Cascade of phase transitions in the training of energy-based models

In this paper, we investigate the feature encoding process in a prototypical energy-based generative model, the Restricted Boltzmann Machine (RBM). We start with an analytical investigation using simplified architectures and data structures, and end with numerical analysis of real trainings on real datasets. Our study tracks the evolution of the model’s weight matrix through its singular value decomposition, revealing a series of thermodynamic phase transitions that shape the principal learning modes of the empirical probability distribution. We first describe this process analytically in several controlled setups that allow us to fully monitor the training dynamics until convergence. We then validate these findings by training the Bernoulli-Bernoulli RBM on real data sets. By studying the phase behavior over data sets of increasing dimension, we show that these phase transitions are genuine in the thermodynamic sense. Moreover, we propose a mean-field finite-size scaling hypothesis, confirming that the initial phase transition, reminiscent of the paramagnetic-to-ferromagnetic phase transition in mean-field ferromagnetism models, is governed by mean-field critical exponents.