Relationships Between Support Vector Classifiers and Generalized Linear Discriminant Analysis on Support Vectors

Abstract

The linear discriminant analysis based on the generalized singular value decomposition (LDA/GSVD) has been introduced to circumvent the nonsingularity restriction inherent in the classical LDA. The LDA/GSVD provides a framework in which a dimension reducing transformation can be effectively obtained for undersampled problems. In this paper, relationships between support vector machines (SVMs) and the generalized linear discriminant analysis applied to the support vectors are studied. Based on the GSVD, the weight vector of the hard-margin SVM is proved to be equivalent to the dimension reducing transformation vector generated by LDA/GSVD applied to the support vectors of the binary class. We also show that the dimension reducing transformation vector and the weight vector of soft-margin SVMs are related when a subset of support vectors are considered. These results can be generalized when kernelized SVMs and the kernelized LDA/GSVD called KDA/GSVD are considered. Through these relationships, it is shown that support vector classification is related to data reduction as well as dimension reduction by LDA/GSVD.