Exploration of the generative capabilities of Boltzmann machines applied to social systems under the majority rule

Energy-based models (EBMs) are flexible generative architectures inspired by statistical physics, but their learning and generative properties remain poorly understood. Here, we analyze a solvable EBM in the high-dimensional limit: the spherical Boltzmann machine (SBM). Combining tools from random matrix theory and dynamical mean-field theory, we: solve exact equations describing the training dynamics of the SBM; compute the Bayesian evidence, which acts as a partition function in parameter space and encodes global properties of the trained model; and uncover cascades of phase transitions that occur both during training and as a function of hyperparameters, related to successive alignment and condensation of the top modes of the coupling matrix to the data. We connect these transitions to sampling-time generative phenomena in a teacher-student scenario, including: sampling temperature tuning, double descent as a function of regularization strength, tempered posterior effects, and out-of-equilibrium effects during training that induce biases in the trained model. We provide numerical evidence demonstrating that all these phenomena appear in standard generative architectures, beyond the SBM.

Fixed points of recurrent neural networks can be leveraged to store and generate information. These fixed points are captured by the Boltzmann-Gibbs measure, which leads to neural Langevin dynamics that relax to those fixed points for generative learning of a real dataset. We call this type of generative model a neural Langevin machine, which derives an asymmetric and firing-rate-speed-adjusted learning rule requiring only local neural signals, thereby bearing biological relevance in terms of local predictive learning. An out-of-equilibrium regime of the generative process is revealed, together with a memorization-to-generalization transition with increasing training data size. The neuro-inspired machine can also realize a continuous exploration of the phase space for different kinds of generative images and can denoise a corrupted image as well.

Deep Belief Networks (DBNs) learn hierarchical generative models without class supervision. Here, we ask whether this purely unsupervised process nevertheless organizes internal representations according to the unknown data classes. We analyze successive layers of DBNs trained on MNIST, Fashion-MNIST, and KMNIST using the Generalized Discrimination Value (GDV), supervised probes applied only after training, a reconstruction-based measure of abstraction distance, effective dimensionality, and free sample generation. Remarkably, class-specific clustering generally increases with depth across datasets and network widths, although no label information is available during DBN training. Control experiments show that this effect depends on the learned feature structure and cannot be explained by random transformations, weight marginals, dimensionality reduction, or sigmoid saturation. The first hidden layers also frequently make class identity more accessible to linear and nonlinear probes. With greater depth, representations become increasingly compact and prototype-like as neurons acquire correlated feature directions. At the same time, GDV and probe accuracy reveal complementary aspects of class structure: improved average clustering can coexist with reduced accessibility for a few difficult class pairs. These findings demonstrate that layer-wise generative learning can spontaneously uncover and progressively amplify class-related structure in unlabeled data.