  Progress in the application of machine learning techniques to the prediction
of solid-state and molecular materials properties has been greatly facilitated
by the development state-of-the-art feature representations and novel deep
learning architectures. A large class of atomic structure representations based
on expansions of smoothed atomic densities have been shown to correspond to
specific choices of basis sets in an abstract many-body Hilbert space.
Concurrently, tensor network structures, conventionally the purview of quantum
many-body physics and quantum information, have been successfully applied in
supervised and unsupervised learning tasks in computer vision and natural
language processing. In this work, we argue that architectures based on tensor
networks are well-suited to machine learning on Hilbert-space representations
of atomic structures. This is demonstrated on supervised learning tasks
involving widely available datasets of density functional theory calculations
of metal and semiconductor alloys. In particular, we show that certain standard
tensor network topologies exhibit strong generalizability even on small
training datasets while being parametrically efficient. We further relate this
generalizability to the presence of complex entanglement in the trained tensor
networks. We also discuss connections to learning with generalized structural
kernels and related strategies for compressing large input feature spaces.
