Augmented James--Stein estimation for leading eigenvectors and eigenspaces in high dimensions

2026-09-28

AI Summary

This study addresses the insufficient estimation accuracy of principal eigenvectors under high-dimensional proportional asymptotics by proposing a data-adaptive augmented James–Stein shrinkage framework. Built upon the generalized spiked model and subspace augmentation techniques, the proposed method incorporates auxiliary information to optimize eigenspace estimation. Notably, it automatically reduces to principal component analysis (PCA) when such auxiliary information is unavailable, thereby ensuring both flexibility and robustness. Theoretical analyses and simulation studies demonstrate that this framework strictly outperforms PCA and existing high-dimension, low-sample-size (HDLSS) methods, yielding substantial improvements in finite-sample settings. Furthermore, the approach exhibits strong robustness against the misspecification of the number of spikes.

Abstract

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.

Problem

• high-dimensional eigenvector estimation
• spiked population model
• principal component analysis
• James-Stein shrinkage
• eigenspace estimation

Innovation

• Augmented James-Stein estimation
• High-dimensional PCA
• Spiked population model
• Data-adaptive shrinkage
• Eigenvector estimation