Exploiting fisher and fukunaga-koontz transforms in chernoff dimensionality reduction

Knowledge discovery from big data demands effective representation of data. However, big data are often characterized by high dimensionality, which makes knowledge discovery more difficult. Many techniques for dimensionality reudction have been proposed, including well-known Fisher's Linear Discriminant Analysis (LDA). However, the Fisher criterion is incapable of dealing with heteroscedasticity in the data. A technique based on the Chernoff criterion for linear dimensionality reduction has been proposed that is capable of exploiting heteroscedastic information in the data. While the Chernoff criterion has been shown to outperform the Fisher's, a clear understanding of its exact behavior is lacking. In this article, we show precisely what can be expected from the Chernoff criterion. In particular, we show that the Chernoff criterion exploits the Fisher and Fukunaga-Koontz transforms in computing its linear discriminants. Furthermore, we show that a recently proposed decomposition of the data space into four subspaces is incomplete. We provide arguments on how to best enrich the decomposition of the data space in order to account for heteroscedasticity in the data. Finally, we provide experimental results validating our theoretical analysis.