Since fitting regression models with many multivariate responses and covariates can be challenging, but such responses and covariates sometimes have tensor-variate structure, the authors extend the classical multivariate regression model to exploit such structure in two ways.
First, they impose four types of low-rank tensor formats on the regression coefficients. Second, they model the errors using the tensor-variate normal distribution that imposes a Kronecker separable format on the covariance matrix. They obtain maximum likelihood estimators via block-relaxation algorithms and derive their computational complexity and asymptotic distributions. Their regression framework enables them to formulate tensor-variate analysis of variance (TANOVA) methodology. This methodology, when applied in a one-way TANOVA layout, enables them to identify cerebral regions significantly associated with the interaction of suicide attempters or non-attempter 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. (Published abstract provided)
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