  The restricted Boltzmann machine is a network of stochastic units with
undirected interactions between pairs of visible and hidden units. This model
was popularized as a building block of deep learning architectures and has
continued to play an important role in applied and theoretical machine
learning. Restricted Boltzmann machines carry a rich structure, with
connections to geometry, applied algebra, probability, statistics, machine
learning, and other areas. The analysis of these models is attractive in its
own right and also as a platform to combine and generalize mathematical tools
for graphical models with hidden variables. This article gives an introduction
to the mathematical analysis of restricted Boltzmann machines, reviews recent
results on the geometry of the sets of probability distributions representable
by these models, and suggests a few directions for further investigation.
