Estimating eigenvectors and eigenspaces of covariance matrices: Optimal Bounds and Conditions for Consistency
Abstract
Let X = [ ξ1, ξ2,... ,ξd]⊤ be a zero-mean random vector of large dimension d (d → ∞) with (hidden) covariance matrix M = (mij)1 ≤ i, j ≤ d, where mij = mji = Cov(ξi, ξj). Let X1, X2, …, Xn be n iid samples of X. Consider the sample covariance matrix M̃ := 1/n ∑i=1n Xi Xi⊤. In practice, one frequently uses the eigenvectors and eigenspaces of M̃ as estimators for those of M. A central task is to provide an error analysis for these estimators. In this paper, we provide an optimal error analysis, obtaining upper and lower bounds of matching order of magnitude, for a wide range of parameters d and n, under mild assumptions on M. As corollaries, we obtain new necessary and sufficient conditions for the consistency of the estimators. In these conditions, we only require the number of samples n to depend linearly on the effective rank of M, which can be much smaller than the dimension d.
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BibTeX
@misc{tran2026estimating,
title = {{Estimating eigenvectors and eigenspaces of covariance matrices: Optimal Bounds and Conditions for Consistency}},
author = {Phuc Tran and Van Vu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.23964},
eprint = {2607.23964},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.23964},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Estimating eigenvectors and eigenspaces of covariance matrices: Optimal Bounds and Conditions for Consistency
AU - Phuc Tran
AU - Van Vu
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2607.23964
UR - https://arxiv.org/abs/2607.23964
AB - Let X = [ ξ_1, ξ_2,... ,ξ_d]^⊤ be a zero-mean random vector of large dimension d (d → ∞) with (hidden) covariance matrix M = (m_ij)_1 ≤ i, j ≤ d, where m_ij = m_ji = Cov(ξ_i, ξ_j). Let X_1, X_2, …, X_n be n iid samples of X. Consider the sample covariance matrix M̃ := 1/n ∑_i=1^n X_i X_i^⊤. In practice, one frequently uses the eigenvectors and eigenspaces of M̃ as estimators for those of M. A central task is to provide an error analysis for these estimators. In this paper, we provide an optimal error analysis, obtaining upper and lower bounds of matching order of magnitude, for a wide range of parameters d and n, under mild assumptions on M. As corollaries, we obtain new necessary and sufficient conditions for the consistency of the estimators. In these conditions, we only require the number of samples n to depend linearly on the effective rank of M, which can be much smaller than the dimension d.
ER -