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Equivalence of restricted Boltzmann machines and tensor network states

Jing Chen, Song Cheng, Haidong Xie, Lei Wang, and Tao Xiang
Phys. Rev. B 97, 085104 – Published 2 February 2018

Abstract

The restricted Boltzmann machine (RBM) is one of the fundamental building blocks of deep learning. RBM finds wide applications in dimensional reduction, feature extraction, and recommender systems via modeling the probability distributions of a variety of input data including natural images, speech signals, and customer ratings, etc. We build a bridge between RBM and tensor network states (TNS) widely used in quantum many-body physics research. We devise efficient algorithms to translate an RBM into the commonly used TNS. Conversely, we give sufficient and necessary conditions to determine whether a TNS can be transformed into an RBM of given architectures. Revealing these general and constructive connections can cross fertilize both deep learning and quantum many-body physics. Notably, by exploiting the entanglement entropy bound of TNS, we can rigorously quantify the expressive power of RBM on complex data sets. Insights into TNS and its entanglement capacity can guide the design of more powerful deep learning architectures. On the other hand, RBM can represent quantum many-body states with fewer parameters compared to TNS, which may allow more efficient classical simulations.

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  • Received 25 January 2017
  • Revised 15 December 2017

DOI:https://doi.org/10.1103/PhysRevB.97.085104

©2018 American Physical Society

Physics Subject Headings (PhySH)

Condensed Matter, Materials & Applied PhysicsQuantum Information, Science & TechnologyNetworks

Authors & Affiliations

Jing Chen1,2,4, Song Cheng1,2, Haidong Xie1,2, Lei Wang1,*, and Tao Xiang1,3,†

  • 1Institute of Physics, Chinese Academy of Sciences, P.O. Box 603, Beijing 100190, China
  • 2School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
  • 3Collaborative Innovation Center of Quantum Matter, Beijing 100190, China
  • 4Center for Computational Quantum Physics, Flatiron Institute, New York 10010, USA

  • *wanglei@iphy.ac.cn
  • txiang@iphy.ac.cn

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Issue

Vol. 97, Iss. 8 — 15 February 2018

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