Non-iterative heteroscedastic linear dimension reduction for two-class data; from Fisher to Chernoff

Linear discriminant analysis (LDA) is a traditional solution to the linear dimension reduction (LDR) problem, which is based on the maximization of the between-class scatter over the within-class scatter. This solution is incapable of dealing with heteroscedastic data in a proper way, because of the implicit assumption that the covariance matrices for all the classes are equal. Hence, discriminatory information in the difference between the covariance matrices is not used and, as a consequence, we can only reduce the data to a single dimension in the two-class case. We propose a fast non-iterative eigenvector-based LDR technique for heteroscedastic two-class data, which generalizes, and improves upon LDA by dealing with the aforementioned problem. For this purpose, we use the concept of directed distance matrices, which generalizes the between-class covariance matrix such that it captures the differences in (co)variances.