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Marc Mézard

Publications and source records attributed to Marc Mézard.

At least 19 recordsLinked to original sources
arXiv2026

First Learn, Then Memorize: The Spectral Bias of Diffusion Models

Diffusion models trained on a finite dataset first learn to generate novel, high-quality samples and only much later collapse onto their training set. We identify the mechanism behind this separation of timescales and the object that probes it. The training dynamics of the score function are governed---exactly, and at any width---by the Gram matrix of the Neural Tangent Kernel (NTK) evaluated on the noisy training data, so the timescales of generalization and of memorization must be encoded in its spectrum. We show that they are, and that the structure responsible has no analogue in standard kernel settings. The use of multiple noise realizations per sample ($m$ noised copies at a fixed noise level) in the score-matching loss is what restructures the Gram matrix spectrum into two distinct parts. The first, of large eigenvalues, carries the global features of the target distribution and is present already for $m=1$. The second, which the repeated noising creates, consists of the smallest eigenvalues and is supported on eigenvectors aligned with the sample-specific noise directions; it sets a memorization timescale parametrically larger in the training set size $n$. We establish this picture on two fronts. Analytically, we solve the spectrum in the lazy high-dimensional limit for both linear ($n \asymp d$) and polynomial ($n \asymp d^k$) sample complexities, and prove through a bias--variance decomposition that the first bulk minimizes the approximation error while the second drives the error associated with memorization. Empirically, we show the same two-bulk structure in Convolutional NTKs on CelebA and in finite-width U-Nets trained well beyond the lazy regime, and we make the link causal: truncating the Gram matrix at rank $r$ tunes the generalization--memorization transition, and an $L_2$ penalty targeting the second bulk suppresses memorization in feature-learning U-Nets.

cs.LG↗
arXiv2026

Double Descent and Malign Overfitting in Diffusion Models

Conventional wisdom in deep learning holds that overparameterization---having more parameters $p$ than training samples $n$---is benign: larger models generalize better and, even without regularization, interpolating models generalize well, the test error following a double-descent curve. One might expect the same benign overfitting for diffusion models, whose training reduces to regression, i.e. to minimizing a quadratic score-matching loss. Yet the opposite is observed: overfitting here is catastrophic, driving the model into a memorization regime. We resolve this paradox by combining experiments on U-Nets trained on CelebA with a random-features model for which we derive closed-form learning curves. We show that with a fixed number $m$ of noise realizations per training sample, an interpolation peak does occur, but at $p\sim nm$ rather than at $p\sim n$ as in standard regression. The rise of the test loss, however, sets in much earlier, at $p\sim n$, independently of $m$. This overfitting is malign because, although the implicit regularization of training is fully at work, it drives the model toward the empirical score, which memorizes the training set, rather than toward the true score. A bias-variance decomposition pinpoints the mechanism: the bias of the score estimator starts to grow at $p\sim n$; past the peak the variance decays, as in regression, whereas the bias keeps growing and both saturate at a large value. Since diffusion models are trained with $m\gg1$, the peak is pushed to very large model sizes, and therefore sit on the rising branch that precedes it, where malign overfitting is already in play. Nevertheless, overparameterization remains beneficial when paired with regularization: in the random-features theory and in U-Net experiments, optimally regularized large models---via a ridge penalty or early stopping, respectively---outperform any unregularized models.

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arXiv2026

Collision Resistance of Single-Layer Neural Nets

We initiate the study of the algorithmic complexity of finding collisions in single-layer binary neural networks. Given a random matrix $\mathbf{A} \in \mathbb{R}^{m\times n}$, an input $\mathbf{x} \in \{-1,1\}^n$ is mapped to a binary output vector $φ(\mathbf{A}\mathbf{x})\in \{-1,1\}^m$, where $φ$ is an activation function with constant behavior on $[κ, \infty)$ for some threshold $κ\geq 0$. We identify the threshold scale $κ=Θ(1/\sqrtα)$, where $α=m/n$, as separating two complementary phenomena. When $κ\ll 1/\sqrtα$, we give a simple online algorithm that efficiently produces extensive collisions. When $κ\gg 1/\sqrtα$, for a natural \emph{randomized} non-periodic activation and suitable oscillation complexity, we prove that the extensive-collision space exhibits an overlap gap property (OGP), yielding an exponential lower bound against online algorithms. Ours is the first work to use the overlap gap property as a rigorous criterion for collision resistance. The key difference between collision finding and average-case search is that collision finding has a new ``worst-case'' aspect: the collision finder has full control over the choice of colliding pairs. Our lower bound is proved in the online model; extending such guarantees to broader classes of algorithms, including spectral, algebraic, lattice-based, or quantum methods, remains an open direction.

cs.CR↗
arXiv2026

Biased Generalization in Diffusion Models

Generalization in generative modeling is defined as the ability to learn an underlying distribution from a finite dataset and produce novel samples, with evaluation largely driven by held-out performance and perceived sample quality. In practice, training is often stopped at the minimum of the test loss, taken as an operational indicator of generalization. We challenge this viewpoint by identifying a phase of biased generalization during training, in which the model continues to decrease the test loss while favoring samples with anomalously high proximity to training data. By training the same network on two disjoint datasets and comparing the mutual distances of generated samples and their similarity to training data, we introduce a quantitative measure of bias and demonstrate its presence on real images. We then study the mechanism of bias, using a controlled hierarchical data model where access to exact scores and ground-truth statistics allows us to precisely characterize its onset. We attribute this phenomenon to the sequential nature of feature learning in deep networks, where coarse structure is learned early in a data-independent manner, while finer features are resolved later in a way that increasingly depends on individual training samples. Our results show that early stopping at the test loss minimum, while optimal under standard generalization criteria, may be insufficient for privacy-critical applications.

cs.LG↗
arXiv2026

Theory of Speciation Transitions in Diffusion Models with General Class Structure

Diffusion Models generate data by reversing a stochastic diffusion process, progressively transforming noise into structured samples drawn from a target distribution. Recent theoretical work has shown that this backward dynamics can undergo sharp qualitative transitions, known as speciation transitions, during which trajectories become dynamically committed to data classes. Existing theoretical analyses, however, are limited to settings where classes are identifiable through first moments, such as mixtures of Gaussians with well-separated means. In this work, we develop a general theory of speciation in diffusion models that applies to arbitrary target distributions admitting well-defined classes. We formalize the notion of class structure through Bayes classification and characterize speciation times in terms of free-entropy difference between classes. This criterion recovers known results in previously studied Gaussian-mixture models, while extending to situations in which classes are not distinguishable by first moments and may instead differ through higher-order or collective features. Our framework also accommodates multiple classes and predicts the existence of successive speciation times associated with increasingly fine-grained class commitment. We illustrate the theory on two analytically tractable examples: mixtures of one-dimensional Ising models at different temperatures and mixtures of zero-mean Gaussians with distinct covariance structures. In the Ising case, we obtain explicit expressions for speciation times by mapping the problem onto a random-field Ising model and solving it via the replica method. Our results provide a unified and broadly applicable description of speciation transitions in diffusion-based generative models.

cs.LG↗
arXiv2025

Dynamical Learning in Deep Asymmetric Recurrent Neural Networks

We investigate recurrent neural networks with asymmetric interactions and demonstrate that the inclusion of self-couplings or sparse excitatory inter-module connections leads to the emergence of a densely connected manifold of dynamically accessible stable configurations. This representation manifold is exponentially large in system size and is reachable through simple local dynamics, despite constituting a subdominant subset of the global configuration space. We further show that learning can be implemented directly on this structure via a fully local, gradient-free mechanism that selectively stabilizes a single task-relevant network configuration. Unlike error-driven or contrastive learning schemes, this approach does not require explicit comparisons between network states obtained with and without output supervision. Instead, transient supervisory signals bias the dynamics toward the representation manifold, after which local plasticity consolidates the attained configuration, effectively shaping the latent representation space. Numerical evaluations on standard image classification benchmarks indicate performance comparable to that of multilayer perceptrons trained using backpropagation. More generally, these results suggest that the dynamical accessibility of fixed points and the stabilization of internal network dynamics constitute viable alternative principles for learning in recurrent systems, with conceptual links to statistical physics and potential implications for biologically motivated and neuromorphic computing architectures.

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arXiv2025

Overlap Gap and Computational Thresholds in the Square Wave Perceptron

Square Wave Perceptrons (SWPs) form a class of neural network models with oscillating activation function that exhibit intriguing ``hardness'' properties in the high-dimensional limit at a fixed constraint density $α= O(1)$. In this work, we examine two key aspects of these models. The first is related to the so-called \emph{overlap-gap property}, that is a disconnectivity feature of the geometry of the solution space of combinatorial optimization problems proven to cause the failure of a large family of solvers, and conjectured to be a symptom of algorithmic hardness. We identify, both in the storage and in the teacher-student settings, the emergence of an overlap gap at a threshold $α_{\mathrm{OGP}}(δ)$, which can be made arbitrarily small by suitably increasing the frequency of oscillations $1/δ$ of the activation. This suggests that in this small-$δ$ regime, typical instances of the problem are hard to solve even for small values of $α$. Second, in the teacher-student setup, we show that the recovery threshold of the planted signal for message-passing algorithms can be made arbitrarily large by reducing $δ$. These properties make SWPs both a challenging benchmark for algorithms and an interesting candidate for cryptographic applications.

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arXiv2025

Are Neural Networks Collision Resistant?

When neural networks are trained to classify a dataset, one finds a set of weights from which the network produces a label for each data point. We study the algorithmic complexity of finding a collision in a single-layer neural net, where a collision is defined as two distinct sets of weights that assign the same labels to all data. For binary perceptrons with oscillating activation functions, we establish the emergence of an overlap gap property in the space of collisions. This is a topological property believed to be a barrier to the performance of efficient algorithms. The hardness is supported by numerical experiments using approximate message passing algorithms, for which the algorithms stop working well below the value predicted by our analysis. Neural networks provide a new category of candidate collision resistant functions, which for some parameter setting depart from constructions based on lattices. Beyond relevance to cryptography, our work uncovers new forms of computational hardness emerging in large neural networks which may be of independent interest.

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arXiv2025

Why Diffusion Models Don't Memorize: The Role of Implicit Dynamical Regularization in Training

Diffusion models have achieved remarkable success across a wide range of generative tasks. A key challenge is understanding the mechanisms that prevent their memorization of training data and allow generalization. In this work, we investigate the role of the training dynamics in the transition from generalization to memorization. Through extensive experiments and theoretical analysis, we identify two distinct timescales: an early time $τ_\mathrm{gen}$ at which models begin to generate high-quality samples, and a later time $τ_\mathrm{mem}$ beyond which memorization emerges. Crucially, we find that $τ_\mathrm{mem}$ increases linearly with the training set size $n$, while $τ_\mathrm{gen}$ remains constant. This creates a growing window of training times with $n$ where models generalize effectively, despite showing strong memorization if training continues beyond it. It is only when $n$ becomes larger than a model-dependent threshold that overfitting disappears at infinite training times. These findings reveal a form of implicit dynamical regularization in the training dynamics, which allow to avoid memorization even in highly overparameterized settings. Our results are supported by numerical experiments with standard U-Net architectures on realistic and synthetic datasets, and by a theoretical analysis using a tractable random features model studied in the high-dimensional limit.

cs.LG↗
arXiv2025

How transformers learn structured data: insights from hierarchical filtering

Understanding the learning process and the embedded computation in transformers is becoming a central goal for the development of interpretable AI. In the present study, we introduce a hierarchical filtering procedure for data models of sequences on trees, allowing us to hand-tune the range of positional correlations in the data. Leveraging this controlled setting, we provide evidence that vanilla encoder-only transformers can approximate the exact inference algorithm when trained on root classification and masked language modeling tasks, and study how this computation is discovered and implemented. We find that correlations at larger distances, corresponding to increasing layers of the hierarchy, are sequentially included by the network during training. By comparing attention maps from models trained with varying degrees of filtering and by probing the different encoder levels, we find clear evidence of a reconstruction of correlations on successive length scales corresponding to the various levels of the hierarchy, which we relate to a plausible implementation of the exact inference algorithm within the same architecture.

cs.LG↗
arXiv2025

Memorization and Generalization in Generative Diffusion under the Manifold Hypothesis

We study the memorization and generalization capabilities of Diffusion Models (DMs) when data lies on a structured latent manifold. Specifically, we consider a set of $P$ data points in $N$ dimensions confined to a latent subspace of dimension $D = α_D N$, following the Hidden Manifold Model (HMM). We analyze the reverse diffusion process using the empirical score function as a proxy, and characterize it in the high-dimensional limit $P = \exp(αN)$, $N \gg 1$, by exploiting a connection with the Random Energy Model (REM). We show that a characteristic time $t_o$ marks the emergence of traps in the time-dependent potential, which however do not affect typical trajectories. The size of their basins of attraction is computed at all times. We derive the collapse time $t_c < t_o$, at which trajectories fall into the basin of a training point, signaling memorization. An explicit formula for $t_c$ as a function of $P$ and $α_D$ shows that the curse of dimensionality is avoided for structured data ($α_D \ll 1$), even with nonlinear manifolds. We also prove that collapse corresponds to the condensation transition in the REM. Generalization is quantified via the Kullback-Leibler divergence between the exact distribution and the reverse one at time $t$. We find a distinct time $t_g < t_c < t_o$ minimizing this divergence. Surprisingly, the best generalization occurs inside the memorization phase. We conclude that generalization in DMs improves with data structure, as $t_g \to 0$ faster than $t_c$ when $α_D \to 0$.

cond-mat.dis-nn↗
arXiv2024

Kernel Density Estimators in Large Dimensions

This paper studies Kernel Density Estimation for a high-dimensional distribution $ρ(x)$. Traditional approaches have focused on the limit of large number of data points $n$ and fixed dimension $d$. We analyze instead the regime where both the number $n$ of data points $y_i$ and their dimensionality $d$ grow with a fixed ratio $α=(\log n)/d$. Our study reveals three distinct statistical regimes for the kernel-based estimate of the density $\hat ρ_h^{\mathcal {D}}(x)=\frac{1}{n h^d}\sum_{i=1}^n K\left(\frac{x-y_i}{h}\right)$, depending on the bandwidth $h$: a classical regime for large bandwidth where the Central Limit Theorem (CLT) holds, which is akin to the one found in traditional approaches. Below a certain value of the bandwidth, $h_{CLT}(α)$, we find that the CLT breaks down. The statistics of $\hatρ_h^{\mathcal {D}}(x)$ for a fixed $x$ drawn from $ρ(x)$ is given by a heavy-tailed distribution (an alpha-stable distribution). In particular below a value $h_G(α)$, we find that $\hatρ_h^{\mathcal {D}}(x)$ is governed by extreme value statistics: only a few points in the database matter and give the dominant contribution to the density estimator. We provide a detailed analysis for high-dimensional multivariate Gaussian data. We show that the optimal bandwidth threshold based on Kullback-Leibler divergence lies in the new statistical regime identified in this paper. As known by practitioners, when decreasing the bandwidth a Kernel-estimated estimated changes from a smooth curve to a collections of peaks centred on the data points. Our findings reveal that this general phenomenon is related to sharp transitions between phases characterized by different statistical properties, and offer new insights for Kernel density estimation in high-dimensional settings.

cs.LG↗
arXiv2024

Dynamical Regimes of Diffusion Models

Using statistical physics methods, we study generative diffusion models in the regime where the dimension of space and the number of data are large, and the score function has been trained optimally. Our analysis reveals three distinct dynamical regimes during the backward generative diffusion process. The generative dynamics, starting from pure noise, encounters first a 'speciation' transition where the gross structure of data is unraveled, through a mechanism similar to symmetry breaking in phase transitions. It is followed at later time by a 'collapse' transition where the trajectories of the dynamics become attracted to one of the memorized data points, through a mechanism which is similar to the condensation in a glass phase. For any dataset, the speciation time can be found from a spectral analysis of the correlation matrix, and the collapse time can be found from the estimation of an 'excess entropy' in the data. The dependence of the collapse time on the dimension and number of data provides a thorough characterization of the curse of dimensionality for diffusion models. Analytical solutions for simple models like high-dimensional Gaussian mixtures substantiate these findings and provide a theoretical framework, while extensions to more complex scenarios and numerical validations with real datasets confirm the theoretical predictions.

cs.LG↗
arXiv2024

The Exponential Capacity of Dense Associative Memories

Recent generalizations of the Hopfield model of associative memories are able to store a number $P$ of random patterns that grows exponentially with the number $N$ of neurons, $P=\exp(αN)$. Besides the huge storage capacity, another interesting feature of these networks is their connection to the attention mechanism which is part of the Transformer architectures widely applied in deep learning. In this work, we study a generic family of pattern ensembles using a statistical mechanics analysis which gives exact asymptotic thresholds for the retrieval of a typical pattern, $α_1$, and lower bounds for the maximum of the load $α$ for which all patterns can be retrieved, $α_c$, as well as sizes of attraction basins. We discuss in detail the cases of Gaussian and spherical patterns, and show that they display rich and qualitatively different phase diagrams.

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arXiv2023

Spin glass theory and its new challenge: structured disorder

This paper first describes, from a high level viewpoint, the main challenges that had to be solved in order to develop a theory of spin glasses in the last fifty years. It then explains how important inference problems, notably those occurring in machine learning, can be formulated as problems in statistical physics of disordered systems. However, the main questions that we face in the analysis of deep networks require to develop a new chapter of spin glass theory, which will address the challenge of structured data.

cond-mat.dis-nn↗
arXiv2023

Generative diffusion in very large dimensions

Generative models based on diffusion have become the state of the art in the last few years, notably for image generation. Here, we analyse them in the high-dimensional limit, where data are formed by a very large number of variables. We use methods from statistical physics and focus on two well-controlled high-dimensional cases: a Gaussian model and the Curie-Weiss model of ferromagnetism. In the latter case, we highlight the mechanism of symmetry breaking in the inverse diffusion, and point out that, in order to reconstruct the relative asymmetry of the two low-temperature states, and thus to obtain the correct probability weights, one needs a database with a number of points much larger than the dimension of each data point. We characterize the scaling laws in the number of data and in the number of dimensions for an efficient generation.

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arXiv2023

The Decimation Scheme for Symmetric Matrix Factorization

Matrix factorization is an inference problem that has acquired importance due to its vast range of applications that go from dictionary learning to recommendation systems and machine learning with deep networks. The study of its fundamental statistical limits represents a true challenge, and despite a decade-long history of efforts in the community, there is still no closed formula able to describe its optimal performances in the case where the rank of the matrix scales linearly with its size. In the present paper, we study this extensive rank problem, extending the alternative 'decimation' procedure that we recently introduced, and carry out a thorough study of its performance. Decimation aims at recovering one column/line of the factors at a time, by mapping the problem into a sequence of neural network models of associative memory at a tunable temperature. Though being sub-optimal, decimation has the advantage of being theoretically analyzable. We extend its scope and analysis to two families of matrices. For a large class of compactly supported priors, we show that the replica symmetric free entropy of the neural network models takes a universal form in the low temperature limit. For sparse Ising prior, we show that the storage capacity of the neural network models diverges as sparsity in the patterns increases, and we introduce a simple algorithm based on a ground state search that implements decimation and performs matrix factorization, with no need of an informative initialization.

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