Tensor-on-tensor regression
This work addresses a domain-specific problem in multiway data analysis, offering an incremental extension to tensor regression methods.
The authors tackled the problem of predicting a tensor from another tensor of arbitrary dimension using a linear framework based on the contracted tensor product, generalizing existing methods and applying it to facial image data with an available R package.
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 .