增强James--Stein估计用于高维主特征向量与特征子空间
Augmented James--Stein estimation for leading eigenvectors and eigenspaces in high dimensions
- Seoul National University（首尔大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出数据自适应的增强James--Stein收缩框架，用于高维尖峰模型下主特征向量和特征子空间的估计，通过结合辅助信息改进PCA，并严格优于现有收缩估计器。
AI中文摘要：
基于James--Stein方法用于主特征向量估计（Goldberg和Kercheval，Proc. Natl. Acad. Sci. USA 120, e2207046120, 2023），我们开发了一个数据自适应的增强James--Stein收缩框架，用于在广义尖峰总体模型下估计主特征向量和特征子空间，在高维情形中，维度$p$和样本量$n$按比例增长。对于每个尖峰特征向量，我们构造一个增强的目标子空间，该子空间结合了辅助信息（来自领域知识或先验信息）与剩余的样本尖峰特征向量。这种增强允许利用样本尖峰分量之间共享的信息，同时保持完全数据驱动的收缩规则。我们证明，当目标子空间包含关于总体特征向量的非消失信息时，所得特征向量估计器严格优于标准PCA，而当目标无信息时，渐近地退化为PCA。个体估计器进一步为所有前导尖峰特征子空间产生嵌套的估计器序列，具有类似的支配性质。所提出的估计器在比例高维情形中也严格优于现有的基于HDLSS的收缩估计器。模拟研究展示了显著的有限样本增益和对尖峰数量误设的稳健性。
英文摘要：
Building on the James--Stein approach to leading eigenvector estimation (Goldberg and Kercheval, Proc. Natl. Acad. Sci. USA 120, e2207046120, 2023), we develop a data-adaptive augmented James--Stein shrinkage framework for estimating leading eigenvectors and eigenspaces under a generalized spiked population model, in the high-dimensional regime where the dimension $p$ and sample size $n$ grow proportionally. For each spiked eigenvector, we construct an augmented target subspace that combines auxiliary information, either from domain knowledge or prior information, with the remaining sample spiked eigenvectors. This augmentation allows information shared across the sample spiked components to be exploited while retaining a fully data-driven shrinkage rule. We show that the resulting eigenvector estimator strictly improves upon standard PCA whenever the target subspace contains nonvanishing information about the population eigenvector, while asymptotically reverting to PCA when the target is uninformative. The individual estimators further yield a nested sequence of estimators for all leading spiked eigenspaces, with analogous dominance properties. The proposed estimator also strictly improves upon the existing HDLSS-motivated shrinkage estimator in the proportional high-dimensional regime. Simulation studies demonstrate substantial finite-sample gains and robustness to misspecification of the number of spikes.
