Adaptive Reference-Guided Estimation of Principal Component Subspace in High Dimensions

Abstract

We propose a novel estimator for the principal component (PC) subspace tailored to the high-dimension, low-sample size (HDLSS) context. The method, termed Adaptive Reference-Guided (ARG) estimator, is designed for data exhibiting spiked covariance structures and seeks to improve upon the conventional sample PC subspace by leveraging auxiliary information from reference vectors, presumed to carry prior knowledge about the true PC subspace. The estimator is constructed by first identifying vectors asymptotically orthogonal to the true PC subspace within a signal subspace, the subspace spanned by the leading sample PC directions and the references, and then taking the orthogonal complement. The estimator is adaptive, as it automatically selects the subspace asymptotically closest to the true PC subspace inside the signal subspace, without requiring parameter tuning. We show that when the reference vectors carry nontrivial information, the proposed estimator asymptotically reduces all principal angles between the estimated and true PC subspaces compared to the naive sample-based estimator. Interestingly, despite being derived from a completely different rationale, the ARG estimator is theoretically equivalent to an estimator based on James-Stein shrinkage. Our results thus establish a theoretical foundation that unifies these two distinct approaches.