A procedure of linear discrimination analysis with detected sparsity structure for high-dimensional multi-class classification

In this article, we consider discrimination analyses in high-dimensional cases where the dimension of the predictor vector diverges with the sample size in a theoretical setting. The emphasis is on the case where the number of classes is bigger than two. We first deal with the asymptotic misclassification rates of linear discrimination rules under various conditions. In practical high-dimensional classification problems, it is reasonable to assume certain sparsity conditions on the class means and the common precision matrix. Our theoretical study reveals that with known sparsity structures an asymptotically optimal linear discrimination rule can be constructed. Motivated by the theoretical result, we propose a linear discrimination rule constructed based on estimated sparsity structures which is dubbed as linear discrimination with detected sparsity (LDwDS). The asymptotic optimality of LDwDS is established. Numerical studies are carried out for the comparison of LDwDS with other existing methods. The numerical studies include a comprehensive simulation study and two real data analyses. The numerical studies demonstrate that the LDwDS has an edge in terms of misclassification rate over all the other methods under consideration in the comparison. (C) 2020 Elsevier Inc. All rights reserved.