Thermodynamics of Restricted Boltzmann Machines and related learning dynamics
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We analyze the learning process of the restricted Boltzmann machine (RBM), a certain type of generative models used in the context of unsupervised learning. In a first step, we investigate the thermodynamics properties by considering a realistic statistical ensemble of RBM, assuming the information content of the RBM to be mainly reflected by spectral properties of its weight matrix W. A phase diagram is obtained which seems at first sight similar to the one of the Sherrington-Kirkpatrick (SK) model with ferromagnetic couplings. The main difference resides in the structure of the ferromagnetic phase which may or may not be of compositional type, depending mainly on the distribution's kurtosis of the singular vectors components of W. In a second step the learning dynamics of an RBM from arbitrary data is studied in thermodynamic limit. A "typical" learning trajectory is shown to solve an effective dynamical equation, based on the aforementioned ensemble average and involving explicitly order parameters obtained from the thermodynamic analysis. This accounts in particular for the dominant singular values evolution and how this is driven by the input data: in the linear regime at the beginning of the learning, they correspond to unstable deformation modes of W reflecting dominant covariance modes of the data. In the non-linear regime it is seen how the selected modes interact in later stages of the learning procedure, by eventually imposing a matching between order parameters with their empirical counterparts estimated from the data. Experiments on both artificial and real data illustrate these considerations, showing in particular how the RBM operates in the ferromagnetic compositional phase.
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