Geometric linear discriminant analysis

When it becomes necessary to reduce the complexity of a classifier, dimensionality reduction can be an effective way to address classifier complexity. Linear Discriminant Analysis (LDA) is one approach to dimensionality reduction that makes use of a linear transformation matrix. The widely used Fisher’s LDA is “sub-optimal” when the sample class covariance matrices are unequal, meaning that another linear transformation exists that produces lower loss in discrimination power. In this paper, we introduce a geometric approach to Linear Discriminant Analysis (GLDA) that can reduce the number of dimensions from n to m for any number of classes. GLDA is able to compute a better linear transformation matrix than Fisher’s LDA for unequal sample class covariance matrices and is equivalent to Fisher’s LDA when those matrices are equal or proportional. The classification problems we present in this paper demonstrate and strongly suggest that geometric LDA can generate the “optimal” classifier in a lower dimension.