Boltzmann machines as two-dimensional tensor networks

Restricted Boltzmann machines (RBMs) and deep Boltzmann machines (DBMs) 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 characterizations of the expressive power of RBMs and DBMs using entanglement structures of the tensor networks, and also provides an efficient tensor network contraction algorithm for the computing partition function of RBMs and DBMs. Using numerical experiments, we show that the proposed algorithm is more accurate than the state-of-the-art machine learning methods in estimating the partition function of RBMs and DBMs, and have potential applications in training DBMs for general machine learning tasks.

Automated summary

Restricted Boltzmann machines (RBMs) and deep Boltzmann machines (DBMs) are important models in machine learning, with recent applications in quantum many-body physics. This study establishes fundamental connections between RBMs and DBMs with tensor networks, and presents an efficient algorithm for computing their partition functions, showing improved accuracy compared to state-of-the-art methods. The research highlights potential applications in training DBMs and estimating the partition function of RBMs.