Abstract
Linear discriminant analysis (LDA) is a well-known classification technique that enjoyed great success in practical applications. Despite its effectiveness for traditional low-dimensional problems, extensions of LDA are necessary in order to classify high-dimensional data. Many variants of LDA have been proposed in the literature. However, most of these methods do not fully incorporate the structure information among predictors when such information is available. In this paper, we introduce a new high-dimensional LDA technique, namely graph-based sparse LDA (GSLDA), that utilizes the graph structure among the features. In particular, we use the regularized regression formulation for penalized LDA techniques, and propose to impose a structure-based sparse penalty on the discriminant vector β . The graph structure can be either given or estimated from the training data. Moreover, we explore the relationship between the within-class feature structure and the overall feature structure. Based on this relationship, we further propose a variant of our proposed GSLDA to utilize effectively unlabeled data, which can be abundant in the semi-supervised learning setting. With the new regularization, we can obtain a sparse estimate of β and more accurate and interpretable classifiers than many existing methods. Both the selection consistency of β estimation and the convergence rate of the classifier are established, and the resulting classifier has an asymptotic Bayes error rate. Finally, we demonstrate the competitive performance of the proposed GSLDA on both simulated and real data studies.