Abstract 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.

中文翻译：

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一种用于高维多类分类的检测稀疏结构的线性判别分析程序

摘要 在本文中，我们考虑了高维情况下的判别分析，其中预测向量的维数在理论环境中随样本大小而发散。重点是类数大于两个的情况。我们首先处理各种条件下线性判别规则的渐近误分类率。在实际的高维分类问题中，对类均值和公共精度矩阵假设一定的稀疏条件是合理的。我们的理论研究表明，利用已知的稀疏结构，可以构建渐近最优的线性判别规则。受理论结果的启发，我们提出了一种基于估计稀疏结构构建的线性判别规则，称为具有检测稀疏性的线性判别（LDwDS）。建立了 LDwDS 的渐近最优性。进行了数值研究以比较 LDwDS 与其他现有方法。数值研究包括综合模拟研究和两个真实数据分析。数值研究表明，LDwDS 在误分类率方面优于比较中考虑的所有其他方法。