A theoretical contribution to the fast implementation of null linear discriminant analysis method using random matrix multiplication with scatter matrices

The null linear discriminant analysis method is a competitive approach for dimensionality reduction. The implementation of this method, however, is computationally expensive. Recently, a fast implementation of null linear discriminant analysis method using random matrix multiplication with scatter matrices was proposed. However, if the random matrix is chosen arbitrarily, the orientation matrix may be rank deficient, and some useful discriminant information will be lost. In this paper, we investigate how to choose the random matrix properly, such that the two criteria of the null LDA method are satisfied theoretically. We give a necessary and sufficient condition to guarantee full column rank of the orientation matrix. Moreover, the geometric characterization of the condition is also described.