L1 least squares for sparse high-dimensional LDA

Abstract

This paper studies high-dimensional linear discriminant analysis (LDA). First, we review the $\ell_{1}$ penalized least square LDA proposed in [10], which could circumvent estimation of the annoying high-dimensional covariance matrix. Then detailed theoretical analyses of this sparse LDA are established. To be specific, we prove that the penalized estimator is $\ell_{2}$ consistent in high-dimensional regime and the misclassification error rate of the penalized LDA is asymptotically optimal under a set of reasonably standard regularity conditions. The theoretical results are complementary to the results to [10], together with which we have more understanding of the $\ell_{1}$ penalized least square LDA (or called Lassoed LDA).