Fast Second-Order Orthogonal Tensor Subspace Analysis for Face Recognition

Tensor subspace analysis (TSA) and discriminant TSA (DTSA) are two effective two-sided projection methods for dimensionality reduction and feature extraction of face image matrices. However, they have two serious drawbacks. Firstly, TSA and DTSA iteratively compute the left and right projection matrices. At each iteration, two generalized eigenvalue problems are required to solve, which makes them inapplicable for high dimensional image data. Secondly, the metric structure of the facial image space cannot be preserved since the left and right projection matrices are not usually orthonormal. In this paper, we propose the orthogonal TSA (OTSA) and orthogonal DTSA (ODTSA). In contrast to TSA and DTSA, two trace ratio optimization problems are required to be solved at each iteration. Thus, OTSA and ODTSA have much less computational cost than their nonorthogonal counterparts since the trace ratio optimization problem can be solved by the inexpensive Newton-Lanczos method. Experimental results show that the proposed methods achieve much higher recognition accuracy and have much lower training cost. 1. Introduction Many applications in the field of information process, such as data mining, information retrieval, machine learning, and pattern recognition, require dealing with high-dimensional data. Dimensionality reduction has been a key technique for achieving high efficiency in manipulating the high-dimensional data. In dimensionality reduction, the high-dimensional data are transformed into a low-dimensional subspace with limited loss of information. Principal component analysis (PCA) [1] and linear discriminant analysis (LDA) [2] are two of the most well-known and widely used dimension reduction methods. PCA is an unsupervised method, which aims to find the projection directions by maximizing variance of features in the low-dimensional subspace. It is also considered as the best data representation method in that the mean squared error between the original data and the data reconstructed using the PCA transform result is the minimum. LDA is a supervised method and is based on the following idea: the transform results of the data points of different classes should be far as much as possible from each other and the transform results of the data points of the same class should be close as much as possible to each other. To achieve this goal, LDA seeks to find optimal linear transformation by minimizing the within-class distance and maximizing the between-class distance simultaneously. The optimal transformation of LDA can be computed by solving a generalized