对利用散射矩阵进行随机矩阵乘法实现null线性判别分析方法的理论贡献
A theoretical contribution to the fast implementation of null linear discriminant analysis method using random matrix multiplication with scatter matrices
AI总结：
本文提出一种理论方法，通过合理选择随机矩阵保证null LDA的列满秩，避免信息丢失，提升计算效率。
摘要：
null线性判别分析方法是一种有竞争力的降维方法，但其实现计算成本高。最近提出了一种利用随机矩阵乘法与散射矩阵的快速实现方法，但若随机矩阵任选，则导向矩阵可能秩不足，导致有用判别信息丢失。本文研究如何合理选择随机矩阵以满足null LDA的两个理论准则，给出了保证导向矩阵列满秩的必要且充分条件，并描述了该条件的几何特性。
英文摘要：
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.
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