MPCA and MDA via Einstein product

This work deals with the problem of multilinear principal component analysis (MPCA) and multilinear discriminant analysis (MDA), that solve for a tensor to tensor projection (TTP) using Einstein product. MPCA and MDA are considered as a higher-order extension of principal component analysis (PCA ) and linear discriminant analysis (LDA), respectively. MPCA seeks to find a low-dimensional representation that captures most of the variation present in the original data tensor. Whereas MDA seeks to find discriminative features that maximize the separation between classes, while preserving the multilinear structure. Specifically, we are interested in finding a projective tensor that maps the original data tensor onto a new lower-dimensional subspace. In this paper, we propose to solve the MPCA problem by employing the global Lanczos procedure via Einstein product for a fourth-order tensor, while solving the MDA problem by combining Newton method and global tensorial Lanczos method. The numerical experiments illustrate the use of these algorithms for face recognition problems, compression and classification.