Boltzmann machines as two-dimensional tensor networks
This work connects machine learning models to tensor network theory, offering improved computational tools for Boltzmann machines, though it is incremental in applying existing tensor network concepts to these models.
The paper demonstrates that restricted and deep Boltzmann machines can be exactly represented as two-dimensional tensor networks, providing insights into their expressive power and enabling a more accurate algorithm for computing partition functions, with numerical experiments showing superior accuracy over state-of-the-art methods.
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.