  We propose a tensor neural network ($t$-NN) framework that offers an exciting
new paradigm for designing neural networks with multidimensional (tensor) data.
Our network architecture is based on the $t$-product (Kilmer and Martin, 2011),
an algebraic formulation to multiply tensors via circulant convolution. In this
$t$-product algebra, we interpret tensors as $t$-linear operators analogous to
matrices as linear operators, and hence our framework inherits mimetic matrix
properties. To exemplify the elegant, matrix-mimetic algebraic structure of our
$t$-NNs, we expand on recent work (Haber and Ruthotto, 2017) which interprets
deep neural networks as discretizations of non-linear differential equations
and introduces stable neural networks which promote superior generalization.
Motivated by this dynamic framework, we introduce a stable $t$-NN which
facilitates more rapid learning because of its reduced, more powerful
parameterization. Through our high-dimensional design, we create a more compact
parameter space and extract multidimensional correlations otherwise latent in
traditional algorithms. We further generalize our $t$-NN framework to a family
of tensor-tensor products (Kernfeld, Kilmer, and Aeron, 2015) which still
induce a matrix-mimetic algebraic structure. Through numerical experiments on
the MNIST and CIFAR-10 datasets, we demonstrate the more powerful
parameterizations and improved generalizability of stable $t$-NNs.
