  We propose a framework for the linear prediction of a multi-way array (i.e.,
a tensor) from another multi-way array of arbitrary dimension, using the
contracted tensor product. This framework generalizes several existing
approaches, including methods to predict a scalar outcome from a tensor, a
matrix from a matrix, or a tensor from a scalar. We describe an approach that
exploits the multiway structure of both the predictors and the outcomes by
restricting the coefficients to have reduced CP-rank. We propose a general and
efficient algorithm for penalized least-squares estimation, which allows for a
ridge (L_2) penalty on the coefficients. The objective is shown to give the
mode of a Bayesian posterior, which motivates a Gibbs sampling algorithm for
inference. We illustrate the approach with an application to facial image data.
An R package is available at https://github.com/lockEF/MultiwayRegression .
