  Higher order data is modeled using matrices whose entries are numerical
arrays of a fixed size. These arrays, called t-scalars, form a commutative ring
under the convolution product. Matrices with elements in the ring of t-scalars
are referred to as t-matrices. The t-matrices can be scaled, added and
multiplied in the usual way. There are t-matrix generalizations of positive
matrices, orthogonal matrices and Hermitian symmetric matrices. With the
t-matrix model, it is possible to generalize many well-known matrix algorithms.
In particular, the t-matrices are used to generalize the SVD (Singular Value
Decomposition), HOSVD (High Order SVD), PCA (Principal Component Analysis),
2DPCA (Two Dimensional PCA) and GCA (Grassmannian Component Analysis). The
generalized t-matrix algorithms, namely TSVD, THOSVD,TPCA, T2DPCA and TGCA, are
applied to low-rank approximation, reconstruction,and supervised classification
of images. Experiments show that the t-matrix algorithms compare favorably with
standard matrix algorithms.
