Tensor-on-Tensor Regression

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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 (L2) 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 https://github.com/lockEF/MultiwayRegression.

Original languageEnglish (US)
Pages (from-to)638-647
Number of pages10
JournalJournal of Computational and Graphical Statistics
Issue number3
StatePublished - Jul 3 2018

Bibliographical note

Publisher Copyright:
© 2018, © 2018 American Statistical Association, Institute of Mathematical Statistics, and Interface Foundation of North America.


  • Multiway data
  • Reduced rank regression
  • Ridge regression


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