Quantifying Error Propagation and Model Collapse in Diffusion Models

Nail B. Khelifa; Richard E. Turner; Ramji Venkataramanan

stat.MLFeb 18, 2026

Quantifying Error Propagation and Model Collapse in Diffusion Models

Authors: Nail B. Khelifa, Richard E. Turner, Ramji Venkataramanan

Organizations: University of Cambridge, Cambridge, United Kingdom.

Abstract

Machine learning models are increasingly trained or fine-tuned on synthetic data. Recursively training on such data has been observed to significantly degrade performance in a wide range of tasks, often characterized by a progressive drift away from the target distribution. In this work, we theoretically analyze this phenomenon in the setting of score-based diffusion models. For a realistic pipeline where each training round uses a combination of synthetic data and fresh samples from the target distribution, we obtain upper and lower bounds on the accumulated divergence between the generated and target distributions. Notably, to the best of our knowledge, this is the first lower bound on the divergence between the learned and target distributions, even for standard diffusion models. Our results allow us to characterize different regimes of drift, depending on the score estimation error and the proportion of fresh data used in each generation. In a certain regime, the accumulated divergence after several retraining rounds can be expressed as a discounted sum of score estimation errors made at each generation. We also provide empirical results on synthetic data and images to illustrate the theory.

Figures & tables

Figure 1: Samples generated by a recursively trained model, projected onto the first two principal components, with $p_{\mathrm{data}}$ chosen to be a 10-dimensional Gaussian mixture. Columns correspond to fresh data fractions $\alpha\in\{0.1,0.5,0.9\}$ (left to right); rows show generations 0 (start), 10 (middle), and 20 (end). At $\alpha=0.1$ (left column), the distribution progressively disperses as errors accumulate without sufficient fresh data. At $\alpha=0.5$ (center), moderate degradation occurs with visible spreading but preserved structure. At $\alpha=0.9$ (right column), the distribution remains stable throughout, demonstrating that high fresh data fractions effectively prevent collapse. Experiment details in Appendix G.1 .

Figure 2: Observability of three classes of score perturbations in CIFAR-10 ( Krizhevsky, 2009 ) diffusion sampling. We inject controlled perturbations

\mathbf{e}_{i}(\mathbf{x},t)

and estimate the observability coefficient

\hat{\eta}_{i}=\mathrm{Var}(\widehat{\mathbb{E}}[M_{i}\mid\mathbf{Y}_{t_{0}}])/\mathrm{Var}(M_{i})

using paired reverse trajectories with shared diffusion noise. Aligned :

\mathbf{e}_{i,t}(\mathbf{x})=\mathbf{w}_{i}\mathbf{x}

with

\mathbf{w}_{i}

chosen so that the error points in a direction correlated with the drift (i.e., errors push trajectories further along where they are already going). Random :

\mathbf{e}_{i,t}(\mathbf{x})=\mathbf{w}\mathbf{x}

with random

\mathbf{w}

independent of the drift direction Time-only :

\mathbf{e}_{i,t}(\mathbf{x})=\xi(t)

, independent of state. State-dependent perturbations yield higher

\hat{\eta}_{i}

, confirming that errors coupled to the sample state are more visible at the terminal distribution.

Figure 3: Empirical verification of intra-generational error bounds for a Gaussian mixture.

p_{\mathrm{data}}\sim\tfrac{1}{5}\sum_{k=1}^{5}\mathcal{N}(\boldsymbol{\mu}_{k},\sigma^{2}\mathbf{I}_{10})\in\mathbb{R}^{10}

. Left: Upper bound verification (Proposition 3.1 ). The KL divergence

\mathrm{KL}(\hat{p}^{i+1}\|q_{i})

between the learned distribution and the training mixture remains bounded by

\frac{1}{2}\hat{\varepsilon}_{i}^{2}

. The bound is tight at early generations and becomes conservative as the system equilibrates. Right: Lower bound verification (Proposition 3.3 ). The

\chi^{2}

divergence

I_{i}=\chi^{2}(\hat{p}^{i+1}\|q_{i})

is bounded below by

\frac{1}{8}\hat{\eta}_{i}\varepsilon_{\star,i}^{2}

, where

\hat{\eta}_{i}

is the estimated observability coefficient. Shaded regions indicate

\pm 1

standard deviation across runs. Both bounds hold consistently across

\alpha\in\{0.1,0.5\}

, validating the theoretical framework. Experiment details are given in Appendix G.1 .

Figure 4: Two-sided control of intra-generation divergence.

p_{\mathrm{data}}\sim\tfrac{1}{5}\sum_{k=1}^{5}\mathcal{N}(\boldsymbol{\mu}_{k},\sigma^{2}\mathbf{I}_{10})\in\mathbb{R}^{10}

. We track the evolution of the divergences

I_{i}=\chi^{2}(\hat{p}^{i+1}\|q_{i})

and

I_{i}^{\mathrm{KL}}=\mathrm{KL}(\hat{p}^{i+1}\|q_{i})

(solid lines with markers) across 20 generations for low (

\alpha=0.1

, left) and high (

\alpha=0.5

, right) fresh data ratios. Consistent with Theorem 3.5 , the intra-generational divergences are effectively bounded below and above by the pathwise score error energy (dashed upper bound

=4\varepsilon_{\star,i}^{2}

) and the observability-weighted lower bound (dotted line

=\frac{1}{4}\hat{\eta}_{i}\varepsilon_{\star,i}^{2}

, where

\hat{\eta}_{i}

is the estimated observability of errors, as detailed in Appendix G.1 ). Shaded regions indicate

\pm 1

standard deviation across runs.

Figure 5: Geometrically-discounted decomposition of accumulated divergence in ( 23 ), for $p_{\mathrm{data}}=\tfrac{1}{5}\sum_{k=1}^{5}\mathcal{N}(\boldsymbol{\mu}_{k},\sigma^{2}\mathbf{I}_{10})\in\mathbb{R}^{10}$ . Each panel shows contributions of $\varepsilon_{\star,i}^{2}$ to the current global divergence $D_{N}=\chi^{2}(\hat{p}^{N}\,\|\,p_{\text{data}})$ , for $i\leq N$ . Left ( $\alpha=0.1$ ): Errors persist across many generations, creating a wide band of contributions. Center ( $\alpha=0.5$ ): Intermediate regime with moderate memory decay. Right ( $\alpha=0.9$ ): Short memory—only the most recent generation dominates, yielding a sharp diagonal structure. The plots are consistent with Theorem 4.2 , confirming that higher fractions of fresh data accelerate the forgetting of past errors. Experiment details in Appendix G.1 .

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 6 — Figure 6: Accumulation of inter-generational divergence in the 10D GMM experiment. Solid lines show the empirical divergence $D_{N}=\chi^{2}(\hat{p}^{N}\|p_{\mathrm{data}})$ averaged over runs, with shaded regions indicating one standard deviation, for values of $\alpha\in\{0.1,0.5,0.9\}$ . In each figure, the dotted curve corresponds to the theoretical lower bound $\frac{\alpha\eta_{\min}}{16(1+(1-\alpha)^{2})}\,S_{N}$ , derived in Theorem F.7 , where $S_{N}=\sum_{k=0}^{N-1}(1-\alpha)^{2(N-1-k)}\varepsilon_{\star,k}^{2}$ is the predicted accumulated error energy. While the lower bound is easy to compute and corresponds to the theoretical lower bound derived in Theorem F.7 , the upper bound, more difficult to compute directly (as it depends on theoretical constants) is obtained by an affine regression of $S_{N}+(1-\alpha)^{2N}D_{0}$ onto the observed divergence. As such, the upper bound is not a theoretical bound and is shown only to illustrate that the functional dependence predicted by the theory accurately captures the observed growth across generations.

Figure 7: Observability coefficient

\hat{\eta}_{i}

across generations in the 10-dimensional Gaussian Mixture setting. The observability coefficient measures the fraction of score error energy that leaves a detectable imprint on the terminal distribution:

\hat{\eta}_{i}=\mathrm{Var}(\widehat{\mathbb{E}}[M_{i}\mid\mathbf{Y}_{t_{0}}^{\star,i}])/\mathrm{Var}(M_{i})

, where

M_{i}=-\int_{0}^{T}\mathbf{e}_{i}\cdot\mathrm{d}\bar{\mathbf{B}}_{t}

is the error martingale. Higher

\alpha

(more fresh data) yields larger

\hat{\eta}_{i}

, indicating that score errors are more strongly coupled to observable features when training distributions are closer to

p_{\mathrm{data}}

. At low

\alpha

, the training mixture

q_{i}

drifts further from the data manifold, causing errors to accumulate in directions that are less distinguishable at the terminal time. Dashed horizontal lines indicate asymptotic means; shaded regions show

\pm 1

standard deviation across runs. The non-zero values confirm that the lower bound

\chi^{2}(\hat{p}^{i+1}\|q_{i})\geq\frac{1}{8}\eta_{i}\varepsilon_{\star,i}^{2}

provides meaningful information about intra-generational error.

Figure 8: Random samples at generations $0$ (left), $10$ , $15$ and generation $25$ (right) following the above recursive pipeline (right) for three fresh-data mixing rates $\alpha\in\{0.1,0.5,0.9\}$ (from top ( $\alpha=0.1$ ) to bottom ( $\alpha=0.9$ ). Smaller values of $\alpha$ (more self-generated data) yield rapid degradation—loss of sharpness and diversity—while larger $\alpha$ stabilizes training and preserves sample quality over generations.

Figure 9: Observability coefficient on CIFAR-10. We estimate

\eta_{i}=\mathrm{Var}(\widehat{\mathbb{E}}[M_{i}\mid\mathbf{Y}_{t_{0}}])/\mathrm{Var}(M_{i})

by regressing the error martingale on terminal sample features (described in p. G.1 ). After an initial transient (generation 0 uses a placeholder value of 1), all three

\alpha

values converge to similar observability levels (

\eta\approx 0.25

–

0.35

), indicating that approximately 30% of score error energy propagates to detectable distributional changes. Unlike the GMM setting where higher

\alpha

yielded higher

\eta

, the CIFAR-10 curves overlap substantially, suggesting that observability in high-dimensional image spaces is governed more by the data manifold geometry than by the training mixture composition. The non-zero values confirm that the lower bound

\chi^{2}\geq\frac{1}{8}\eta\varepsilon_{\star}^{2}

provides meaningful information in the neural network setting.

Panel of Figure 10 — Figure 10: Memory structure of error accumulation on Fashion-MNIST. Each panel shows absolute (left) and fractional (right) contributions of generation- $k$ errors to generation- $N$ divergence, computed as $(1-\alpha)^{2(N-1-k)}\varepsilon_{\star,k}^{2}$ . The structural predictions of Theorem 4.2 are clearly validated: (a) At $\alpha=0.1$ , errors persist across the full 10-generation horizon, with early generations (particularly generation 1) contributing substantially even at generation 10—the wide triangular structure indicates long memory and susceptibility to collapse. (b) At $\alpha=0.5$ , intermediate memory decay creates a band approximately 3–4 generations wide, balancing error persistence with forgetting. (c) At $\alpha=0.9$ , the sharp diagonal structure confirms that only the most recent generation contributes meaningfully—past errors are rapidly forgotten, preventing accumulation. These patterns match the GMM results (Figure 5 ) despite the 78 $\times$ increase in dimension, confirming that memory structure is a universal feature of recursive diffusion training independent of data complexity.

Model Collapse

Model collapse studies how recursive training or feedback can erode a model’s diversity, accuracy, or fidelity to the target distribution.

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Jun 11, 2026

Recursively Trained Diffusion Models: Limiting Collapse Distribution and Spectral Characterization

Recursive training of generative models on their own outputs can lead to model collapse, a compounding drift away from the true data distribution. Existing theoretical works bound finite-round error accumulation in the context of diffusion models, but two questions remain open:~what distribution does the recursion converge to, and how fast? We answer both, isolating a mechanism distinct from imperfect learning: even with perfect score estimation and exact sampling, the early stopping of the reverse diffusion (required for numerical stability) drives a progressive drift away from the data distribution. We prove that this recursion converges geometrically to a unique limiting distribution, which admits a closed-form characterization as an infinite mixture of increasingly Gaussian-smoothed versions of the data distribution. A Hermite spectral decomposition of this limit reveals that recursive training acts as a low-pass filter: higher-order modes, which encode fine non-Gaussian structure, are attenuated much more strongly than coarse modes. This spectral picture motivates annealed truncation schedules that progressively shrink truncation times across retraining rounds; we prove that any schedule converging to

0

asymptotically eliminates recursive compounding. Finally, we show our idealized characterization is robust: in the presence of discretization and score estimation errors, the learned distribution remains in a Wasserstein-2 ball around the ideal limit, with mode-dependent contraction rates that contract high-order errors faster than low-order ones. We validate the theory on synthetic Gaussian mixtures and CIFAR-10.

Naïl B. Khelifa, Richard E. Turner, Ramji Venkataramanan

University of Cambridge

Diffusion Sampling Model Collapse Generative Modeling Diffusion Models

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Jun 4, 2026

Diffusion Models Observe Only Gradients: A Geometric Perspective on Score Matching Errors

Score-based diffusion models are typically trained by minimizing the

L^2

score matching error, and standard theoretical analyses rely on this quantity to bound the sampling discrepancy between the learned and target distributions. We show the

L^2

score error is not the right intrinsic measure of marginal distributional quality: a learned diffusion model can incur arbitrarily large

L^2

score error while perfectly matching the target distribution. By decomposing score errors into a gradient and a solenoidal component (a Helmholtz-Hodge decomposition), we identify the geometric reason behind this: only the gradient component enters the marginal Fokker-Planck dynamics, while the solenoidal component is structurally invisible. We make this precise in three results. First, building on the corrected geometry, we prove an impossibility result: no monotone function of the

L^2

score error can uniformly lower bound any divergence between the learned and target distributions. Second, we derive an upper bound on the Kullback-Leibler divergence that depends only on the observable gradient component of the error, tightening the standard Girsanov bound for generic score networks, and identifying its looseness as the cost of operating on path-space rather than marginal-space dynamics. Third, we give a tractable estimator of the gradient component via a dual Sobolev identity, which is shown to empirically correlate substantially better with sample quality than the full

L^2

error.

Naïl B. Khelifa, Richard E. Turner, Ramji Venkataramanan

University of Cambridge

Score Matching Hodge Decomposition Score-Based Generative Modeling Diffusion Models

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Jul 25, 2026

From Score Learning to Discretized Sampling: An End-to-End Generalization Analysis of Diffusion Models

Despite the empirical success of score-based diffusion models, a complete theoretical understanding of how finite-sample learning, network parameterization, and numerical discretization jointly dictate generative quality remains underdeveloped. Existing sampling analyses often evaluate the generative performance conditional on an oracle score or a pre-specified error threshold. In this work, we establish a unified convergence and generalization framework for score-based diffusion models parameterized by practical ResNet-type architectures. We analyze the generalization and convergence properties from the practical finite-sample, discrete-time learning problem of the score function to the ideal continuous-time, population-level objective. Based on the generalization result of the learning problem of score function, we analyze the sampling process induced by the learned score function and provide an end-to-end total variation distance estimate for the generated terminal distribution. This estimate explicitly decomposes the overall generative error into four interpretable components: the truncation error of the forward process, the reverse-time discretization error, the generalization error incorporating both finite data and forward-time discretization, and the training optimization gap. Our results quantitatively characterize how the training sample size, temporal discretization grids, and optimization accuracy jointly control the final fidelity of samples generated by diffusion models.

Jinshu Huang, Yiming Jiang, Chunlin Wu

School of Mathematical Sciences, Nankai University, Tianjin, China

Diffusion Model Sampling Score-Based Generative Modeling Diffusion Models

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Abstract

Model Collapse

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