研究高维高斯混合分类的泛化误差与变量选择，提出新估计方法。
Statistical Inference in Classification of High-dimensional Gaussian Mixture
- 用统计物理的副本法分析高维极限下正则化分类器行为。
- 提出去偏估计器，可准确识别重要变量，错误率随维度上升仍可控。
- 适用于高维生物数据、金融建模等需特征筛选的场景。
研究高维两分类高斯混合模型在一般协方差结构下的分类问题。通过统计物理中的副本法，分析当样本量n与维度p同时趋于无穷、其比值α = n/p保持固定时，一类广义正则化凸分类器的渐近性质，重点关注泛化误差与变量选择能力。基于分类器的分布极限，构造去偏估计器，通过合适的假设检验实现变量选择。以L1正则化逻辑回归为例，大量数值实验验证了理论分析结果在有限系统中依然成立。还探讨了协方差结构对去偏估计器性能的影响。
原文摘要 · Abstract (English)
We consider the classification problem of a high-dimensional mixture of two Gaussians with general covariance matrices. Using the replica method from statistical physics, we investigate the asymptotic behavior of a general class of regularized convex classifiers in the high-dimensional limit, where both the sample size $n$ and the dimension $p$ approach infinity while their ratio $α=n/p$ remains fixed. Our focus is on the generalization error and variable selection properties of the estimators. Specifically, based on the distributional limit of the classifier, we construct a de-biased estimator to perform variable selection through an appropriate hypothesis testing procedure. Using $L_1$-regularized logistic regression as an example, we conducted extensive computational experiments to confirm that our analytical findings are consistent with numerical simulations in finite-sized systems. We also explore the influence of the covariance structure on the performance of the de-biased estimator.
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