Training Linear Discriminant Analysis in Linear Time

Abstract

Linear Discriminant Analysis (LDA) has been a popular method for extracting features which preserve class separability. 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 eigen-decomposition which can be computationally expensive both in time and memory. Specifically, LDA has O(mnt + t <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ) time complexity and requires O(mn + mt + nt) memory, where m is the number of samples, n is the number of features and t = min (m,n). 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 which facilitates both efficient computation and the use of regularization techniques. Our theoretical analysis shows that SRDA can be computed with O(ms) time and O(ms) memory, where s(les n) is the average number of non-zero features in each sample. Extensive experimental results on four real world data sets demonstrate the effectiveness and efficiency of our algorithm.