  Tensor-network techniques have enjoyed outstanding success in physics, and
have recently attracted attention in machine learning, both as a tool for the
formulation of new learning algorithms and for enhancing the mathematical
understanding of existing methods. Inspired by these developments, and the
natural correspondence between tensor networks and probabilistic graphical
models, we provide a rigorous analysis of the expressive power of various
tensor-network factorizations of discrete multivariate probability
distributions. These factorizations include non-negative tensor-trains/MPS,
which are in correspondence with hidden Markov models, and Born machines, which
are naturally related to local quantum circuits. When used to model probability
distributions, they exhibit tractable likelihoods and admit efficient learning
algorithms. Interestingly, we prove that there exist probability distributions
for which there are unbounded separations between the resource requirements of
some of these tensor-network factorizations. Particularly surprising is the
fact that using complex instead of real tensors can lead to an arbitrarily
large reduction in the number of parameters of the network. Additionally, we
introduce locally purified states (LPS), a new factorization inspired by
techniques for the simulation of quantum systems, with provably better
expressive power than all other representations considered. The ramifications
of this result are explored through numerical experiments. Our findings imply
that LPS should be considered over hidden Markov models, and furthermore
provide guidelines for the design of local quantum circuits for probabilistic
modeling.
