Poster
Expressive power of tensor-network factorizations for probabilistic modeling
Ivan Glasser · Ryan Sweke · Nicola Pancotti · Jens Eisert · Ignacio Cirac

Wed Dec 11th 05:00 -- 07:00 PM @ East Exhibition Hall B + C #96

Tensor-network techniques have recently proven useful 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 the probabilistic interpretation of 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. Of particular interest, 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.

Author Information

Ivan Glasser (Max Planck Institute of Quantum Optics)
Ryan Sweke (Freie Universitaet Berlin)
Nicola Pancotti (Max Planck Institute of Quantum Optics)
Jens Eisert (Freie Universitaet Berlin)
Ignacio Cirac (Max-Planck Institute of Quantum Optics)