Beatrice Achilli

Enrico Ventura, Beatrice Achilli, Luca Ambrogioni et al.

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.

Beatrice Achilli, Marco Benedetti, Giulio Biroli et al.

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.

Beatrice Achilli, Enrico Ventura, Gianluigi Silvestri et al.

Generative diffusion processes are state-of-the-art machine learning models deeply connected with fundamental concepts in statistical physics. Depending on the dataset size and the capacity of the network, their behavior is known to transition from an associative memory regime to a generalization phase in a phenomenon that has been described as a glassy phase transition. Here, using statistical physics techniques, we extend the theory of memorization in generative diffusion to manifold-supported data. Our theoretical and experimental findings indicate that different tangent subspaces are lost due to memorization effects at different critical times and dataset sizes, which depend on the local variance of the data along their directions. Perhaps counterintuitively, we find that, under some conditions, subspaces of higher variance are lost first due to memorization effects. This leads to a selective loss of dimensionality where some prominent features of the data are memorized without a full collapse on any individual training point. We validate our theory with a comprehensive set of experiments on networks trained both in image datasets and on linear manifolds, which result in a remarkable qualitative agreement with the theoretical predictions.