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pdf:docinfo:title: Expressive power of tensor-network factorizations for probabilistic modeling
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subject: Neural Information Processing Systems http://nips.cc/
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dc:title: Expressive power of tensor-network factorizations for probabilistic modeling
Book: Advances in Neural Information Processing Systems 32
pdf:docinfo:custom:Date: 2019
modified: 2020-06-16T14:09:53Z
Description-Abstract: 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.
cp:subject: Neural Information Processing Systems http://nips.cc/
pdf:docinfo:subject: Neural Information Processing Systems http://nips.cc/
pdf:docinfo:custom:Created: 2019
pdf:docinfo:creator: Ivan Glasser, Ryan Sweke, Nicola Pancotti, Jens Eisert, Ignacio Cirac
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pdf:docinfo:custom:Type: Conference Proceedings
Editors: H. Wallach and H. Larochelle and A. Beygelzimer and F. d'Alché-Buc and E. Fox and R. Garnett
Author: Ivan Glasser, Ryan Sweke, Nicola Pancotti, Jens Eisert, Ignacio Cirac
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pdf:docinfo:custom:Description: Paper accepted and presented at the Neural Information Processing Systems Conference (http://nips.cc/)
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dc:description: Neural Information Processing Systems http://nips.cc/
Description: Paper accepted and presented at the Neural Information Processing Systems Conference (http://nips.cc/)
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dc:creator: Ivan Glasser, Ryan Sweke, Nicola Pancotti, Jens Eisert, Ignacio Cirac
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dcterms:created: 2020-06-16T14:09:53Z
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title: Expressive power of tensor-network factorizations for probabilistic modeling
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pdf:docinfo:custom:Book: Advances in Neural Information Processing Systems 32
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creator: Ivan Glasser, Ryan Sweke, Nicola Pancotti, Jens Eisert, Ignacio Cirac
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pdf:docinfo:custom:Editors: H. Wallach and H. Larochelle and A. Beygelzimer and F. d'Alché-Buc and E. Fox and R. Garnett
pdf:docinfo:custom:Description-Abstract: 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.
pdf:docinfo:custom:Published: 2019
Published: 2019
pdf:docinfo:custom:Publisher: Curran Associates, Inc.
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