Tensor Regression with Applications in Neuroimaging Data Analysis

Modern technologies are producing a wealth of data with complex structures. For instance in medical sciences, brain images of individuals are collected for large neuroimaging studies. These image data often take the form of 2D, 3D or even higher dimensional arrays, also known as tensors. To address the scientific questions arising from the data, new regression methods that take arrays as covariates are needed. Simply turning an image array into a vector leads to ultra-high dimensionality in classical regression methods. More seriously, it would destroy the inherent spatial structure of the data, causing loss of valuable information. To keep the structural integrity of the data, we propose a flexible tensor regression model based on the Tucker decomposition of the regression coefficient array. It effectively exploits information in the tensor covariates, reduces the ultra-high dimensionality to a manageable level, and thus results in efficient estimation. We demonstrate the effectiveness of the method by applying it to identify brain regions associated with attention deficit hyperactivity disorder (ADHD) from magnetic resonance images (MRI). In addition, we compare our method with another type of tensor regression based on CANDECOMP/PARAFAC (CP) decomposition.