Tensor-on-Tensor Regression

2017/01/31 by Eric F. Lock · 8 citations
Engineering · Mathematics · Medicine · #Advanced Neuroimaging Techniques and Applications #Sparse and Compressive Sensing Techniques #Tensor decomposition and applications

paper · doi:10.1080/10618600.2017.1401544

openalex publication_date 2017/11/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

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.

Citations

• Some Mathematical Notes on Three-Mode Factor Analysis
• Tensor Decompositions and Applications
• Algorithm 862
• Tensor Regression with Applications in Neuroimaging Data Analysis
• Ridge Regression: Biased Estimation for Nonorthogonal Problems

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