T-2DPCA, a novel approach considering the third-order tensors as linear operators on the space of oriented matrices, benefits from treating a 2D image as an inherently integrated object, has been proposed recently and showed better performance than traditional matrix PCA in image analysis and recognition. In T-2DPCA, a reconstructing tubal coefficient is obtained from the defined tensor product, called T-product, of a 2D image and a 2D basis element. In this study, by assuming that an eigenvector of the covariance tensor of the 2D training images is the tensor linear combination, called T-linear combination, of the training images, the T-2DPCA is improved to a new version with better performance. The improved method is further extended to a nonlinear version by using the general kernel trick in machine learning field, but with a new inner product called inside product defined with the T-product in the third-order tensor spaces, and simultaneously the general inner product defended in vector spaces. The effectiveness of the proposed algorithms is tested by face recognition experiment results.
Linear operators and positive semidefiniteness of symmetric tensor spaces