Memorisation, convergence and generalisation in generative models

Abstract: Generative neural networks learn how to produce highly realistic images from a large, but finite number of examples - or do they simply memorise their training set? To settle this question, Kadkhodaie, Guth, Simoncelli and Mallat (ICLR '24) trained diffusion models independently on disjoint subsets of a dataset and showed that they converge to nearly the same density when the number of training images is large enough. This result raises two basic questions: how much data do you need for convergence, and what does convergence capture about learning the data distribution? Here, we address these questions by providing an exact analytical characterisation of the transition from memorisation to generalisation in linear generative models. We find that these models memorise at small load, while convergence emerges continuously when the number of samples is linear in the input dimension. Strikingly, we find that convergence is insensitive to recovery of the principal latent factors of the data, which are recovered in a sharp transition. After extending our approach to data with power-law spectra, we find the same distinction between convergence and latent recovery in our experiments with convolutional denoisers and in the data of Kadkhodaie et al. We thus show that generalisation in generative models decomposes into at least two distinct objectives: matching the bulk of the data distribution and recovering the principal latent factors. These objectives correspond to two different distances between true and learnt data distribution, and only the first one is captured by convergence.

• The paper develops an asymptotically exact framework to quantify memorisation, convergence, and generalisation in linear and spectral generative models.
• It reveals distinct regimes from overfitting to convergence, and identifies a sharp BBP-style phase transition in latent recovery.
• Empirical validation using diffusion models shows that convergence in output does not ensure accurate recovery of latent structure.

Memorisation, Convergence, and Generalisation in Generative Models: A Theoretical Dissection

The paper "Memorisation, convergence and generalisation in generative models" (2605.21402) examines the statistical learning dynamics governing modern generative models, with a particular focus on disentangling the often conflated notions of memorisation, convergence, and generalisation. The authors develop an asymptotically exact theoretical framework that formalizes and quantifies these phenomena in linear and spectral generative models, then validate and extend their conclusions to high-dimensional real-world data such as images, which are characterized by power-law covariance spectra.

Analytical Framework and Definitions

The investigation is centered on generative models parameterized by empirical covariances of high-dimensional Gaussian samples. The data are assumed to arise from a spiked covariance model, $$\Sigma^\star = I_d + \beta \mathbf{u} \mathbf{u}^\top$$, capturing data with a low-rank latent signal embedded in noise, a ubiquitous structure in both synthetic and real-world domains.

Three distinct macroscopic order parameters are introduced:

• Memorisation ($m$): The normalized maximum overlap between a model's generated sample and any training datum.
• Convergence ($q$): The mean overlap between outputs of two independently trained models (on disjoint datasets) when conditioned on the same latent variable.
• Latent Recovery ($Q$): The overlap between the leading eigenvectors ("principal components") of the empirical covariances produced by two independently trained models; directly probes recovery of the top latent factor.

Through exact random matrix-theoretic analysis, the study quantifies how these overlaps transition as the sample size $n$ (relative to ambient data dimension $d$) increases.

Transition Regimes and Key Theoretical Results

The analysis reveals three sharply distinct regimes as a function of sample complexity $\gamma = n/d$:

• Low Sample (Memorisation/Overfitting): For $n = \Theta(1)$, generative models essentially memorize training samples ($m \approx 1$), and convergence between independently trained models is negligible ($q \sim 1/\sqrt{d}$).
• Intermediate Sample (Convergence): At $m$0, memorisation vanishes ($m$1), but convergence $m$2 approaches $m$3 continuously — the models produce nearly identical outputs from a given latent vector, as previously observed empirically in diffusion models.
• High Sample (Latent Recovery): Crucially, convergence ($m$4) does not imply successful learning of the principal latent components. The recovery of these components ($m$5) undergoes a sharp BBP-style (Baik–Ben Arous–Péché) phase transition, controlled by the signal-to-noise ratio $m$6 and sample complexity. The thresholds for convergence and subspace recovery can be separated, so that models may converge before they consistently recover latent factors.

Figure 2: Convergence ($m$7) and latent recovery ($m$8) decoded as distinct statistical distances; $m$9 governs the decay of KL divergence, $q$0 that of maximum-sliced (MS) distance.

Overlaps, Statistical Distances, and Implications for Generalisation

The meaning and functional roles of $q$1 and $q$2 are further clarified by their relationships to principled statistical distances:

• Convergence Overlap Controls KL Divergence: The excess Kullback-Leibler (KL) divergence between the learned and ground-truth distributions is tightly bounded by the convergence overlap $q$3. Specifically, $q$4 for $q$5 approaching $q$6.
• Latent Recovery Controls Maximum-Sliced (MS) Distance: The decay of the maximum-sliced distance, a stringent integral probability metric over all linear projections, is governed by $q$7: $q$8 as $q$9. This highlights that while $Q$0 measures distributional similarity in the bulk, $Q$1 probes alignment of principal subspaces, i.e., recovery of the informative latent variables.

Figure 1: Theoretical and empirical phase transitions in latent recovery as quantified by the “rotated” subspace overlap $Q$2 on real and synthetic power-law data.

Power-Law Spectra and the Rotated Overlap

Direct application of these overlap metrics to real-world data (e.g., images) is complicated by their power-law spectral decay, which blurs the distinction between bulk and spike. The authors introduce a novel "rotated subspace overlap" $Q$3, computed after transforming data into a basis (e.g., Fourier) where the power-law structure is manifest and factored out. $Q$4 exhibits a sharp phase transition on both real and synthetic power-law data, faithfully matching the theoretical predictions for latent factor recovery.

Figure 3: $Q$5, $Q$6, and $Q$7 measured in samples from state-of-the-art diffusion models, showing clear separation between convergence and true latent recovery.

Empirical Validation in Neural Diffusion Models

The theoretical picture is validated empirically by training denoising diffusion probabilistic models (DDPM) on subsets of CelebA and ImageNet. Overlaps $Q$8, $Q$9, $n$0, and $n$1 are computed on generated samples as a function of training data size. The findings corroborate the predicted sequence: memorization at low sample complexity, convergence at moderate sample complexity, with sharp latent recovery detected by $n$2 at a distinctly higher sample complexity.

Figure 4: Spectral structure of image correlation matrices in Fourier space, further substantiating the separation of bulk and spike phenomena in real data.

Broader Implications and Theoretical Perspectives

This rigorous separation between convergence and latent recovery has substantial implications:

• Generalisation in Generative Models is Multi-Objective: High-fidelity sample generation (convergence) does not guarantee correct inference or generation of novel structured content (latent recovery).
• Evaluation Protocols Must Go Beyond Bulk Statistics: KL-divergence or FID-type metrics may be insufficient for tasks where recovery of structured latent variables is critical (e.g., scientific modelling, causal inference).
• Architectural Biases and Inductive Priors: CNN-based denoisers’ translation/locality biases accelerate convergence in image domains but may not enhance latent recovery equivalently. The effects of other inductive biases warrant systematic study.
• Applications Beyond Vision: Analogous phenomena are predicted in graph generative models (e.g., stochastic block models) and other structured domains with low-rank signals embedded in heavy-tailed background variation.

Future Directions

Noteworthy research avenues highlighted include:

• Extension to non-Gaussian, nonlinear generative models (e.g., random features, ICA, autoencoders).
• Characterization of the dynamics of these overlap parameters during training and sampling.
• Adapting these theoretical metrics to autoregressive and transformer-based models, which present additional compositional and temporal dependencies.

Conclusion

This paper establishes a comprehensive, quantitative framework delineating memorisation, convergence, and generalisation in high-dimensional generative models, rigorously distinguishing agreement on bulk statistics from recovery of informative latent variables. The resulting insights clarify both the limitations of conventional evaluation metrics and the need for multidimensional generalisation criteria, particularly in models intended for scientific understanding or structured synthesis tasks. The theoretical predictions, robust across synthetic and deep generative models, lay the groundwork for further systematic investigations into the statistical physics of modern generative modeling.