$\mathcal{DBSDA}$ : Lowering the Bound of Misclassification Rate for Sparse Linear Discriminant Analysis via Model Debiasing

Linear discriminant analysis (LDA) is a well-known technique for linear classification, feature extraction, and dimension reduction. To improve the accuracy of LDA under the high dimension low sample size (HDLSS) settings, shrunken estimators, such as Graphical Lasso, can be used to strike a balance between biases and variances. Although the estimator with induced sparsity obtains a faster convergence rate, however, the introduced bias may also degrade the performance. In this paper, we theoretically analyze how the sparsity and the convergence rate of the precision matrix (also known as inverse covariance matrix) estimator would affect the classification accuracy by proposing an analytic model on the upper bound of an LDA misclassification rate. Guided by the model, we propose a novel classifier, <inline-formula> <tex-math notation="LaTeX">$\mathcal {DBSDA}$ </tex-math></inline-formula>, which improves classification accuracy through <italic>debiasing</italic>. Theoretical analysis shows that <inline-formula> <tex-math notation="LaTeX">$\mathcal {DBSDA}$ </tex-math></inline-formula> possesses a reduced upper bound of misclassification rate and better asymptotic properties than sparse LDA (SDA). We conduct experiments on both synthetic datasets and real application datasets to confirm the correctness of our theoretical analysis and demonstrate the superiority of <inline-formula> <tex-math notation="LaTeX">$\mathcal {DBSDA}$ </tex-math></inline-formula> over LDA, SDA, and other downstream competitors under HDLSS settings.