The joint density of top k sample eigenvectors and principal subspace inference
Abstract
In this paper, the exact joint density of the top k sample eigenvectors is derived for any p× p population covariance matrix varSigma, sample size n > p −1, and 1 ≤ k ≤ p. Using symmetric functions such as zonal polynomials and a dual summation identity by I. G. Macdonald, the result is expressed as a series in terms of determinants of differential operators. A matrix Kummer transformation converts the resulting local alternating expansion into a globally absolutely convergent series over the entire positive definite matrix space. For k=2, the ordered eigenvalue integrals reduce to the Gauss hypergeometric function 2F1 with a simple closed form when n=p+1. Previously, explicit formulas were only available for a scalar matrix varSigma = σ2 Ip or a general matrix varSigma with p=2 or k=1, obtained by T. W. Anderson and T. Sugiyama, respectively. The frame law also induces an exact Grassmann density that provides a finite-sample benchmark for principal subspace inference.
Related
- Two-sample tests for principal eigenvalues and eigenvectors in high-dimensional elliptical factor models
- Geometric Fluctuations of the sinΘ Distance in High-Dimensional Principal Subspace Estimation
- On the sample complexity of the active subspace method
- Risk-Averse Inference-Time Diffusion Model Alignment with Finite-Sample Guarantees
- High-Dimensional Two-Sample Inference via Marginal Likelihood-Ratio Rank Statistics
BibTeX
@misc{shimizu2026joint,
title = {{The joint density of top k sample eigenvectors and principal subspace inference}},
author = {Koki Shimizu and Haoming Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07028},
eprint = {2609.07028},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07028},
}
RIS
TY - UNPB
TI - The joint density of top k sample eigenvectors and principal subspace inference
AU - Koki Shimizu
AU - Haoming Wang
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07028
UR - https://arxiv.org/abs/2609.07028
AB - In this paper, the exact joint density of the top k sample eigenvectors is derived for any p× p population covariance matrix varSigma, sample size n > p −1, and 1 ≤ k ≤ p. Using symmetric functions such as zonal polynomials and a dual summation identity by I. G. Macdonald, the result is expressed as a series in terms of determinants of differential operators. A matrix Kummer transformation converts the resulting local alternating expansion into a globally absolutely convergent series over the entire positive definite matrix space. For k=2, the ordered eigenvalue integrals reduce to the Gauss hypergeometric function _2F_1 with a simple closed form when n=p+1. Previously, explicit formulas were only available for a scalar matrix varSigma = σ^2 I_p or a general matrix varSigma with p=2 or k=1, obtained by T. W. Anderson and T. Sugiyama, respectively. The frame law also induces an exact Grassmann density that provides a finite-sample benchmark for principal subspace inference.
ER -