  Unsupervised neural network learning extracts hidden features from unlabeled
training data. This is used as a pretraining step for further supervised
learning in deep networks. Hence, understanding unsupervised learning is of
fundamental importance. Here, we study the unsupervised learning from a finite
number of data, based on the restricted Boltzmann machine learning. Our study
inspires an efficient message passing algorithm to infer the hidden feature,
and estimate the entropy of candidate features consistent with the data. Our
analysis reveals that the learning requires only a few data if the feature is
salient and extensively many if the feature is weak. Moreover, the entropy of
candidate features monotonically decreases with data size and becomes negative
(i.e., entropy crisis) before the message passing becomes unstable, suggesting
a discontinuous phase transition. In terms of convergence time of the message
passing algorithm, the unsupervised learning exhibits an easy-hard-easy
phenomenon as the training data size increases. All these properties are
reproduced in an approximate Hopfield model, with an exception that the entropy
crisis is absent, and only continuous phase transition is observed. This key
difference is also confirmed in a handwritten digits dataset. This study
deepens our understanding of unsupervised learning from a finite number of
data, and may provide insights into its role in training deep networks.
