We consider the problem of high-dimensional classification between two groups with unequal covariance matrices. Rather than estimating the full quadratic discriminant rule, we propose to perform simultaneous variable selection and linear dimension reduction on the original data, with the subsequent application of quadratic discriminant analysis on the reduced space. In contrast to quadratic discriminant analysis, the proposed framework doesn't require the estimation of precision matrices; it scales linearly with the number of measurements, making it especially attractive for the use on high-dimensional datasets. We support the methodology with theoretical guarantees on variable selection consistency, and empirical comparisons with competing approaches. We apply the method to gene expression data of breast cancer patients, and confirm the crucial importance of the ESR1 gene in differentiating estrogen receptor status.

中文翻译：

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通过线性降维的稀疏二次分类规则

我们考虑协方差矩阵不等的两组之间的高维分类问题。我们建议对原始数据同时进行变量选择和线性降维，然后在缩减的空间上应用二次判别分析，而不是估计完整的二次判别规则。与二次判别分析相比，所提出的框架不需要估计精度矩阵；它随着测量数量线性扩展，这使得它对于高维数据集的使用特别有吸引力。我们通过变量选择一致性的理论保证以及与竞争方法的实证比较来支持该方法。我们将该方法应用于乳腺癌患者的基因表达数据，并证实了 ESR1 基因在区分雌激素受体状态中的至关重要性。