  We consider restricted Boltzmann machine (RBMs) trained over an unstructured
dataset made of blurred copies of definite but unavailable ``archetypes'' and
we show that there exists a critical sample size beyond which the RBM can learn
archetypes, namely the machine can successfully play as a generative model or
as a classifier, according to the operational routine. In general, assessing a
critical sample size (possibly in relation to the quality of the dataset) is
still an open problem in machine learning. Here, restricting to the random
theory, where shallow networks suffice and the grand-mother cell scenario is
correct, we leverage the formal equivalence between RBMs and Hopfield networks,
to obtain a phase diagram for both the neural architectures which highlights
regions, in the space of the control parameters (i.e., number of archetypes,
number of neurons, size and quality of the training set), where learning can be
accomplished. Our investigations are led by analytical methods based on the
statistical-mechanics of disordered systems and results are further
corroborated by extensive Monte Carlo simulations.
