Eigenvector overlaps in large sample covariance matrices and nonlinear shrinkage estimators

2024/04/28 by Lin, Zeqin, Pan, Guangming · 2 citations
#FOS: Mathematics #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.2404.18173

Abstract

Consider a data matrix Y = [y₁, ⋯, y_N] of size M × N, where the columns are independent observations from a random vector y with zero mean and population covariance Σ. Let u_i and v_j denote the left and right singular vectors of Y, respectively. This study investigates the eigenvector/singular vector overlaps ⟨ u_i, D₁ u_j ⟩, ⟨ v_i, D₂ v_j ⟩ and ⟨ u_i, D₃ v_j ⟩, where D_k are general deterministic matrices with bounded operator norms. We establish the convergence in probability of these eigenvector overlaps toward their deterministic counterparts with explicit convergence rates, when the dimension M scales proportionally with the sample size N. Building on these findings, we offer a more precise characterization of the loss for Ledoit and Wolf's nonlinear shrinkage estimators of the population covariance Σ.

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• Eigenvector overlaps of sample covariance matrices with intersecting time periods
• Data-Driven Matrix Recovery via Optimal Shrinkage and Spatially Resolved Singular Vector Denoising under High-Dimensional Separable Noise

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