training has become both a practical and a theoretical necessity with societal and technical implications. Shumailov et al. (2024) (see also Dohmatob et al. (2024b)) showed that this feedback loop degrades successive models until the tails of the original distribution disappear, a phenomenon they called model collapse. Their main protocol is one of recursive replacement: each generation is trained on the output of the previous one, and the real dataset is gone after the first step. Subsequent work reached conclusions that appear to contradict this one. Gerstgrasser et al. (2024) argued that if synthetic data accumulate alongside the real data instead of replacing it collapse no longer occurs. Seddik et al. (2024) made collapse depend on the synthetic-to-real ratio rather than on the recursion itself, and Barzilai and Shamir (2026) showed that for maximum likelihood estimation collapse can be avoided even when the fraction of real data vanishes, while also exhibiting cases where it cannot. Zhu et al. (2025) approach the question from the mitigation side, editing human-produced text rather than regenerating it. Closer to us in model class, Yoon et al. (2025) and Bohacek and Farid (2025) document collapse in finetuned Stable Diffusion over short chains. We investigate a recursive-training protocol that is intermediate between those of Shumailov et al. (2024) and Gerstgrasser et al. (2024). As in the latter, synthetic data are accumulated rather than replacing the original dataset. However, unlike that setting, here the training-set size is kept fixed: at each generation, a constant number of examples is resampled from the growing data pool. This regime, which naturally arises under a finite training budget, was introduced as accumulate-subsample by Kazdan et al. (2025) and examined empirically for diffusion models by Shi et al. (2025); we complement these works with a per-feature theoretical analysis. Our contributions are both empirical and theoretical. On the empirical side, we focus on the class of diffusion models that has received comparatively less attention in the model-collapse literature. We recursively train diffusion models over several generations, first on a toy 2D spiral distribution and then on MNIST, Fashion-MNIST & CIFAR-10. Under total replacement, we recover the rapid deterioration reported in previous work. Under accumulation with a fixed dataset size and constant training budget, the deterioration is only partial and affects specific features, leading to a behavior that differs both from complete collapse and from the bounded-error regime predicted under unbounded accumulation. On the theoretical side, we introduce a system of equations describing the multi-generational parameter dynamics of recursively trained diffusion models. Like Bertrand et al. (2024), we linearize the retraining map in parameter space; unlike their stability analysis, which requires a large enough fraction of real data, our model covers a real fraction vanishing as 1/(n+1)1/(n+1) and a random re-initialization at each generation. Using stochastic analysis, we establish rigorous convergence results that explain both the rapid degradation observed under total replacement and, in the fixed-budget accumulation regime, the heterogeneous evolution of reconstruction quality across different features. We also derive per-feature halving times which shows that recursive training first eliminates ’fragile’ features providing thus quantitative guidance on when real-data refreshes and related mitigation measures become necessary. A detailed positioning relative to prior work is presented in Appendix A. In short, the works differ less about the recursion itself than about how the training set is built at each step; the fixed training-set size removes the vanishing-noise-share weighting on which the protective effect of accumulation relies. 2 Methodology We train diffusion models with DDPM training (Ho et al., 2020) and DDIM sampling (Song et al., 2021). See appendix for architecture details. Each model is trained on a dataset and results in a new model called a "generation". Let D be the real dataset (also denoted 0D_0), d the number of model parameters and let Sn∈ℝdS_n ^d denote the generative model parameters at generation n; once SnS_n is obtained, a synthetic dataset of size ndn_d, denoted nD_n, is sampled using the SnS_n as model parameters. Two data protocols are used: the last generation protocol where generation n+1n+1 is trained on nD_n alone and the fixed budget protocol where generation n+1n+1 is trained on a sample of fixed size ndn_d drawn from the pool ∪1∪⋯∪nD _1∪·s _n; the pool grows but the training set does not, so the probability that a training example is real decreases with n although no data is ever discarded. This protocol can describe for instance the data accumulation on internet, each new model trains only on a part of available corpora. All models start from a set of parameters ZnZ_n sampled at random from some distribution ν. As datasets we use a toy 2D spiral dataset and then MNIST, Fashion-MNIST & CIFAR-10. Parameters We use nd∼104n_d 10^4 samples, nepochs∼100−1000n_epochs 100-1000 epochs and ngen∼20−100n_gen 20-100 generations, cf. Appedix for details. To sample from the model we use a Exponential Model average with parameter 0.80.8 that smooths model parameters. Metrics. To evaluate the model collapse for image datasets we use three metrics: Fréchet Inception Distance (FID), pixel-FID (same formula as FID but without using Inception features) and Wasserstein-2 metric, denoted 2W_2 at pixel level. For the simple 2D spiral dataset neither the Inception features nor the pixel-space Fréchet distance make sense so we use 2W_2 computed exactly by solving the discrete optimal transport problem; in this case the Fréchet distance is computed between the Gaussian approximations of the two clouds in ℝ2R^2. 3 Modeling the multi-generational dynamics and Diffusion Diffusion models are procedures that define two mappings: - thse ’training’ phase T that has as input a couple (initial value Z, distribution D) and as output a parameter denoted here S=Z,∈ℝdS=T_Z,D ^d - and an inverse mapping called ’sampling’ S that associates to a parameter S∈ℝdS ^d a distribution denoted SS_S. Both phases involves a neural network whose parameters are S∈ℝdS ^d. Note that in the following we will not make a distinction between the dataset D and the distribution it generates. Also note that we fixed the architecture of the neural network and the a priori distribution from which the sampling is performed to obtain SS_S. We will suppose in the following that Diffusion model hypothesis : for any dataset of interest there exists an unique (but unknown) model hypothesis : for any dataset of interest D there exists an unique (but unknown) parameter S∈ℝd such that =S. S ^d such that D=S_S. (1) Of course, the hypothesis 1 is too strong to be true in general but if the neural network capacity is large enough one can train a diffusion model to represent the dataset D with enough accuracy so we expect the error in (1) to be negligible. 3.1 The simplest Diffusion training equations To model the diffusion training we need to know how the neural network parameter S depends, through the training, on the training (target) dataset D and on the initialization Z. Note that because of the hypothesis (1) we can restrict ourselves to mappings involving only elements in ℝdR^d by replacing D by its representative T such that T=S_T=D. Our goal is not to describe the full nonlinear dependence of Strain=Z,=TS_train=T_Z,D=S_T on (Z,T)(Z,T), but rather to derive a crude first-order approximation in the spirit of Neural Tangent Kernel (NTK) Jacot et al. (2018) analyses. In particular, we seek a model that captures the leading-order effect of the initialization while aggregating all higher-order phenomena into a residual term. Observe first that when the initialization Z coincides with the target parameter T, the procedure is already at equilibrium and therefore if Z=TZ=T then Strain∼TS_train T with possible small oscillations due to noise terms. Hence, the map Φ:Z↦Strain :Z S_train (2) admits an almost ’fixed point’ at T in the sense that Φ(T)=T+ε (T)=T+ where ε is some noise taken here with zero average and independent of T. Assuming that Φ is sufficiently regular, a first-order Taylor expansion around y yields Strain−T=Φ(Z)−Φ(T)≈JT(Z−T), S_train-T= (Z)- (T)≈ J_T(Z-T), (3) where JTJ_T denotes the Jacobian of Φ evaluated at T. The diffusion dynamics are stochastic and only approximately captured by their linearization. Moreover, higher-order nonlinear effects are neglected in the above expansion. We therefore introduce an additive fluctuation term and write Strain−T=M(Z−T)+ε,ε∼μ,[ε]=0,Var[ε]<∞,S_train-T=M(Z-T)+ ,\ μ,\ E[ ]=0,\ Var[ ]<∞, (4) where ε is a centered random variable accounting for both the intrinsic diffusion noise and the approximation error induced by linearization. We assume it follows some centered distribution μ with finite second order moment. The matrix M should be interpreted as a local sensitivity operator describing how a perturbation of the initialization propagates to the final parameter. Under a local quadratic approximation of the discrepancy functional around T, the dominant second-order information is encoded by a Hessian-like operator. This motivates restricting M to be symmetric positive definite. Such a choice is consistent with locally stable gradient-driven dynamics and excludes purely rotational components that are not expected to appear at first order. Moreover, since the training is supposed to improve the initial quality we can assume that M is somehow a contraction which means that its spectrum is in ]0,1[]0,1[. Consequently, among simple first-order models, the linear-response ansatz: Strain−T=M(Z−T)+ε,M=MT,spectrum(M)∈]0,1[,ε∼μ,[ε]=0,Var[ε]<∞, S_train-T=M(Z-T)+ ,\ M=M^T,\ spectrum(M)∈]0,1[,\ μ,\ E[ ]=0,\ Var[ ]<∞, (5) appears as a natural approximation. The deterministic component captures the leading-order dependence on the initialization, while the noise term absorbs higher-order nonlinear effects and the stochasticity intrinsic to the diffusion procedure. 3.2 The simplest model for dataset evolution: Linearity with Respect to Data Mixtures The last ingredient of out model is a way to describe how the dataset evolves from one generation to the other and more precisely how its representative in ℝdR^d evolves. For the ”last generation” protocol (cf. Section 2), the dependence is easy to set: if the previous generation produces parameter S then the dataset will correspond to S (at first order). For the ”fixed budget” protocol we need to explain how the new dataset depends on the old one and on the newly generated data. We next consider a target distribution obtained as a mixture of two datasets i.e. a dataset taking λ percentage of data from dataset uS_u and the rest from vS_v. As distributions this mixture is formally denoted: λu+(1−λ)v,λ∈[0,1] _u+(1-λ)S_v,\ λ∈[0,1]. Within the same first-order approximation framework, we assume that the map S↦S _S is locally linear around the parameters of interest. Under this assumption, convex combinations of distributions are approximately represented by convex combinations in parameter space. Thus, in the same spirit as NTK-type linearizations, convex interpolation in data space induces, to first order, a convex interpolation in parameter space. and consequently, the parameter associated with the mixed dataset is approximated by λu+(1−λ)vλ u+(1-λ)v. More precisely, up to some negligible error, we will assume: λu+(1−λ)v=λu+(1−λ)v. _u+(1-λ)S_v=S_λ u+(1-λ)v. (6) 4 Theoretical results : convergence and large perturbation analysis We analyze here the convergence dynamics of the multi-generation diffusion models under the two data protocols. Denote Sn∈ℝdS_n ^d the parameters of the diffusion neural networks at generation n, Tn∈ℝdT_n ^d the parameters of the target distribution at generation n, ZnZ_n the initialization of the generation n+1n+1 and εn _n the noise at generation n+1n+1 training. Under the assumptions in Section 3 we obtain the following recursion: Training equation (both protocols) : Sn+1−Tn=M(Zn−Tn)+εn equation (both protocols) : S_n+1-T_n=M(Z_n-T_n)+ _n (7) εn∼μ i.i.d., ∀n:[εn]=0,Var[εn]<∞ _n μ i.i.d., \ \ ∀ n:E[ _n]=0,\ Var[ _n]<∞ Fixed budget protocol : (n+1)Tn+1=Tnn+Sn+1 Fixed budget protocol : (n+1)T_n+1=T_nn+S_n+1 (8) Last generation protocol : Tn+1=Sn+1. Last generation protocol : T_n+1=S_n+1. (9) Note that after a change of basis we can assume matrix M diagonal, all diagonal elements are in ]0,1[]0,1[; we will refer to this basis as a ’feature basis’. We state now the main convergence results, one for each protocol. To this end we need to choose a regime for the initialization of the diffusion training. We will therefore suppose that the initialization is non-biased, has finite variance and all generations are independently initialized i.e., Zn∼ν i.i.d., ∀n:[Zn]=0,Var[Zn]<∞. Z_n ν i.i.d., \ \ ∀ n:E[Z_n]=0,\ Var[Z_n]<∞. (10) Theorem 4.1. Under the last generation protocol (10), (7), (9) both the target parameter TnT_n and the model parameters SnS_n converge to some T∞T_∞ with (stationary) distribution πT∞LG _T_∞^LG with zero average but non-null limit variance. Moreover, for a feature eλe_λ (eigenvector of M) corresponding to eigenvalue λ∈]0,1[λ∈]0,1[ the halving time of the average mn=[⟨Tn,eλ⟩]m_n=E[ T_n,e_λ ] is h1/2lastgen=⌈log2−log|1−λ|⌉. h_1/2^last\ gen= 2- |1-λ| . (11) Here, ⌈x⌉ x denotes the smallest integer greater than or equal to x and halving time is the minimal integer n such that mn≤12m0m_n≤ 12m_0. Proof. See appendix C. ∎ Theorem 4.2. Under the fixed budget protocol (10), (7), (8) the target parameter TnT_n converges to zero pointwise. Moreover, for a feature eλe_λ (eigenvector of M) corresponding to eigenvalue λ∈]0,1[λ∈]0,1[ the halving time of the average mn=[⟨Tn,eλ⟩]m_n=E[ T_n,e_λ ] is h1/2fixedbudget=e−γ21/λ(1+O(λ)), h_1/2^fixed\ budget=e^-γ2^1/λ(1+O(λ)), (12) with γ the Euler constant. Proof. See appendix C. ∎ Remark 4.3. Both Theorem 4.2 and 4.1 still hold true with straightforward modification if we allow the possibility for some centered error into the dataset representation, i.e. for fixed budget protocol (n+1)Tn+1=Tnn+Sn+1+ζn(n+1)T_n+1=T_nn+S_n+1+ _n or Tn+1=Sn+1+ζnT_n+1=S_n+1+ _n for the last generation protocol. Remark 4.4. As a further intuition, note that large eigenvalues in the contraction matrix M correspond to features that are lost fast because along eigenvector (feature) eλe_λ with eigenvalue λ∈]0,1[λ∈]0,1[ the initial data decay involves a factor 1−λ1-λ; larger λ means faster decay and conversely, small λ correspond to directions that are more resilient. So λ∼0λ 0 correspond to "robust" features while λ∼1λ 1 to "fragile" feature, consistent with analysis above. 5 Interpretation of the theoretical results The last generation protocol : In this case there is no convergence of the model parameters over the generations, there is just a limit distribution πT∞LG _T_∞^LG different from the original dataset distribution D or the initialization ν. The convergence is in distribution and not point-wise which means that model parameters will take some ergodic walk following πT∞LG _T_∞^LG. The variance is not vanishing to zero but tends to some positive constant. Note that the average goes to zero exponentially fast if ZnZ_n are centered. Let us discuss this more in detail: equation (11) gives an estimate of the number of generations required for some feature in the dataset (more precisely its average) to decay by a factor of 22. This depends on the eigenvalue associated to the feature. For instance, when this eigenvalue is close to 11, say λ=0.99λ=0.99 the number of generations required is h1/2lastgen=log2−log|1−0.99|=0.15≈1h_1/2^last\ gen= 2- |1-0.99|=0.15≈ 1; when the eigenvalue is close to 00, say λ=0.01λ=0.01 the number of generations is h1/2lastgen=log2−log|1−0.01|=68.97≈69h_1/2^last\ gen= 2- |1-0.01|=68.97≈ 69. So even for small eigenvalues the memory is erased after a limited number of generations (about 70). The fixed budget case : Here the situation is quite different as all model parameter trajectories converge to zero, same as the variance ; in particular the average also goes to zero but the memory of the initial data behaves as follows: when λ=0.99λ=0.99 we obtain h1/2fixedbudget∼e−γ21/0.99∼1.13≈1h_1/2^fixed\ budget e^-γ2^1/0.99 1.13≈ 1 ; in sharp contrast with previous case, when λ=0.01λ=0.01 we obtain h1/2fixedbudget∼e−γ21/0.01∼7.12×1029h_1/2^fixed\ budget e^-γ2^1/0.01 7.12× 10^29. So for small eigenvalues the number of generations required to erase the memory of the initial feature in the dataset is extremely high, in practice never lost. The dataset characteristics are therefore partitioned into a set of fragile features (eλe_λ with λ∼1λ 1) that are lost early and robust features (eλe_λ with λ∼0λ 0) that remain for long periods (some of them practically forever). To summarize, while both protocols rapidly forget high-eigenvalue (fragile) features, they differ sharply for low-eigenvalue features: the last-generation protocol erases memory of a feature within a finite number of generations, whereas the fixed-budget protocol preserves some robust features for extremely long times, leading to a practical persistence of part of the initial dataset information. 6 Empirical results The results can be reproduced using the github available at https://github.com/gabriel-turinici/model_collapse_features. We plot here the graphical qualitative results depicting the comparison between the initial dataset and the final dataset (scatter plot for 2D and sample images for the others) for both training protocols and refer to Appendix D for the quantitative metrics results. For the "last generation" protocol the dataset comparison in Figure 1 and metrics evolution in Figure 2 show that deterioration is very fast, in agreement with the literature and theoretical results in Section 5. Reference dataset Last generation protocol Fixed-budget protocol Spiral MNIST FMNIST CIFAR-10 Figure 1: For each dataset we plot the initial data (left column of plots), the samples at the end of the “last generation” protocol (center column) and at the end of the “fixed budget” protocol (right column). For the “last generation” protocol all features are lost leading to an almost white noise random generation, while for the “fixed budget” protocol some features are lost but some are kept like the structure of digits (MNIST), clothes (FMNIST) or texture of scenes (CIFAR-10). For the "fixed budget" protocol the dataset comparison in Figure 1 and the metrics evolution in Figure 3 show that deterioration is partial and affects only some features as predicted by the theoretical results in Section 5. The metric evolution is also very discriminative, with last-generation protocol reaching much higher values of FID, that is departing farther from the original dataset, cf. figures in Appendix D. 7 Discussion and limitations What the analysis contributes as principle. While the two protocols are usually compared through the speed of degradation, we place the difference at the per-feature dynamics and obtain theoretical results concerning the feature decay (11)–(12) which are coherent with the experiments in Section 6. The apparently divergent conclusions of Shumailov et al. (2024), Gerstgrasser et al. (2024), Seddik et al. (2024) and Barzilai and Shamir (2026) can now be reconciled for the diffusion model settings. Why these regimes matter in practice. The fragile/robust partition identified theoretically explains which features are at risk. Fragile features are those associated in earlier work with the tails and low-density regions of the distribution (rare and under-represented cases). For anyone training on web-scale corpora that already contain generated content, this turns a diffuse concern into a concrete encouragement to enforce provenance tracking, synthetic-content disclosure, or refreshing the data with real examples. In general we hope these insights can help asses and mitigate the effects of model collapse by strengthening preferentially the ’fragile’ features. On the other hand, the existence of robust features helps explain why collapse in practice is often partial rather than total, potentially even creating a false sense of reassurance that the procedure remains healthy. Limitations. Our theory is a first-order, NTK-style linear response resting on assumptions that real recursive training meets only approximately, and the identification of the eigenvectors of M with semantic attributes is an interpretation rather than a consequence of the model. Empirically, we retrain small models from scratch with unconditional DDIM sampling, without sweeping the budget size or curating the pool, and our metrics measure distributional distance only. See Appendix E for the additional discussion on the limitations. 8 Conclusion We studied recursive diffusion models training in two regimes: total replacement ("last generation" protocol) and a "fixed budget" protocol where synthetic data accumulate but the training budget and dataset size stay fixed. This latter regime, introduced as accumulate-subsample by Kazdan et al. (2025), is relevant for finite-budget pipelines. Empirically, on a 2D spiral and on image benchmarks (MNIST, Fashion-MNIST & CIFAR-10) degradation under a fixed budget is partial and feature-selective, unlike the general rapid collapse of pure replacement. Theoretically, we gave a linear-response framework for this setting, complementary to the stability analysis of Bertrand et al. (2024), and show rigorously that the two protocols differ in kind, not degree, and the dataset characteristics split into fragile features, forgotten within a few generations, and robust features that a fixed budget preserves over practically unbounded horizons. For the principled study of generative modeling, this reconciles previously discordant results; for the trustworthy and sustainable use of these models, it provides a way to anticipate which content, especially the tails and under-represented, fragile, modes, is lost under recursive training on accumulating data, and how many generations that takes. Identifying features semantically, scaling to large pretrained models and conditional sampling, modeling curation of the pool, and relating halving times to explicit fairness criteria are the natural next steps. References Barzilai and Shamir (2026) D. Barzilai and O. Shamir When models don’t collapse: on the consistency of iterative MLE. 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In Advances in Neural Information Processing Systems, Cited by: Appendix A, §1. Shumailov et al. (2024) I. Shumailov, Z. Shumaylov, Y. Zhao, Y. Gal, N. Papernot, and R. Anderson The curse of recursion: training on generated data makes models forget. External Links: 2305.17493, Link Cited by: Appendix A, §1, §1, §7. Song et al. (2021) J. Song, C. Meng, and S. Ermon Denoising diffusion implicit models. In International Conference on Learning Representations, Cited by: §2. Yoon et al. (2025) Y. Yoon, D. Hu, I. Weissburg, Y. Qin, and H. Jeong Model collapse in the self-consuming chain of diffusion finetuning: a novel perspective from quantitative trait modeling. External Links: 2407.17493, Link Cited by: Appendix A, 3rd item, 4th item, §1. Zhu et al. (2025) X. Zhu, D. Cheng, H. Li, K. Zhang, E. Hua, X. Lv, N. Ding, Z. Lin, Z. Zheng, and B. Zhou How to synthesize text data without model collapse?. In Proceedings of the 42nd International Conference on Machine Learning, A. Singh, M. Fazel, D. Hsu, S. Lacoste-Julien, F. Berkenkamp, T. Maharaj, K. Wagstaff, and J. Zhu (Eds.), Proceedings of Machine Learning Research, Vol. 267, p. 79746–79771. External Links: Link Cited by: Appendix A, §1. Appendix A Related literature and positioning We recall below some of the works that documented model collapse. The works disagree less about the recursion itself than about how the training set is built at each step. Shumailov et al. (2024) test on VAE, GMM, and language models and refit each generation on a finite sample drawn from its predecessor, with no real data retained. We will call this protocol the "last generation" protocol. They distinguish an early phase, in which the low-probability regions of the distribution are lost, from a late phase, in which modes merge and the generated distribution concentrates on a point estimate, and they show that the finite sample size alone suffices to produce this, independently of any limitation in the expressivity of the model. We retain the finite per-generation sample, which is the mechanism they identify, but not the deletion of the real data. The three works of Dohmatob et al. (2024a); Dohmatob et al. (2024b); Dohmatob et al. (2025) analyze model collapse through tractable regression and generative models. They show that recursive training on synthetic data changes the relevant scaling laws, primarily because rare events and distributional tails are progressively lost, leading to bias, un-learning, and eventual collapse. While the first two works focus mainly on recursive self-training and also consider mixtures of real and synthetic data, Dohmatob et al. (2025) specifically studies a growing data set with a fixed nonzero synthetic proportion. It shows that performance may fail to improve, or may even collapse, as the total data size grows, since the synthetic fraction does not vanish. The first two works are therefore closest to our “last generation” protocol, whereas the latter provides a comparison with growing-data protocols. Gerstgrasser et al. (2024) discuss LLMs and generative language models in general and let the training set grow: generation n is fit on the union of the real dataset and all previous synthetic generations. The protection they establish comes from this growth, since a generation that contributes a vanishing share of the training set also contributes a vanishing share of the noise. It is arguably not a consequence of the presence of the real data as such. We depart from it on one point: instead of training on the accumulated pool we only train on a fixed size sample from the generated pool. Note in particular that the fixed size removes the weighting that makes their argument work. Kazdan et al. (2025) introduce exactly our fixed budget protocol under the name accumulate-subsample, with the same finite-compute motivation, and report across five generative settings a test loss lying between those of replacement and accumulation and typically reaching a plateau. Our experiments are consistent with this picture. What we add is a per-feature account of it: the plateau reflects robust features, whose halving time grows as 21/λ2^1/λ, while fragile features are lost as fast as under replacement. Seddik et al. (2024) focus on LLM (autoregressive language models) and predict a threshold phenomenon, with a maximal proportion of synthetic data below which collapse is avoided, and show in the purely synthetic regime that the learned distribution converges to a point mass supported on an outcome that was likely under the original distribution. Because our protocol drives the real proportion to zero, their framework is the one that most directly predicts what we observe, and their tails-first ordering is the discrete counterpart of the feature-selective degradation reported in Section 6. Barzilai and Shamir (2026) work with generic generative models under maximum likelihood estimation in an accumulating setting and avoid collapse even as the real fraction vanishes, but under two conditions that our setting does not meet: the number of samples drawn per generation must grow with the number of generations, whereas ours is constant, and the family of distributions must be smooth enough, failing which they construct consistent families that collapse arbitrarily fast. DDIM sampling has no maximum likelihood property to begin with, so their positive result does not transfer, while their negative results show that accumulation alone is not a guarantee. Bertrand et al. (2024) study likelihood-based generative models, diffusion models among them, retrained on a mixture of their own samples and a fixed proportion of the real data. They linearize the retraining map in parameter space around the optimum and prove stability when the initial model is accurate enough and the real proportion stays above a threshold. Our framework rests on the same kind of first-order linearization but differs in three respects: under the fixed budget the real proportion vanishes as 1/(n+1)1/(n+1), so their threshold is eventually crossed; each generation is re-initialized at random, which is the source of the noise term MZnMZ_n; and we derive per-feature halving times rather than a stability condition, which is what separates fragile from robust features. Applied to autoregressive transformers LLMs, Zhu et al. (2025) study mixtures of real and synthetic text at a fixed synthetic proportion rather than long recursive chains, and propose token-level editing of human-produced data as a remedy. Their observation that synthetic data cover only a narrowed portion of the original distribution is the language-model analogue of the loss of low-density features we report, obtained without any recursion at all. We now discuss three works that study diffusion models rather than language models. Yoon et al. (2025) finetune Stable Diffusion v1.5 with LoRA on its own outputs, restarting each iteration from the same pretrained weights on a fixed prompt set with no real data retained: a last generation protocol with a deterministic and strongly informative initialization. Their central finding is that the classifier-free guidance scale governs both the speed and the direction of degradation, low guidance producing blurring and high guidance high-frequency artifacts. This mechanism has no counterpart here, where sampling is unconditional, which is one reason our degradation is feature-selective rather than directional. Their quantitative-trait analysis yields a drifting mean with variance stabilizing at a positive value; Theorem 4.1 reaches a comparable non-degenerate limit through an unrelated mechanism. Bohacek and Farid (2025) retrain the denoising network of Stable Diffusion on faces of its own creation over five iterations, at fixed synthetic proportions from 3.3%3.3\% to 100%100\%. Collapse occurs at every proportion, and curation of the retraining set does not slow it down. Their real proportion stays constant across iterations rather than being driven to zero as in our fixed budget protocol, so their result bounds how little synthetic data suffices but leaves the long-horizon asymptotics open. Their healing experiment, where FID recovers while artifacts persist, indicates that part of the information survives the chain, consistent with the robust features of Section 5. Both works finetune a large pretrained model over few iterations, whereas we retrain small models from scratch over one hundred generations, so the pretrained weights play in their setting the role that the random initialization ZnZ_n plays in ours. Shi et al. (2025) retrain DDPMs from scratch on CIFAR-10, FFHQ and MNIST under both replacement and accumulate-subsample, over fewer than ten iterations. They identify a transition from generalization to memorization driven by the decreasing entropy of the training set, and propose an entropy-based selection of the training subset as a remedy. Their study is empirical and measures sample novelty; ours follows the same protocol over one hundred generations and gives a theoretical account of which features survive. Recall on positioning. Our contributions are empirical and theoretical. Empirically, most model-collapse works study VAEs, GMMs or language models, while we work with diffusion models (DDPM training, DDIM sampling), which have received far less attention. Theoretically, we introduce a new vector-valued mathematical description of the multi-generational parameter dynamics and prove convergence results explaining both behaviors. In particular we recover the rapid collapse of total replacement but find that a fixed training budget causes only partial, feature-selective degradation. Appendix B Classical stochastic recursion convergence results We recall here some classical stochastic analysis results that will be used to prove the main results. Lemma B.1. Consider the recursion Yn+1=(I−1nM)Yn+1nϵn, Y_n+1= (I- 1nM )Y_n+ 1n _n, (13) where M is symmetric positive definite with eigenvalues λi(M)∈]0,1[ _i(M)∈]0,1[, and (ϵn)∼μ( _n) μ are i.i.d. and Var[μ]<∞Var[μ]<∞; denote μ¯=[μ] μ=E[μ]. One has Yn→n→∞a.s.M−1μ¯. Y_n [n→∞]a.s.M^-1 μ. (14) In particular, if [ϵ1]=0E[ _1]=0, then Yn→n→∞a.s.0. Y_n [n→∞]a.s.0. (15) Proof. We use (Borkar, 2022, Chapters 2 and 4) more precisely convergence theorem Theorem 2.1 page 17 (we will check conditions (A1)-(A3) page 11) and boundedness Theorem 4.1 page 54, (we check condition (A9) page 50). Here h(x)=Mxh(x)=Mx. Apply the convergence theorem to Yn+1=Yn+1n[−(MYn−μ¯)+(ϵn−μ¯)]. Y_n+1=Y_n+ 1n [-(MY_n- μ)+( _n- μ) ]. (16) The associated ODE is y˙=−My+μ¯ y=-My+ μ, whose unique globally asymptotically stable equilibrium is y∗=M−1μ¯y =M^-1 μ. Since ∑n=1∞n−1=∞ _n=1^∞n^-1=∞, ∑n=1∞n−2<∞ _n=1^∞n^-2<∞, and ϵn−μ¯ _n- μ has finite second moment, the theorem yields Yn⟶M−1μ¯almost surely. Y_n M^-1 μ surely. (17) Alternative proofs can follow the arguments in Duflo (1997); Kushner and Yin (2003). ∎ Lemma B.2 (Stable linear stochastic recursion). Let (Xn)n≥0(X_n)_n≥ 0 be defined by Xn+1=AXn+Un, X_n+1=AX_n+U_n, (18) where Xn,Un∈ℝdX_n,U_n ^d, (Un)n≥0(U_n)_n≥ 0 are i.i.d. random vectors with ‖U0‖2<∞, \|U_0\|^2<∞, (19) and A∈ℝd×dA ^d× d a symetric positive definite matrix with spectrum in ]0,1[]0,1[. Then there exists a unique stationary distribution π, and for every initial condition X0X_0, Xn→n→∞X∞∼π with X∞:=d∑k=0∞AkUk, X_n [n→∞]dX_∞ π\ with X_∞ d:= _k=0^∞A^kU_k, (20) where the series converges in L2L^2, and π is the unique distribution such that: X∞=dAX∞+U, X_∞ d=AX_∞+U, (21) with U distributed as U0U_0 and independent of X∞X_∞. In addition: [X∞]=(I−A)−1[U0]. [X_∞]=(I-A)^-1E[U_0]. (22) If ΣU=Cov(U0) _U=Cov(U_0), then the covariance matrix Σ∞=Cov(X∞) _∞=Cov(X_∞) is the unique solution of the discrete Lyapunov equation Σ∞=AΣ∞A⊤+ΣU. _∞=A _∞A + _U. (23) Proof. This is the standard stationary solution theorem for a stable VAR(1) process; see, e.g., Lütkepohl (Lütkepohl, 2005, Sections 2.1.1–2.1.4). The contraction property of A is one of the main ingredient for the existence and uniqueness of the distribution π. Note that only the convergence in distribution hold, not pointwise; moreover the formula X∞:=d∑k=0∞AkUkX_∞ d:= _k=0^∞A^kU_k has only a meaning in distribution sense and not pointwise, in particular the order of UkU_k has no importance (can be permuted) as they do not correspond necessary to the increments in recursion (18). ∎ Appendix C Proof of the main results C.1 Proof of Theorem 4.1 Proof. We prove convergence for TnT_n, the convergence of SnS_n being immediate from (9). Under the last generation protocol after replacing (9) in (7) we obtain: Tn+1=ATn+Un, with A:=I−M,Un:=MZn+εn. T_n+1=AT_n+U_n,\ with A:=I-M,\ U_n:=MZ_n+ _n. (24) We can use now Lemma B.2 to obtain the convergence result. For the second part, let us now inquire about the halving time h1/2lastgenh_1/2^last\ gen of the average mn=[⟨Tn,eλ⟩]m_n=E[ T_n,e_λ ]. Then, since UnU_n is centered, the recursion for the average mnm_n of this feature can be written mn+1=(1−λ)mnm_n+1=(1-λ)m_n . This means mn=(1−λ)nm0m_n=(1-λ)^nm_0. The time required to reach half of the value m0m_0 will be the smallest n such that (1−λ)n≤1/2(1-λ)^n≤ 1/2 which gives equation (11). ∎ C.2 Proof of Theorem 4.2 Proof. We consider the fixed budget protocol (10), (7), (8). After replacing (8) in (7) we obtain: Tn+1=Tn−Mn+1Tn+Un, with Un:=MZn+εn+1. T_n+1=T_n- Mn+1T_n+U_n,\ with U_n:= MZ_n+ _nn+1. (25) We can use now Lemma B.1 to obtain that the target parameter TnT_n converges to zero pointwise. For the second part, we study the halving time h1/2fixedbudgeth_1/2^fixed\ budget of the average mn=[⟨Tn,eλ⟩]m_n=E[ T_n,e_λ ]. Since UnU_n is centered, the recursion for mnm_n can be written mn+1=(1−λn+1)mnm_n+1= (1- λn+1 )m_n which yields mn/m0=∏k=1n(1−λk)=Γ(n+1−λ)Γ(n+1)Γ(1−λ)∼n−λΓ(1−λ)m_n/m_0= _k=1^n (1- λk )= (n+1-λ) (n+1) (1-λ) n^-λ (1-λ) which show that halving time if e−γ21/λ(1+O(λ))e^-γ2^1/λ(1+O(λ)) where Γ is the Euler function. ∎ Appendix D Metrics results Figure 2: Last generation training protocol cf. Section 2. We plot the evolution of the FID, pixel-FID and Wasserstein-2 metrics. Distance rapidly increases in all cases. From top to bottom: 2D spiral, MNIST, Fashion-MNIST, CIFAR-10. Figure 3: Fixed-budget generation training protocol cf. Section 2. We plot the evolution of the FID, pixel-FID and Wasserstein-2 metrics. Distance increases in all cases but the increase is more tempered than for the last generation protocol in Figure 2. From top to bottom: 2D spiral, MNIST, Fashion-MNIST, CIFAR-10. Appendix E Detailed limitations • Modelling assumptions. The theory is a first-order, NTK-style linear response. It assumes an exact parameter representative for every dataset (1), linear interpolation in parameter space for data mixtures, a symmetric positive-definite contraction M with a fixed spectrum, and centered i.i.d. noise and initialization independent across generations. Real recursive training meets none of these exactly; the remark after Theorem 4.2 only absorbs additive representation error. So it should be rather be read as the simplest linear model that can be relevant to this setting, not the most precise one. • Feature identification. The features are eigenvectors of M. Their identification with semantic attributes of the data, or with distribution tails, is an interpretation supported by the experiments and by analogy with Seddik et al. (2024), not something the model establishes. • Sampler and setting. We use unconditional DDIM sampling, which has no maximum-likelihood property, so the positive result of Barzilai and Shamir (2026) does not apply and ours does not transfer to guided or conditional generation, where Yoon et al. (2025) show the guidance scale drives degradation. • Scale. We retrain small models from scratch, with ndn_d samples on several datasets. The behavior of the fragile/robust partition for large pretrained models finetuned over few iterations (Bohacek and Farid, 2025; Yoon et al., 2025) is not tested here. • Pipeline realism. We do not sweep the budget size or the pool-mixing ratio, and we do no model curation or filtering of the pool, which Bohacek and Farid (2025) report can change the picture. • Metrics. FID, pixel-FID and 2W_2 measure distributional distance, not downstream utility or any explicit fairness criterion; connecting halving times to the latter is left open. Appendix F Common architecture and code flow F.1 Recursive training protocol. Generation 0 is trained from scratch on nd=10 000n_d=10\,000 real images (see Section G for 2D-spiral dataset). After training, the EMA model generates another ndn_d synthetic samples. For generation g>0g>0, a pool is constructed from ooln=∪1∪⋯∪n−1, _n=D _1∪… _n-1, (26) where nD_n denotes the synthetic dataset produced at generation n. For the fixed budget protocol a random subset of ndn_d images is sampled from this pool and used to train a newly initialized diffusion model. For the last generation protocol n−1D_n-1 is used to train generation n. The trained model then generates a new synthetic dataset nD_n, which is added to the pool for the next generation. This process is repeated for 100 generations (see Section G for 2D-spiral dataset). Sample quality is monitored through the Fréchet Inception Distance (FID) computed against the original real dataset. F.2 DDPM training, DDIM sampling The sampling principles are common to all datasets. Generation uses deterministic DDIM sampling. Starting from xT∼(0,I), x_T (0,I), (27) the model predicts x^0=xt−1−α¯tϵθ(xt,t)α¯t, x_0= x_t- 1- α_t\, _θ(x_t,t) α_t, (28) and performs the deterministic DDIM update xt′=α¯t′x^0+1−α¯t′ϵθ(xt,t). x_t = α_t \, x_0+ 1- α_t \, _θ(x_t,t). (29) Although the diffusion process is trained with T=1000T=1000 steps, inference uses 100100 or 200200 DDIM steps (depending on dataset). Appendix G Specific architecture and code flow: 2D spiral G.1 Dataset The real dataset is a two-dimensional Archimedean spiral. For ndn_d points we take θ uniformly spaced on [0,2πτ][0,2πτ] and set r=r0+sθ,x=rcosθ,y=rsinθ, r=r_0+s\,θ, x=r θ, y=r θ, (30) with r0=0r_0=0, spacing s=1.5s=1.5 and τ=2τ=2 turns. The sampling is deterministic given the seed, and no normalisation is applied, so the data are supported on a one-dimensional curve embedded in ℝ2R^2 with a total variance of order 10210^2. G.2 Network Architecture The denoising network operates directly on the two-dimensional coordinates and predicts a two-dimensional output. Since the data carry no spatial structure, the convolutional U-Net used for the image datasets is replaced by a fully connected network. G.3 Time Conditioning Diffusion timesteps are encoded using a sinusoidal positional embedding of dimension 3232, followed by a two-layer MLP: et=MLP(SinusoidalEmbedding(t)). e_t=MLP (SinusoidalEmbedding(t) ). (31) The projected time representation has dimension dt=64. d_t=64. (32) Unlike the image architectures, time information is not injected through FiLM-style modulation. The time embedding is concatenated to the input coordinates, h0=[xt;et]∈ℝ2+dt, h_0=[\,x_t\,;\,e_t\,] ^2+d_t, (33) and the resulting vector is passed to the backbone. G.4 MLP Backbone The backbone is a four-layer perceptron with ReLU activations and constant hidden width dh=256d_h=256: • Linear: (2+dt)→dh(2+d_t)→ d_h, ReLU, • Linear: dh→dhd_h→ d_h, ReLU, • Linear: dh→dhd_h→ d_h, ReLU, • Linear: dh→2d_h→ 2. No residual connection, no normalisation layer and no attention mechanism are used. G.5 Summary of Main Architectural Hyperparameters • Input dimension: 22. • Time embedding dimension: 3232. • Time MLP dimension: 6464. • Hidden width: 256256. • Number of hidden layers: 33. • Activation function: ReLU. • Output dimension: 22. G.6 Parameter Count The model contains 1.56×105 1.56× 10^5 (34) trainable parameters (exactly 155 522155\,522), of which 6 2726\,272 belong to the time-embedding MLP, a size comparable to that of the MNIST and Fashion-MNIST denoiser. G.7 Main hyperparameters For reproducibility, the most important parameters in the code are: • Dataset: 2D Archimedean spiral, 22 turns. • Training set size: 10001000 points. • Number of generations : 2020 • Batch size: 256256. • Diffusion steps: T=1000T=1000, linear schedule βt∈[10−4,2×10−2] _t∈[10^-4,2× 10^-2]. • DDIM inference steps: 100100. • Optimiser: Adam. • Learning rate: 10−310^-3 • EMA decay: 0.80.8. • Training epochs: 20002000. G.8 Metrics In dimension two neither the Inception features nor the pixel-space Fréchet distance are meaningful, so the metrics of Section 2 are instantiated as follows: 2W_2 is computed exactly by solving the discrete optimal transport problem between the real and generated point clouds, and the Fréchet distance is computed between the Gaussian approximations of the two clouds in ℝ2R^2. Appendix H Specific architecture and code flow: MNIST and Fashion-MNIST H.1 Network Architecture The denoising network is a compact U-Net operating on MNIST/ Fashion-MNIST images of size 28×28×128× 28× 1. The model predicts a single-channel output having the same spatial resolution as the input. H.2 Time Conditioning Diffusion timesteps are encoded using a sinusoidal positional embedding of dimension 3232, followed by a two-layer MLP: et=MLP(SinusoidalEmbedding(t)). e_t=MLP (SinusoidalEmbedding(t) ). (35) The projected time representation has dimension dt=64. d_t=64. (36) Time information is injected into every residual block through FiLM-style conditioning. The time embedding is mapped to channel-wise scale and shift vectors, (γt,βt)=Wet, ( _t, _t)=We_t, (37) which modulate the intermediate feature maps according to h←h⊙(1+γt)+βt. h← h (1+ _t)+ _t. (38) H.3 Residual Blocks Each residual block contains 1. a 3×33× 3 convolution, 2. ReLU activation, 3. FiLM conditioning from the timestep embedding, 4. a second 3×33× 3 convolution, 5. ReLU activation, 6. a residual skip connection. Whenever the channel dimensions differ between input and output, the shortcut path uses a 1×11× 1 convolution for channel projection. H.4 U-Net Backbone The architecture follows a two-level encoder-decoder design with skip connections. Encoder. Starting from a 28×28×128× 28× 1 image, the encoder consists of • Residual block: 1→321→ 32 channels, • 2×22× 2 max pooling, • Residual block: 32→6432→ 64 channels, • 2×22× 2 max pooling. The bottleneck representation therefore has spatial resolution 7×77× 7 and 6464 channels. Bottleneck. The latent representation is processed by a single residual block 64→64. 64→ 64. (39) No self-attention mechanism is used. Decoder. The decoder mirrors the encoder through nearest-neighbor upsampling and skip connections: • Upsample: 7×7→14×147× 7→ 14× 14, • Concatenate encoder features, • Residual block: (64+64)→64(64+64)→ 64, • Upsample: 14×14→28×2814× 14→ 28× 28, • Concatenate encoder features, • Residual block: (64+32)→32(64+32)→ 32. The final prediction layer is a 1×11× 1 convolution mapping 32→1 32→ 1 (40) channel, producing the image-sized output. H.5 Summary of Main Architectural Hyperparameters • Input resolution: 28×2828× 28. • Number of input channels: 11. • Time embedding dimension: 3232. • Time MLP dimension: 6464. • Encoder channels: 32→6432→ 64. • Bottleneck channels: 6464. • Decoder channels: 64→3264→ 32. • Activation function: ReLU. • Downsampling: MaxPool(2)(2). • Upsampling: nearest-neighbor interpolation. • Skip connections: U-Net concatenation skips at both resolutions. • Output channels: 11. H.6 Parameter Count The model contains approximately 1.87×105 1.87× 10^5 (41) trainable parameters (about 187187k parameters), making it substantially smaller than the CIFAR-10 attention U-Net used in the companion experiment. H.7 Main hyperparameters • Training set size: 10,00010,000 points. • Number of generations : 100100 • Batch size: 256256. • Diffusion steps: T=1000T=1000, linear schedule βt∈[10−4,2×10−2] _t∈[10^-4,2× 10^-2]. • DDIM inference steps: 100100. • Optimiser: Adam. • Learning rate: 10−410^-4 . • EMA decay: 0.80.8. • Training epochs: 10001000. Appendix I Specific architecture and code flow: CIFAR-10 I.1 Diffusion Backbone We employ a denoising diffusion model operating on CIFAR-10 images of size 32×32×332× 32× 3. The network predicts the additive Gaussian noise ϵθ(xt,t) _θ(x_t,t) from a noisy image xtx_t and diffusion timestep t. Training uses the standard noise-prediction objective ℒ=x0,ϵ,t[‖ϵ−ϵθ(xt,t)‖22], =E_x_0,ε,t [ \|ε- _θ(x_t,t) \|_2^2 ], (42) where xt=α¯tx0+1−α¯tϵ,ϵ∼(0,I). x_t= α_t\,x_0+ 1- α_t\,ε, ε (0,I). (43) A linear diffusion schedule is used with T=1000T=1000 timesteps and βt∈[10−4,2×10−2]. _t∈[10^-4,2× 10^-2]. (44) I.2 Time Conditioning Diffusion timesteps are encoded using sinusoidal embeddings followed by a two-layer MLP: et=MLP(SinusoidalEmbedding(t)). e_t=MLP(SinusoidalEmbedding(t)). (45) The model uses: • sinusoidal embedding dimension: 6464, • projected time dimension: 3232. Time information is injected into every residual block through FiLM-style conditioning. For a hidden activation h, the time embedding produces channel-wise scale and shift vectors (γt,βt)=Wet, ( _t, _t)=We_t, (46) and conditioning is applied as h←h⊙(1+γt)+βt. h← h (1+ _t)+ _t. (47) I.3 Residual Block Each residual block consists of 1. 3×33× 3 convolution, 2. GroupNorm, 3. SiLU activation, 4. FiLM conditioning from the timestep embedding, 5. second 3×33× 3 convolution, 6. GroupNorm, 7. SiLU activation, 8. residual skip connection. If the input and output channel dimensions differ, a 1×11× 1 convolution is used in the residual shortcut. I.4 Attention U-Net Architecture The denoiser follows a compact U-Net architecture with encoder, bottleneck, self-attention layer, and symmetric decoder. For the experiments reported in the code, the base width is D=8. D=8. (48) The channel progression is therefore 8→16→32. 8→ 16→ 32. (49) Encoder. • Input: 32×32×332× 32× 3 • Two residual blocks: 3→D→D3→ D→ D • Downsampling by 2×22× 2 max pooling • Two residual blocks: D→2D→2D→ 2D→ 2D • Downsampling • Two residual blocks: 2D→4D→4D2D→ 4D→ 4D • Downsampling This produces a bottleneck representation of spatial size 4×44× 4 with 4D4D channels. Bottleneck. The bottleneck contains two residual blocks 4D→4D→4D. 4D→ 4D→ 4D. (50) A multi-head self-attention layer is then applied after flattening the 4×44× 4 spatial grid into 1616 tokens. • embedding dimension: 4D=324D=32, • number of heads: 44. The attention output is added back through a residual connection. Decoder. The decoder mirrors the encoder: • Upsample 4×4→8×84× 4→ 8× 8 • Concatenate encoder skip features • Residual blocks: (4D+4D)→4D→4D(4D+4D)→ 4D→ 4D • Upsample 8×8→16×168× 8→ 16× 16 • Concatenate skip features • Residual blocks: (4D+2D)→2D→2D(4D+2D)→ 2D→ 2D • Upsample 16×16→32×3216× 16→ 32× 32 • Concatenate skip features • Residual blocks: (2D+D)→D→D(2D+D)→ D→ D Finally, a 1×11× 1 convolution maps the hidden representation back to three channels to predict noise: D→3. D→ 3. (51) I.5 Main hyperparameters • Training subset size: 10,00010,000 images. • Batch size: 256256. • Diffusion steps: T=1000T=1000. • DDIM inference steps: 200200. • Learning rate: 5×10−45× 10^-4. • EMA decay: 0.80.8. • Base channel width: D=8D=8. • Time embedding dimension: 6464. • Projected time dimension: 3232. • Attention heads: 44. • Training epochs: 10001000.