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Tensor-network techniques have enjoyed outstanding success in physics, andhave recently attracted attention in machine learning, both as a tool for theformulation of new learning algorithms and for enhancing the mathematicalunderstanding of existing methods. Inspired by these developments, and thenatural correspondence between tensor networks and probabilistic graphicalmodels, we provide a rigorous analysis of the expressive power of varioustensor-network factorizations of discrete multivariate probabilitydistributions. These factorizations include non-negative tensor-trains/MPS,which are in correspondence with hidden Markov models, and Born machines, whichare naturally related to local quantum circuits. When used to model probabilitydistributions, they exhibit tractable likelihoods and admit efficient learningalgorithms. Interestingly, we prove that there exist probability distributionsfor which there are unbounded separations between the resource requirements ofsome of these tensor-network factorizations. Particularly surprising is thefact that using complex instead of real tensors can lead to an arbitrarilylarge reduction in the number of parameters of the network. Additionally, weintroduce locally purified states (LPS), a new factorization inspired bytechniques for the simulation of quantum systems, with provably betterexpressive power than all other representations considered. The ramificationsof this result are explored through numerical experiments. Our findings implythat LPS should be considered over hidden Markov models, and furthermoreprovide guidelines for the design of local quantum circuits for probabilisticmodeling.

Expressive power of tensor-network factorizations for probabilistic  modeling, with applications from hidden Markov models to quantum machine  learning