In this work, we analytically study the peaking phenomenon in the context of linear discriminant analysis in the multivariate Gaussian model under the assumption of a common known covariance matrix. The focus is finite-sample setting where the sample size and observation dimension are comparable. Therefore, in order to study the phenomenon in such a setting, we use an asymptotic technique whereby the number of sample points is kept comparable in magnitude to the dimensionality of observations. The analysis provides a more thorough picture of the phenomenon. In particular, the analysis shows that as long as the Relative Cumulative Efficacy of an additional Feature set (RCEF) is greater (less) than the size of this set, the expected error of the classifier constructed using these additional features will be less (greater) than the expected error of the classifier constructed without them. Our result highlights underlying factors of the peaking phenomenon relative to the classifier used in this study and, at the same time, calls into question the classical wisdom around the peaking phenomenon.

中文翻译：

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分类中尖峰现象的理论分析

在这项工作中，我们在已知协方差矩阵的假设下，分析研究了多元高斯模型中线性判别分析背景下的峰值现象。重点是有限样本设置，其中样本大小和观察维度具有可比性。因此，为了研究这种情况下的现象，我们使用了一种渐近技术，其中样本点的数量在数量上与观察的维度保持可比性。该分析提供了对该现象的更全面的描述。特别是，分析表明，只要附加特征集（RCEF）的相对累积功效大于（小于）该集的大小，使用这些附加特征构建的分类器的预期误差将小于（大于）没有它们构建的分类器的预期误差。我们的结果突出了与本研究中使用的分类器相关的峰值现象的潜在因素，同时，对峰值现象的经典智慧提出了质疑。