  Matrix product states (MPS), a tensor network designed for one-dimensional
quantum systems, has been recently proposed for generative modeling of natural
data (such as images) in terms of `Born machine'. However, the exponential
decay of correlation in MPS restricts its representation power heavily for
modeling complex data such as natural images. In this work, we push forward the
effort of applying tensor networks to machine learning by employing the Tree
Tensor Network (TTN) which exhibits balanced performance in expressibility and
efficient training and sampling. We design the tree tensor network to utilize
the 2-dimensional prior of the natural images and develop sweeping learning and
sampling algorithms which can be efficiently implemented utilizing Graphical
Processing Units (GPU). We apply our model to random binary patterns and the
binary MNIST datasets of handwritten digits. We show that TTN is superior to
MPS for generative modeling in keeping correlation of pixels in natural images,
as well as giving better log-likelihood scores in standard datasets of
handwritten digits. We also compare its performance with state-of-the-art
generative models such as the Variational AutoEncoders, Restricted Boltzmann
machines, and PixelCNN. Finally, we discuss the future development of Tensor
Network States in machine learning problems.

Machine Learning;Statistical Mechanics;Machine Learning;Quantum Physics