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

TL;DR

This paper provides a theoretical framework for selecting appropriate random matrices in fast null linear discriminant analysis, ensuring full rank and preserving discriminant information during dimensionality reduction.

Contribution

It introduces necessary and sufficient conditions for choosing random matrices that guarantee the full rank of the orientation matrix in null LDA.

Findings

Established conditions for random matrix selection ensuring full rank

Provided geometric interpretation of the rank condition

Enhanced the theoretical understanding of fast null LDA implementation

Abstract

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.