Tensor-on-tensor regression

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 .

33 pages, 3 figures

References in corpus (4)

• Multilinear tensor regression for longitudinal relational data
• Supervised multiway factorization
• Discriminating sample groups with multi-way data
• Array Variate Skew Normal Random Variables with Multiway Kronecker Delta Covariance Matrix Structure

Cited by in corpus (10)

• Reduced-Rank Tensor-on-Tensor Regression and Tensor-variate Analysis of Variance
• Broadcasted Nonparametric Tensor Regression
• Supervised tensor decomposition with features on multiple modes
• CP Degeneracy in Tensor Regression
• A Statistically Identifiable Model for Tensor-Valued Gaussian Random Variables
• Jointly Modeling and Clustering Tensors in High Dimensions
• Bayesian Methods in Tensor Analysis
• A Doubly-Enhanced EM Algorithm for Model-Based Tensor Clustering
• Latent Functional PARAFAC for modeling multidimensional longitudinal data
• An Augmented Regression Model for Tensors with Missing Values