  Tensor networks, a model that originated from quantum physics, has been
gradually generalized as efficient models in machine learning in recent years.
However, in order to achieve exact contraction, only tree-like tensor networks
such as the matrix product states and tree tensor networks have been
considered, even for modeling two-dimensional data such as images. In this
work, we construct supervised learning models for images using the projected
entangled pair states (PEPS), a two-dimensional tensor network having a similar
structure prior to natural images. Our approach first performs a feature map,
which transforms the image data to a product state on a grid, then contracts
the product state to a PEPS with trainable parameters to predict image labels.
The tensor elements of PEPS are trained by minimizing differences between
training labels and predicted labels. The proposed model is evaluated on image
classifications using the MNIST and the Fashion-MNIST datasets. We show that
our model is significantly superior to existing models using tree-like tensor
networks. Moreover, using the same input features, our method performs as well
as the multilayer perceptron classifier, but with much fewer parameters and is
more stable. Our results shed light on potential applications of
two-dimensional tensor network models in machine learning.
