SRDA: An Efficient Algorithm for Large-Scale

Linear Discriminant Analysis (LDA) has been a popular method for extracting features that preserves class separability. The projection functions of LDA are commonly obtained by maximizing the between-class covariance and simultaneously minimizing the within-class covariance. It has been widely used in many fields of information processing, such as machine learning, data mining, information retrieval, and pattern recognition. However, the computation of LDA involves dense matrices eigendecomposition, which can be computationally expensive in both time and memory. Specifically, LDA has Oðmnt þ t3Þ time complexity and requires Oðmn þ mt þ ntÞ memory, where m is the number of samples, n is the number of features, and t ¼ minðm; nÞ. When both m and n are large, it is infeasible to apply LDA. In this paper, we propose a novel algorithm for discriminant analysis, called Spectral Regression Discriminant Analysis (SRDA). By using spectral graph analysis, SRDA casts discriminant analysis into a regression framework that facilitates both efficient computation and the use of regularization techniques. Specifically, SRDA only needs to solve a set of regularized least squares problems, and there is no eigenvector computation involved, which is a huge save of both time and memory. Our theoretical analysis shows that SRDA can be computed with OðmsÞ time and OðmsÞ memory, where sð� nÞ is the average number of nonzero features in each sample. Extensive experimental results on four real-world data sets demonstrate the effectiveness and efficiency of our algorithm.