# Equivalence of restricted Boltzmann machines and tensor network states

## TL;DR

This paper establishes a formal equivalence between restricted Boltzmann machines and tensor network states, enabling cross-disciplinary insights and more efficient representations in deep learning and quantum physics.

## Contribution

It provides algorithms to convert between RBMs and TNS, and establishes conditions for their transformability, bridging deep learning and quantum many-body physics.

## Key findings

- RBM can be rigorously analyzed using TNS entanglement entropy bounds.
- TNS can be efficiently represented by RBMs with fewer parameters.
- Insights from TNS entanglement capacity inform deep learning architecture design.

## 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.