Tensor-on-tensor regression

TL;DR

This paper introduces a tensor-on-tensor regression framework that predicts multi-way array outcomes from other tensors, leveraging reduced CP-rank and Bayesian inference, with applications demonstrated on facial image data.

Contribution

It generalizes existing methods for tensor prediction, proposing a novel efficient algorithm with Bayesian interpretation and practical implementation in R.

Findings

Effective prediction of tensor data demonstrated on facial images.

The proposed method outperforms traditional approaches in capturing multiway structures.

An R package facilitates practical application of the method.

Abstract

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 .