  Revealing hidden features in unlabeled data is called unsupervised feature
learning, which plays an important role in pretraining a deep neural network.
Here we provide a statistical mechanics analysis of the unsupervised learning
in a restricted Boltzmann machine with binary synapses. A message passing
equation to infer the hidden feature is derived, and furthermore, variants of
this equation are analyzed. A statistical analysis by replica theory describes
the thermodynamic properties of the model. Our analysis confirms an entropy
crisis preceding the non-convergence of the message passing equation,
suggesting a discontinuous phase transition as a key characteristic of the
restricted Boltzmann machine. Continuous phase transition is also confirmed
depending on the embedded feature strength in the data. The mean-field result
under the replica symmetric assumption agrees with that obtained by running
message passing algorithms on single instances of finite sizes. Interestingly,
in an approximate Hopfield model, the entropy crisis is absent, and a
continuous phase transition is observed instead. We also develop an iterative
equation to infer the hyper-parameter (temperature) hidden in the data, which
in physics corresponds to iteratively imposing Nishimori condition. Our study
provides insights towards understanding the thermodynamic properties of the
restricted Boltzmann machine learning, and moreover important theoretical basis
to build simplified deep networks.
