Optimality in high-dimensional tensor discriminant analysis

Abstract

Tensor discriminant analysis is an important topic in tensor data analysis. However, given the many proposals for tensor discriminant analysis methods, there lacks a systematic theoretical study, especially concerning optimality. We fill this gap by providing the minimax lower bounds for the estimation and prediction errors under the tensor discriminant analysis model coupled with the sparsity assumption. We further show that one existing high-dimensional tensor discriminant analysis estimator has matching upper bounds, and is thus optimal. Our results apply to tensors with arbitrary orders and ultra-high dimensions. If one focuses on one-way tensors (i.e., vectors), our results further provide strong theoretical justifications for several popular sparse linear discriminant analysis methods. Numerical studies are also presented to support our theoretical results.