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. 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 demonstrate and strongly suggest that geometric LDA can generate the, "optimal" classifier in a lower dimension