Tensor-on-Tensor Regression

2018-06-06T15:16:57Z (GMT) by Eric F. Lock
<p>I propose a framework for the linear prediction of a multiway array (i.e., a tensor) from another multiway 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. I describe an approach that exploits the multiway structure of both the predictors and the outcomes by restricting the coefficients to have reduced PARAFAC/CANDECOMP rank. I propose a general and efficient algorithm for penalized least-squares estimation, which allows for a ridge (<i>L</i><sub>2</sub>) penalty on the coefficients. The objective is shown to give the mode of a Bayesian posterior, which motivates a Gibbs sampling algorithm for inference. I illustrate the approach with an application to facial image data. An R package is available at <i><a href="https://github.com/lockEF/MultiwayRegression" target="_blank">https://github.com/lockEF/MultiwayRegression</a></i>.</p>