  Large-scale neuroimaging studies have been collecting brain images of study
individuals, which take the form of two-dimensional, three-dimensional, or
higher dimensional arrays, also known as tensors. Addressing scientific
questions arising from such data demands new regression models that take
multidimensional arrays as covariates. Simply turning an image array into a
long vector causes extremely high dimensionality that compromises classical
regression methods, and, more seriously, destroys the inherent spatial
structure of array data that possesses wealth of information. In this article,
we propose a family of generalized linear tensor regression models based upon
the Tucker decomposition of regression coefficient arrays. Effectively
exploiting the low rank structure of tensor covariates brings the ultrahigh
dimensionality to a manageable level that leads to efficient estimation. We
demonstrate, both numerically that the new model could provide a sound recovery
of even high rank signals, and asymptotically that the model is consistently
estimating the best Tucker structure approximation to the full array model in
the sense of Kullback-Liebler distance. The new model is also compared to a
recently proposed tensor regression model that relies upon an alternative
CANDECOMP/PARAFAC (CP) decomposition.
