An efficient algorithm for generalized discriminant analysis using incomplete Cholesky decomposition

Abstract

Generalized discriminant analysis (GDA) has provided an extremely powerful approach to extracting nonlinear features via kernel trick. And it has been suggested for a number of applications, such as classification problem. Whereas the GDA could be solved by the utilization of Mercer kernels, a drawback of the standard GDA is that it may suffer from computational problem for large scale data set. Besides, there is still attendant problem of numerical accuracy when computing the eigenvalue problem of large matrices. Also, the GDA would occupy large memory (to store the kernel matrix). To overcome these deficiencies, we use Gram–Schmidt orthonormalization and incomplete Cholesky decomposition to find a basis for the entire training samples, and then formulate GDA as another eigenvalue problem of matrix whose size is much smaller than that of the kernel matrix by using the basis, while still working out the optimal discriminant vectors from all training samples. The theoretical analysis and experimental results on both artificial and real data set have shown the superiority of the proposed method for performing GDA in terms of computational efficiency and even the recognition accuracy, especially when the training samples size is large.