Optimal fisher discriminant analysis using the rank decomposition

摘要

The Fisher optimal discriminant vector is a very efficient means in high-dimensional pattern analysis. In the case of a small number of samples, the within-class scatter matrix Sw is singular; therefore, the calculation of the Fisher optimal discriminant vector becomes very important. In this paper, the conception of the rank decomposition of matrices is first introduced, and a new method for calculating the Fisher optimal discriminant vector is presented which is particularly well suited to the case of a small number of samples in the sense that the scatter matrices are rank-deficient. The new method is compared with the pseudoinverse and perturbation methods in Tian (J. Opt. Soc. Am. A 5, 1670–1672 (1988)) and Hong and Yang (Pattern Recognition 24, 317–324 (1991)). An important conclusion is proved: in the three methods, our method is the best one for calculating the Fisher optimal discriminant vectors in the cases of both a large and a small number of samples. The experimental results have also shown that our method can give the highest recognition rate in the case of a small number of samples among the three methods.