Multilinear Discriminant Analysis for Higher-Order Tensor Data Classification

In the past decade, great efforts have been made to extend linear discriminant analysis for higher-order data classification, generally referred to as multilinear discriminant analysis (MDA). Existing examples include general tensor discriminant analysis (GTDA) and discriminant analysis with tensor representation (DATER). Both the two methods attempt to resolve the problem of tensor mode dependency by iterative approximation. GTDA is known to be the first MDA method that converges over iterations. However, its performance relies highly on the tuning of the parameter in the scatter difference criterion. Although DATER usually results in better classification performance, it does not converge, yet the number of iterations executed has a direct impact on DATER´s performance. In this paper, we propose a closed-form solution to the scatter difference objective in GTDA, namely, direct GTDA (DGTDA) which also gets rid of parameter tuning. We demonstrate that DGTDA outperforms GTDA in terms of both efficiency and accuracy. In addition, we propose constrained multilinear discriminant analysis (CMDA) that learns the optimal tensor subspace by iteratively maximizing the scatter ratio criterion. We prove both theoretically and experimentally that the value of the scatter ratio criterion in CMDA approaches its extreme value, if it exists, with bounded error, leading to superior and more stable performance in comparison to DATER.