  Restricted Boltzmann machines (RBM) and deep Boltzmann machines (DBM) are
important models in machine learning, and recently found numerous applications
in quantum many-body physics. We show that there are fundamental connections
between them and tensor networks. In particular, we demonstrate that any RBM
and DBM can be exactly represented as a two-dimensional tensor network. This
representation gives an understanding of the expressive power of RBM and DBM
using entanglement structures of the tensor networks, also provides an
efficient tensor network contraction algorithm for the computing partition
function of RBM and DBM. Using numerical experiments, we demonstrate that the
proposed algorithm is much more accurate than the state-of-the-art machine
learning methods in estimating the partition function of restricted Boltzmann
machines and deep Boltzmann machines, and have potential applications in
training deep Boltzmann machines for general machine learning tasks.
