We investigate the thermodynamic properties of a Restricted Boltzmann Machine (RBM), a simple energy-based generative model used in the context of unsupervised learning. Assuming the information content of this model to be mainly reflected by the spectral properties of its weight matrix W , we try to make a realistic analysis by averaging over an appropriate statistical ensemble of RBMs.First, a phase diagram is derived. Otherwise similar to that of the Sherrington-Kirkpatrick (SK) model with ferromagnetic couplings, the RBM's phase diagram presents a ferromagnetic phase which may or may not be of compositional type depending on the kurtosis of the distribution of the components of the singular vectors of W .Subsequently, the learning dynamics of the RBM is studied in the thermodynamic limit. A "typical" learning trajectory is shown to solve an effective dynamical equation, based on the aforementioned ensemble average and explicitly involving order parameters obtained from the thermodynamic analysis. In particular, this let us show how the evolution of the dominant singular values of W , and thus of the unstable modes, is driven by the input data. At the beginning of the training, in which the RBM is found to operate in the linear regime, the unstable modes reflect the dominant covariance modes of the data. In the non-linear regime, instead, the selected modes interact and eventually impose a matching of the order parameters to their empirical counterparts estimated from the data.Finally, we illustrate our considerations by performing experiments on both artificial and real data, showing in particular how the RBM operates in the ferromagnetic compositional phase. arXiv:1803.01960v2 [cond-mat.dis-nn]

The Restricted Boltzmann Machine (RBM), an important tool used in machine learning in particular for unsupervized learning tasks, is investigated from the perspective of its spectral properties. Starting from empirical observations, we propose a generic statistical ensemble for the weight matrix of the RBM and characterize its mean evolution. This let us show how in the linear regime, in which the RBM is found to operate at the beginning of the training, the statistical properties of the data drive the selection of the unstable modes of the weight matrix. A set of equations characterizing the non-linear regime is then derived, unveiling in some way how the selected modes interact in later stages of the learning procedure and defining a deterministic learning curve for the RBM.

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