Recovery Limits for Eigenvector Alignment in HDLSS Spiked Covariance Models
Abstract
In a spiked covariance model, let ℋ and ℬ be selected sample and population eigenvector matrices. Their alignment matrix Ξ=ℋ⊤ℬ is a latent oracle diagnostic of direction-specific correspondence beyond subspace overlap. We study its uniform pathwise recovery in a nested high-dimensional, low-sample-size experiment with fixed sample size and growing dimension. At every fixed model, an observable spectral contrast consistently estimates the row Gram matrix ΞΞ⊤. For two or more spikes, however, no measurable estimator sequence achieves uniformly reliable almost-sure path recovery of Ξ over a family having common spike strengths and a common signal subspace but different ordered eigenframes. For every 0<ε<1/√2, a finite orbit forces each estimator to have some orbit point at which the approximate path-recovery probability is at most a variational bound u(ε), where u(ε) tends to zero as ε tends to zero. The worst-model exact-recovery probability over a countable orbit is zero. Thus the Gram summary is recoverable model by model, while the full alignment matrix is not uniformly recoverable along the nested path.
Related
- Eigenvector distributions and optimal shrinkage estimators for large covariance and precision matrices
- Exact Limits of Random Projections for Preserving Geometry: Distance Recovery, Nearest-Neighbor Rankings, and Covariance Shape in Gaussian Models
- Local Law and Outlier Eigenvalues of Spiked Separable Covariance Matrices
- On Eigenvector Statistics of Large-Dimensional Sample Covariance Matrices with Applications
- Multilevel limits of spiked random matrix corners processes
BibTeX
@misc{gurdogan2026recovery,
title = {{Recovery Limits for Eigenvector Alignment in HDLSS Spiked Covariance Models}},
author = {Hubeyb Gurdogan and Alex Shkolnik},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23081},
eprint = {2609.23081},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23081},
}
RIS
TY - UNPB
TI - Recovery Limits for Eigenvector Alignment in HDLSS Spiked Covariance Models
AU - Hubeyb Gurdogan
AU - Alex Shkolnik
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2609.23081
UR - https://arxiv.org/abs/2609.23081
AB - In a spiked covariance model, let ℋ and ℬ be selected sample and population eigenvector matrices. Their alignment matrix Ξ=ℋ^⊤ℬ is a latent oracle diagnostic of direction-specific correspondence beyond subspace overlap. We study its uniform pathwise recovery in a nested high-dimensional, low-sample-size experiment with fixed sample size and growing dimension. At every fixed model, an observable spectral contrast consistently estimates the row Gram matrix ΞΞ^⊤. For two or more spikes, however, no measurable estimator sequence achieves uniformly reliable almost-sure path recovery of Ξ over a family having common spike strengths and a common signal subspace but different ordered eigenframes. For every 0<ε<1/√2, a finite orbit forces each estimator to have some orbit point at which the approximate path-recovery probability is at most a variational bound u(ε), where u(ε) tends to zero as ε tends to zero. The worst-model exact-recovery probability over a countable orbit is zero. Thus the Gram summary is recoverable model by model, while the full alignment matrix is not uniformly recoverable along the nested path.
ER -