  Tensors offer a natural representation for many kinds of data frequently
encountered in machine learning. Images, for example, are naturally represented
as third order tensors, where the modes correspond to height, width, and
channels. Tensor methods are noted for their ability to discover
multi-dimensional dependencies, and tensor decompositions in particular, have
been used to produce compact low-rank approximations of data. In this paper, we
explore the use of tensor contractions as neural network layers and investigate
several ways to apply them to activation tensors. Specifically, we propose the
Tensor Contraction Layer (TCL), the first attempt to incorporate tensor
contractions as end-to-end trainable neural network layers. Applied to existing
networks, TCLs reduce the dimensionality of the activation tensors and thus the
number of model parameters. We evaluate the TCL on the task of image
recognition, augmenting two popular networks (AlexNet, VGG). The resulting
models are trainable end-to-end. Applying the TCL to the task of image
recognition, using the CIFAR100 and ImageNet datasets, we evaluate the effect
of parameter reduction via tensor contraction on performance. We demonstrate
significant model compression without significant impact on the accuracy and,
in some cases, improved performance.
