The Dimensionality Reduction of Feature Vectors by Generalized Cross Product

Abstract

Fisher¿s discriminant requires the inverse operation of high-order within-class scatter matrix [S <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">w</sub> ] in the dimensionality reduction of feature vectors. The results may be inaccurate if [S <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">w</sub> ] is close to singular. This paper presents another classification-oriented mapping method for the dimensionality reduction of high-dimensional feature vectors, based on the generalized cross product of multi-vectors. The mapped feature vector is transformed into a cross matrix to generate a product vector, whose robustness depends on both the orthogonality and the norm-homogeneousness of the cross matrix, for pattern classification. To insure the within-class congregation and between-class separability of the mapping of feature vectors, it is proved that the optimum cross matrix is merely the orthonormalized basis of 2 reference vectors of sorted sample sets according to the robustness theorem of generalized cross product proposed in this paper. Numerical experiments showed that the proposed method has a better separability and better robustness of separability than Fisher¿s method in the dimensionality reduction of high-dimension feature vectors.