Abstract Linear discriminant analysis (LDA) is an important conventional model for data classification. Classical theory shows that LDA is Bayes consistent for a fixed data dimensionality p and a large training sample size n. However, in high-dimensional settings when p ≫ n, LDA is difficult due to the inconsistent estimation of the covariance matrix and the mean vectors of populations. Recently, a linear programming discriminant (LPD) rule was proposed for high-dimensional linear discriminant analysis, based on the sparsity assumption over the discriminant function. It is shown that the LPD rule is Bayes consistent in high-dimensional settings. In this paper, we further show that the LPD rule is sign consistent under the sparsity assumption. Such sign consistency ensures the LPD rule to select the optimal discriminative features for high-dimensional data classification problems. Evaluations on both synthetic and real data validate our result on the sign consistency of the LPD rule.

中文翻译：

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线性规划判别规则的符号一致性

摘要 线性判别分析（LDA）是一种重要的传统数据分类模型。经典理论表明，对于固定的数据维数 p 和大的训练样本大小 n，LDA 是贝叶斯一致的。然而，在高维设置中，当 p ≫ n 时，由于协方差矩阵和总体平均向量的估计不一致，LDA 很困难。最近，基于对判别函数的稀疏假设，提出了用于高维线性判别分析的线性规划判别（LPD）规则。结果表明，LPD 规则在高维设置中是贝叶斯一致的。在本文中，我们进一步证明了 LPD 规则在稀疏假设下是符号一致的。这种符号一致性确保了 LPD 规则为高维数据分类问题选择最佳判别特征。对合成数据和真实数据的评估验证了我们关于 LPD 规则符号一致性的结果。