Geometric Fluctuations of the sinΘ Distance in High-Dimensional Principal Subspace Estimation
Abstract
We investigate the geometric fluctuations of principal subspaces for high-dimensional covariance matrices through the squared Frobenius sinΘ distance between the sample and population eigenspaces associated with the rp largest eigenvalues. An explicit first-order expansion and a central limit theorem are established for this subspace distance. The theory allows the subspace dimension to diverge subject to rp=o(n), where n is the sample size. It also permits a diverging spectral norm of the population covariance matrix, population spikes of different orders, and repeated or closely spaced spikes. This sharp characterisation captures features of the subspace estimation error that are not reflected in existing perturbation bounds. As applications, we derive an explicit asymptotic expansion for the expected PCA excess risk and a refined error bound for distributed PCA. In both cases, existing upper bounds can increase with the spiked-block condition number when some leading spikes become stronger, whereas our results show that the corresponding estimation errors need not increase and may instead decrease. Numerical experiments reproduce this contrasting behaviour and demonstrate the finite-sample accuracy of our theoretical findings.
Related
- Intrinsic Non-Reductive Geometric Invariant Theory and Moduli Spaces of Principal Bundles with Non-Reductive Structure Groups
- Characterizing Survivor Principal Strata Over Time: Identification, Efficient Estimation, and Sensitivity Analysis
- Two-sample tests for principal eigenvalues and eigenvectors in high-dimensional elliptical factor models
- Connes spectral distance as a quantum distance measure and a geometric measure of entanglement
- Geometric progressions in distance sets in odd dimensions over prime fields
BibTeX
@misc{hu2026geometric,
title = {{Geometric Fluctuations of the $\sinΘ$ Distance in High-Dimensional Principal Subspace Estimation}},
author = {Yanlin Hu and Xiao Han and Qing Yang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07751},
eprint = {2609.07751},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07751},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Geometric Fluctuations of the sinΘ Distance in High-Dimensional Principal Subspace Estimation
AU - Yanlin Hu
AU - Xiao Han
AU - Qing Yang
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07751
UR - https://arxiv.org/abs/2609.07751
AB - We investigate the geometric fluctuations of principal subspaces for high-dimensional covariance matrices through the squared Frobenius sinΘ distance between the sample and population eigenspaces associated with the r_p largest eigenvalues. An explicit first-order expansion and a central limit theorem are established for this subspace distance. The theory allows the subspace dimension to diverge subject to r_p=o(n), where n is the sample size. It also permits a diverging spectral norm of the population covariance matrix, population spikes of different orders, and repeated or closely spaced spikes. This sharp characterisation captures features of the subspace estimation error that are not reflected in existing perturbation bounds. As applications, we derive an explicit asymptotic expansion for the expected PCA excess risk and a refined error bound for distributed PCA. In both cases, existing upper bounds can increase with the spiked-block condition number when some leading spikes become stronger, whereas our results show that the corresponding estimation errors need not increase and may instead decrease. Numerical experiments reproduce this contrasting behaviour and demonstrate the finite-sample accuracy of our theoretical findings.
ER -