Generalized Visual Information Analysis via Tensorial Algebra
This work provides a novel mathematical framework for visual data analysis, potentially benefiting researchers in computer vision and machine learning, though it appears incremental as it extends existing matrix methods.
The paper tackles the problem of analyzing higher-order data by introducing a tensorial algebra framework using t-matrices, which generalizes standard matrix algorithms like SVD and PCA, and shows that these generalized algorithms perform favorably in image tasks such as low-rank approximation and classification.
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.