  In contemporary scientific research, it is of great interest to predict a
categorical response based on a high-dimensional tensor (i.e. multi-dimensional
array) and additional covariates. This mixture of different types of data leads
to challenges in statistical analysis. Motivated by applications in science and
engineering, we propose a comprehensive and interpretable discriminant analysis
model, called CATCH model (in short for Covariate-Adjusted Tensor
Classification in High-dimensions), which efficiently integrates the covariates
and the tensor to predict the categorical outcome. The CATCH model jointly
models the relationships among the covariates, the tensor predictor, and the
categorical response. More importantly, it preserves and utilizes the
structures of the data for maximum interpretability and optimal prediction. To
tackle the new computational and statistical challenges arising from the
intimidating tensor dimensions, we propose a penalized approach to select a
subset of tensor predictor entries that has direct discriminative effect after
adjusting for covariates. We further develop an efficient algorithm that takes
advantage of the tensor structure. Theoretical results confirm that our method
achieves variable selection consistency and optimal classification error, even
when the tensor dimension is much larger than the sample size. The superior
performance of our method over existing methods is demonstrated in extensive
simulated and real data examples.
