发表机构
The Australian National University; Macquarie University（澳大利亚国立大学; 麦考瑞大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文利用随机矩阵理论推导了高维线性判别分析（LDA）在欠参数化和过参数化情形下的渐近误分类风险，完整刻画了其双重下降风险曲线，并通过模拟和ARCENE数据集实验验证了理论结果。
AI 中文摘要
现代机器学习模型中观察到的双重下降现象挑战了经典的偏差-方差权衡，表明当模型复杂度超过插值阈值时，预测误差可以再次下降。尽管这一现象已在深度神经网络和其他高容量模型中得到广泛研究，但其对经典统计方法的影响仍未被充分理解。本文研究了线性判别分析（LDA）中的双重下降现象，LDA是一种用于二分类的基本分类方法。利用随机矩阵理论，我们在维度（p）和样本量（n）满足（p/n \to \u03b3）的高维情形下推导了LDA的渐近误分类风险。具体而言，当（\u03b3 \u2208 (0,1)）时，我们获得了具有一般总体协方差矩阵的LDA风险的显式渐近表达式；当（\u03b3 \u2208 (1,\u221e)）时，我们获得了在各向同性协方差模型下基于Moore-Penrose伪逆的LDA分类器的显式渐近表达式。这些结果共同提供了LDA在欠参数化和过参数化两种情形下双重下降风险曲线的完整刻画。模拟研究和在ARCENE癌症分类数据集上的实验与理论预测高度一致。
英文摘要
The double-descent phenomenon observed in modern machine learning models has challenged the classical bias-variance trade-off by showing that prediction error can decrease again as model complexity exceeds the interpolation threshold. Although this phenomenon has been extensively studied for deep neural networks and other high-capacity models, its implications for classical statistical methods remain less well understood. In this paper, we investigate double descent in linear discriminant analysis (LDA), a fundamental classification method for binary classification. Using random matrix theory, we derive the asymptotic misclassification risk of LDA in the high-dimensional regime where the dimension (p) and sample size (n) satisfy (p/n \to γ). Specifically, we obtain an explicit asymptotic expression for the LDA risk with a general population covariance matrix when (γ\in (0,1)), and for the Moore-Penrose pseudo-inverse LDA classifier under an isotropic covariance model when (γ\in (1,\infty)). Together, these results provide a complete characterization of the double-descent risk curve of LDA across both the under- and over-parameterized regimes. Simulation studies and experiments on the ARCENE cancer classification dataset demonstrate close agreement with the theoretical predictions.
Comments28 pages, 17 figures, 1 table
