Abstract: Discriminant analyses of multidimensional array data (i.e., tensors) are of substantial interest in numerous statistics and engineering research problems, such as signal processing, imaging, genetics, and brain-computer interfaces. In this study, we consider a multi-class discriminant analysis with a tensor-variate predictor and a categorical response. To overcome the high dimensionality and to exploit the tensor correlation structure, we propose the discriminant analysis with tensor envelope (DATE) model for simultaneous dimension reduction and classification. We extend the notion of tensor envelopes from regression to discriminant analysis and develop two complementary estimation procedures: DATE-L is a likelihood-based estimator that is shown to be asymptotically efficient when the sample size goes to infinity and the tensor dimension is fixed; DATE-D is a novel decomposition-based estimator suitable for high-dimensional problems. Interestingly, we show that DATE-D is still root-n consistent, even when the tensor dimensions on each model grow arbitrarily fast, but at a similar rate. We demonstrate the robustness and efficiency of our estimators using extensive simulations and real-data examples.

Key words and phrases: Dimension reduction, linear discriminant analysis, tensor.