  Medical imaging studies have collected high dimensional imaging data to
identify imaging biomarkers for diagnosis, screening, and prognosis, among many
others. These imaging data are often represented in the form of a
multi-dimensional array, called a tensor. The aim of this paper is to develop a
tensor partition regression modeling (TPRM) framework to establish a
relationship between low-dimensional clinical outcomes (e.g., diagnosis) and
high dimensional tensor covariates. Our TPRM is a hierarchical model and
efficiently integrates four components: (i) a partition model, (ii) a canonical
polyadic decomposition model, (iii) a principal components model, and (iv) a
generalized linear model with a sparse inducing normal mixture prior. This
framework not only reduces ultra-high dimensionality to a manageable level,
resulting in efficient estimation, but also optimizes prediction accuracy in
the search for informative sub-tensors. Posterior computation proceeds via an
efficient Markov chain Monte Carlo algorithm. Simulation shows that TPRM
outperforms several other competing methods. We apply TPRM to predict disease
status (Alzheimer versus control) by using structural magnetic resonance
imaging data obtained from the Alzheimer's Disease Neuroimaging Initiative
(ADNI) study.
