Reduced-Rank Tensor-on-Tensor Regression and Tensor-Variate Analysis of Variance

Abstract

Fitting regression models with many multivariate responses and covariates can be challenging, but such responses and covariates sometimes have tensor-variate structure. We extend the classical multivariate regression model to exploit such structure in two ways: first, we impose four types of low-rank tensor formats on the regression coefficients. Second, we model the errors using the tensor-variate normal distribution that imposes a Kronecker separable format on the covariance matrix. We obtain maximum likelihood estimators via block-relaxation algorithms and derive their computational complexity and asymptotic distributions. Our regression framework enables us to formulate tensor-variate analysis of variance (TANOVA) methodology. This methodology, when applied in a one-way TANOVA layout, enables us to identify cerebral regions significantly associated with the interaction of suicide attempters or non-attemptor ideators and positive-, negative-or death-connoting words in a functional Magnetic Resonance Imaging study. Another application uses three-way TANOVA on the Labeled Faces in the Wild image dataset to distinguish facial characteristics related to ethnic origin, age group and gender. A R package totr implements the methodology.

A Tensor‐based Regression Approach for Human Motion Prediction

“…A field of application of the IHOPLS is to reconstruct 3D motion trajectories from video and ECG stream signals. 48 Lock 13 provides a Tensor-on-Tensor Regression method with a CP structure of the regression parameters using least squares, and Llosa et al 49 use the Tucker structure of the coefficients. Gahrooei et al 50 propose a functional regression method in which a high-dimensional response is estimated and predicted by a set of informative and non-informative high-dimensional covariates through a set of low-dimensional smooth basis functions.…”

“…A field of application of the IHOPLS is to reconstruct 3D motion trajectories from video and ECG stream signals 48 . Lock 13 provides a Tensor‐on‐Tensor Regression method with a CP structure of the regression parameters using least squares, and Llosa et al 49 . use the Tucker structure of the coefficients.…”