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On Convergence Rates of Spiked Eigenvalue Estimates: A General Study of Global and Local Laws in Sample Covariance Matrices

Bing-Yi Jing, Weiming Li, Jiahui Xie, Yangchun Zhang, Wang Zhou

TL;DR

This work develops a unified spectral theory for high-dimensional sample covariance matrices under general growth of dimensions, allowing $\log M$ and $\log N$ to be comparable and $\phi=M/N$ to vanish, stay finite, or diverge. It establishes global laws with MP-type fixed-point equations for the limiting Stieltjes transform, and sharp local laws with eigenvalue rigidity that hold in the bulk and extend outside the spectrum, across all $\phi$ regimes via resolvent analysis and fluctuation averaging. The results yield precise convergence rates for spiked-eigenvalue estimation, including an averaged Bai–Ding type estimator with rate $|\hat{\alpha}_{B,\ell}-\alpha_\ell|\prec N^{-1/2}$, and a Mestre contour estimator with matching efficiency; the framework covers general covariance structures and provides practical guidance for spike detection in spiked covariance models. Overall, the paper unifies and extends classical random-matrix results to broad growth regimes, with direct implications for high-dimensional statistics and spike inference in complex data. The approach hinges on Stieltjes-transform methods, resolvent identities, and fluctuation averaging to link global distributions, local fluctuations, and spike behavior in a single coherent theory.

Abstract

This paper investigates global and local laws for sample covariance matrices with general growth rates of dimensions. The sample size $N$ and population dimension $M$ can have the same order in logarithm, which implies that their ratio $M/N$ can approach zero, a constant, or infinity. These theories are utilized to determine the convergence rate of spiked eigenvalue estimates.

On Convergence Rates of Spiked Eigenvalue Estimates: A General Study of Global and Local Laws in Sample Covariance Matrices

TL;DR

This work develops a unified spectral theory for high-dimensional sample covariance matrices under general growth of dimensions, allowing and to be comparable and to vanish, stay finite, or diverge. It establishes global laws with MP-type fixed-point equations for the limiting Stieltjes transform, and sharp local laws with eigenvalue rigidity that hold in the bulk and extend outside the spectrum, across all regimes via resolvent analysis and fluctuation averaging. The results yield precise convergence rates for spiked-eigenvalue estimation, including an averaged Bai–Ding type estimator with rate , and a Mestre contour estimator with matching efficiency; the framework covers general covariance structures and provides practical guidance for spike detection in spiked covariance models. Overall, the paper unifies and extends classical random-matrix results to broad growth regimes, with direct implications for high-dimensional statistics and spike inference in complex data. The approach hinges on Stieltjes-transform methods, resolvent identities, and fluctuation averaging to link global distributions, local fluctuations, and spike behavior in a single coherent theory.

Abstract

This paper investigates global and local laws for sample covariance matrices with general growth rates of dimensions. The sample size and population dimension can have the same order in logarithm, which implies that their ratio can approach zero, a constant, or infinity. These theories are utilized to determine the convergence rate of spiked eigenvalue estimates.

Paper Structure

This paper contains 16 sections, 28 theorems, 190 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Global laws
  3. Global laws for sample covariance matrices
  4. Proof of Theorems \ref{['mp1']} and \ref{['mp2']}.
  5. Local laws
  6. Notation and basic tools
  7. Main result
  8. Proof of Theorem \ref{['thm: locallaw_phi>1']}
  9. Case $\phi\gtrsim1$
  10. Case $\phi\ll 1$
  11. Proofs of Theorem \ref{['thm: Rigidity']} and Theorem \ref{['thm: Local law outside the spectrum']}
  12. Proof of Theorem \ref{['thm: Rigidity']}
  13. Proof of Theorem \ref{['thm: Local law outside the spectrum']}
  14. Application to spiked covariance model
  15. Estimation of spiked eigenvalues
  16. ...and 1 more sections

Key Result

Theorem 2.1

Suppose that Assumptions (A1)-(A2)-(A3) hold. Then, there exists a deterministic function $m_0(z)$ such that In particular, the function $m=m_0(z)$ is the unique solution to on the set $\{z: z\in \mathbb C_+, -(1-\phi)/z+\phi m(z)\in \mathbb C_+\}$.

Figures (1)

  • Figure 1: Eigenvalues of the sample covariance matrix with dimensions $(M,N)$: (a) $(400,40000)$, (b) $(400,400)$, and (c) $(40000,400)$.

Theorems & Definitions (48)

  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4
  • Remark 2.5
  • Lemma 2.6
  • Definition 3.1: Stochastic domination
  • Lemma 3.2: Concentration inequality
  • Lemma 3.3: Resolvent
  • Lemma 3.4: Large deviation bounds
  • ...and 38 more