Quantum mechanics

Constructive

Boltzmann machine

Process (computing)

Pure mathematics

Discrete mathematics

Equivalence (formal languages)

Entropy (arrow of time)

Theoretical computer science

Operating system

Computer science

Statistical physics

Boltzmann constant

Tensor (intrinsic definition)

Physics

Quantum entanglement

Restricted Boltzmann machine

Mathematics

Artificial intelligence

Quantum

Deep learning

Physical review. B./Physical review. B

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.

Equivalence of restricted Boltzmann machines and tensor network states