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.

This work (DOI: 10.1109/TPAMI.2022.3164836) is joint with Carlos Llosa-Vite of Sandia National Laboratories and was supported in part by the National Institute of Justice (NIJ) under Grants No. 2015-DN-BX-K056, 2018-R2-CX-0034 and 15PNIJ-21-GG-04141-RESS, the National Institutes of Health (NIH) under Grant R21EB016212, and the United States Department of Agriculture (USDA) National Institute of Food and Agriculture (NIFA) Hatch project IOW03717. The content of this paper is however solely the responsibility of the authors and does not represent the official views of the NIJ, the NIH or the USDA.