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Enrico Ventura

Publications and source records attributed to Enrico Ventura.

12 recordsLinked to original sources
arXiv2026

Emergence of Distortions in High-Dimensional Guided Diffusion Models

Classifier-free guidance (CFG) is the de facto standard for conditional sampling in diffusion models, yet it often reduces sample diversity. Using tools from statistical physics, we analyze the emergence of generative distortions induced by CFG, namely the mismatch between the CFG sampling distribution and the true conditional distribution. We study this phenomenon in analytically tractable settings with exact score functions, characterizing its dependence on data dimensionality and the number of classes. For high-dimensional Gaussian mixtures, we use dynamic mean-field theory to show that distortions arise when the number of classes scales exponentially with the data dimension, whereas they vanish in the sub-exponential regime due to a dynamical phase transition. We further prove that, in the infinite-class limit, distortions remain unavoidable regardless of dimensionality because of the increasing density of classes. Finally, we show that standard CFG schedules cannot prevent variance shrinkage, and we propose a theoretically grounded guidance schedule incorporating a negative-guidance window that improves both class separability and sample diversity in real-world latent diffusion models.

stat.ML↗
arXiv2026

The Interplay of Data Structure and Imbalance in the Learning Dynamics of Diffusion Models

Real-world datasets are inherently heterogeneous, yet how per-class structural differences and sampling imbalance shape the training dynamics of diffusion models-and potentially exacerbate disparities-remains poorly understood. While models typically transition from an initial phase of generalization to memorizing the training set, existing theory assumes homogeneous data, leaving open how class imbalance and heterogeneity reshape these dynamics. In this work, we develop a high-dimensional analytical framework to study class-dependent learning in score-based diffusion models. Analyzing a random-features model trained on Gaussian mixtures, we derive the feature-covariance spectrum to characterize per-class generalization and memorization times. We reveal the explicit hierarchy governing these dynamics: class variance is the primary determinant of learning order-consistently favoring higher-variance classes-while centroid geometry plays a secondary role. Sampling imbalance acts as a modulator that can reverse this ordering and, under strong imbalance, forces minority classes to acquire distinct, delayed speciation times during backward diffusion. Together, these results suggest that diffusion models can memorize some classes while others remain insufficiently learned. We validate our theoretical predictions empirically using U-Net models trained on Fashion MNIST.

stat.ML↗
arXiv2026

Losing dimensions: Geometric memorization in generative diffusion

Diffusion models power leading generative AI, but when and how they memorize training data, especially on low-dimensional manifolds, remains unclear. We find memorization emerges gradually, not abruptly: as data become scarce, diffusion models experience a smooth collapse where their capacity to vary across independent directions diminishes. Measuring latent dimensionality via the learned score field, we reveal how generative behavior increasingly centers on a few examples while other variations "freeze out". We propose a geometric memorization theory, showing that salient features collapse first, then finer details, leading to near point-wise replication. This mirrors physical systems condensing into a few low-energy configurations. Our theoretical predictions align with both synthetic and real data, identifying geometric memorization as a distinct phase between generalization and exact copying.

stat.ML↗
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↗
arXiv2025

Manifolds, Random Matrices and Spectral Gaps: The geometric phases of generative diffusion

In this paper, we investigate the latent geometry of generative diffusion models under the manifold hypothesis. For this purpose, we analyze the spectrum of eigenvalues (and singular values) of the Jacobian of the score function, whose discontinuities (gaps) reveal the presence and dimensionality of distinct sub-manifolds. Using a statistical physics approach, we derive the spectral distributions and formulas for the spectral gaps under several distributional assumptions, and we compare these theoretical predictions with the spectra estimated from trained networks. Our analysis reveals the existence of three distinct qualitative phases during the generative process: a trivial phase; a manifold coverage phase where the diffusion process fits the distribution internal to the manifold; a consolidation phase where the score becomes orthogonal to the manifold and all particles are projected on the support of the data. This `division of labor' between different timescales provides an elegant explanation of why generative diffusion models are not affected by the manifold overfitting phenomenon that plagues likelihood-based models, since the internal distribution and the manifold geometry are produced at different time points during generation.

stat.ML↗
arXiv2024

Learning and Unlearning: Bridging classification, memory and generative modeling in Recurrent Neural Networks

The human brain is a complex system that is fascinating scientists since a long time. Its remarkable capabilities include categorization of concepts, retrieval of memories and creative generation of new examples. At the same time, modern artificial neural networks are trained on large amounts of data to accomplish these same tasks with a considerable degree of precision. By contrast with biological systems, machines appear to be either significantly slow and energetically expensive to train, suggesting the need for a paradigmatic change in the way they learn. We here review a general learning prescription that allows to perform classification, memorization and generation of new examples in bio-inspired artificial neural networks. The training procedure can be split into a prior Hebbian learning phase and a subsequent anti-Hebbian one (usually referred to as Unlearning). The separation of training in two epochs allows the algorithm to go fully unsupervised while partially aligning with some modern biological theories of learning.

cond-mat.dis-nn↗
arXiv2024

Training neural networks with structured noise improves classification and generalization

The beneficial role of noise-injection in learning is a consolidated concept in the field of artificial neural networks, suggesting that even biological systems might take advantage of similar mechanisms to optimize their performance. The training-with-noise algorithm proposed by Gardner and collaborators is an emblematic example of a noise-injection procedure in recurrent networks, which can be used to model biological neural systems. We show how adding structure to noisy training data can substantially improve the algorithm performance, allowing the network to approach perfect retrieval of the memories and wide basins of attraction, even in the scenario of maximal injected noise. We also prove that the so-called Hebbian Unlearning rule coincides with the training-with-noise algorithm when noise is maximal and data are stable fixed points of the network dynamics.

cond-mat.dis-nn↗
arXiv2024

Unlearning regularization for Boltzmann Machines

Boltzmann Machines (BMs) are graphical models with interconnected binary units, employed for the unsupervised modeling of data distributions. When trained on real data, BMs show the tendency to behave like critical systems, displaying a high susceptibility of the model under a small rescaling of the inferred parameters. This behaviour is not convenient for the purpose of generating data, because it slows down the sampling process, and induces the model to overfit the training-data. In this study, we introduce a regularization method for BMs to improve the robustness of the model under rescaling of the parameters. The new technique shares formal similarities with the unlearning algorithm, an iterative procedure used to improve memory associativity in Hopfield-like neural networks. We test our unlearning regularization on synthetic data generated by two simple models, the Curie-Weiss ferromagnetic model and the Sherrington-Kirkpatrick spin glass model. We show that it outperforms $L_p$-norm schemes and discuss the role of parameter initialization. Eventually, the method is applied to learn the activity of real neuronal cells, confirming its efficacy at shifting the inferred model away from criticality and coming out as a powerful candidate for actual scientific implementations.

cond-mat.dis-nn↗
arXiv2024

Demolition and Reinforcement of Memories in Spin-Glass-like Neural Networks

Statistical mechanics has made significant contributions to the study of biological neural systems by modeling them as recurrent networks of interconnected units with adjustable interactions. Several algorithms have been proposed to optimize the neural connections to enable network tasks such as information storage (i.e. associative memory) and learning probability distributions from data (i.e. generative modeling). Among these methods, the Unlearning algorithm, aligned with emerging theories of synaptic plasticity, was introduced by John Hopfield and collaborators. The primary objective of this thesis is to understand the effectiveness of Unlearning in both associative memory models and generative models. Initially, we demonstrate that the Unlearning algorithm can be simplified to a linear perceptron model which learns from noisy examples featuring specific internal correlations. The selection of structured training data enables an associative memory model to retrieve concepts as attractors of a neural dynamics with considerable basins of attraction. Subsequently, a novel regularization technique for Boltzmann Machines is presented, proving to outperform previously developed methods in learning hidden probability distributions from data-sets. The Unlearning rule is derived from this new regularized algorithm and is showed to be comparable, in terms of inferential performance, to traditional Boltzmann-Machine learning.

cond-mat.dis-nn↗
arXiv2023

Memory Storage and Retrieval in Sparsely Connected Balanced Networks

Ever since the last two decades of the past century pioneering studies in the field of statistical physics had focused their efforts on developing models of neural networks that could display memory storage and retrieval. Though many associative memory models were easy to handle and still quite effective to explain the basic memory retrieval processes in the brain, they were not satisfactory under the biological point of view. It became clear to scientists that a biologically realistic neural network should have respected typical features that were observed in experiments of neurophysiology. This aspect has led to the introduction of Balanced Networks, systems where excitatory and inhibitory neurons balance their effect on each other as an emergent property of the network dynamics. One of such models is the exhibition of a mean level of neuronal activity (i.e. the average spiking rate of neurons) that is univocally defined by a linear equation in the external input. This aspect might help to reproduce what is measured in particular areas devoted to memory storage, i.e. a persistent activity during the memory retrieval performance. Even though progresses in the matter of balanced networks where achieved in the last two decades, there is still no complete theory that conciliates memory retrieval and balance in a network of neurons. The aim of this work is to develop a biologically plausible model that presents both balance and memory retrieval, building on a framework of mean field equations that can predict the theoretical behaviour of the network under the choice of a set of control parameters. We will thus measure the critical capacity of the system as a function of these parameters, comparing the theoretical results with the numerical simulations.

cond-mat.dis-nn↗
arXiv2022

Supervised perceptron learning vs unsupervised Hebbian unlearning: Approaching optimal memory retrieval in Hopfield-like networks

The Hebbian unlearning algorithm, i.e. an unsupervised local procedure used to improve the retrieval properties in Hopfield-like neural networks, is numerically compared to a supervised algorithm to train a linear symmetric perceptron. We analyze the stability of the stored memories: basins of attraction obtained by the Hebbian unlearning technique are found to be comparable in size to those obtained in the symmetric perceptron, while the two algorithms are found to converge in the same region of Gardner's space of interactions, having followed similar learning paths. A geometric interpretation of Hebbian unlearning is proposed to explain its optimal performances. Because the Hopfield model is also a prototypical model of disordered magnetic system, it might be possible to translate our results to other models of interest for memory storage in materials.

cond-mat.dis-nn↗