Boltzmann machines as two-dimensional tensor networks

TL;DR

This paper establishes a fundamental connection between Boltzmann machines and tensor networks, showing they can be exactly represented as two-dimensional tensor networks, which enhances understanding and computational efficiency.

Contribution

It introduces an exact tensor network representation of RBM and DBM, linking their expressive power to entanglement structures and providing an efficient contraction algorithm.

Findings

The tensor network representation improves partition function estimation accuracy.

The proposed algorithm outperforms existing machine learning methods.

Potential applications in training deep Boltzmann machines for various tasks.

Abstract

Restricted Boltzmann machines (RBM) and deep Boltzmann machines (DBM) are important models in machine learning, and recently found numerous applications in quantum many-body physics. We show that there are fundamental connections between them and tensor networks. In particular, we demonstrate that any RBM and DBM can be exactly represented as a two-dimensional tensor network. This representation gives an understanding of the expressive power of RBM and DBM using entanglement structures of the tensor networks, also provides an efficient tensor network contraction algorithm for the computing partition function of RBM and DBM. Using numerical experiments, we demonstrate that the proposed algorithm is much more accurate than the state-of-the-art machine learning methods in estimating the partition function of restricted Boltzmann machines and deep Boltzmann machines, and have potential applications in training deep Boltzmann machines for general machine learning tasks.