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EIGENVECTOR OVERLAPS IN LARGE SAMPLE COVARIANCE MATRICES AND NONLINEAR SHRINKAGE ESTIMATORS
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DOI：10.1214/25-AOS2593.png)
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Abstract
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Consider a data matrix Y = [y1, ...,yN] of size M & times; N, where the columns are independent observations from a random vector y with zero mean and population covariance Sigma. Let ui and vj denote the left and right singular vectors of Y, respectively. This study investigates the eigenvector/singular vector overlaps (ui, D1uj ), (vi, D2vj ) and (ui, D3vj), where Dk are general deterministic matrices with bounded operator norms. In the high-dimensional regime, where the dimension M scales proportionally with the sample size N, we establish the convergence in probability of these eigenvector overlaps towards their deterministic counterparts with explicit convergence rates. Building upon these findings, we offer a more precise characterization of the loss associated with Ledoit and Wolf's nonlinear shrinkage estimators of the population covariance Sigma.
Keywords:
Sample covariance matrices
nonlinear shrinkage estimators
eigenvector overlaps
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