Universality of covariance matrices

Abstract

In this paper we prove the universality of covariance matrices of the form $H_{N\times N}=X^{†}X$ where $X$ is an $M\times N$ rectangular matrix with independent real valued entries $x_{ij}$ satisfying $\mathbb{E}{x}_{ij}=0$ and $\mathbb{E}{x}_{ij}^2=\frac{1}{M}$, $N$, $M\to \infty$. Furthermore it is assumed that these entries have sub-exponential tails or sufficiently high number of moments. We will study the asymptotics in the regime $N/M=d_N\in (0,\infty ),{\lim }_{N\to \infty }{d}_N\ne 0,\infty$. Our main result is the edge universality of the sample covariance matrix at both edges of the spectrum. In the case ${\lim }_{N\to \infty }{d}_N=1$, we only focus on the largest eigenvalue. Our proof is based on a novel version of the Green function comparison theorem for data matrices with dependent entries. En route to proving edge universality, we establish that the Stieltjes transform of the empirical eigenvalue distribution of $H$ is given by the Marcenko-Pastur law uniformly up to the edges of the spectrum with an error of order $(N\eta {)}^{-1}$ where $\eta$ is the imaginary part of the spectral parameter in the Stieltjes transform. Combining these results with existing techniques we also show bulk universality of covariance matrices. All our results hold for both real and complex valued entries.

Published in at http://dx.doi.org/10.1214/13-AAP939 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (5)

• High Dimensional Statistical Inference and Random Matrices
• Multivariate analysis and Jacobi ensembles: largest eigenvalue, Tracy--Widom limits and rates of convergence
• A Necessary and Sufficient Condition for Edge Universality of Wigner matrices
• Edge universality of correlation matrices
• Local Marchenko-Pastur Law at the Hard Edge of Sample Covariance Matrices

Cited by in corpus (61)

• OptShrink: An algorithm for improved low-rank signal matrix denoising by optimal, data-driven singular value shrinkage
• Averaging Fluctuations in Resolvents of Random Band Matrices
• Universality for the largest eigenvalue of sample covariance matrices with general population
• Lectures on the local semicircle law for Wigner matrices
• Edge universality of correlation matrices
• A necessary and sufficient condition for edge universality at the largest singular values of covariance matrices
• Fluctuations of the free energy of the spherical Sherrington-Kirkpatrick model
• Spectral radius of random matrices with independent entries
• Large complex correlated Wishart matrices: Fluctuations and asymptotic independence at the edges
• Edge universality of separable covariance matrices
• Bounds for the Stieltjes Transform and the Density of States of Wigner Matrices
• Convergence of eigenvector empirical spectral distribution of sample covariance matrices
• On the convergence of the extremal eigenvalues of empirical covariance matrices with dependence
• Convergence rates of eigenvector empirical spectral distribution of large dimensional sample covariance matrix
• Singular vector and singular subspace distribution for the matrix denoising model
• Eigenvalue variance bounds for covariance matrices
• Isotropic Local Laws for Sample Covariance and Generalized Wigner Matrices
• On the principal components of sample covariance matrices
• Spiked separable covariance matrices and principal components
• Anisotropic local laws for random matrices
• Random band matrices in the delocalized phase, III: Averaging fluctuations
• Universal halting times in optimization and machine learning
• Sample canonical correlation coefficients of high-dimensional random vectors: local law and Tracy-Widom limit
• Convergence rate to the Tracy-Widom laws for the largest eigenvalue of Wigner matrices
• Singular vector distribution of sample covariance matrices
• Estimation of the number of spiked eigenvalues in a covariance matrix by bulk eigenvalue matching analysis
• Eigenvectors of random matrices: A survey
• Detecting Rare and Weak Spikes in Large Covariance Matrices
• Sample canonical correlation coefficients of high-dimensional random vectors with finite rank correlations
• Random weighted projections, random quadratic forms and random eigenvectors
• The Local Semicircle Law for a General Class of Random Matrices
• Local Circular Law for Random Matrices
• Infinite stable Boltzmann planar maps are subdiffusive
• High dimensional deformed rectangular matrices with applications in matrix denoising
• Quantitative Universality for the Largest Eigenvalue of Sample Covariance Matrices
• Linear spectral statistics of eigenvectors of anisotropic sample covariance matrices
• Degrees of freedom for combining regression with factor analysis
• Limiting distributions for eigenvalues of sample correlation matrices from heavy-tailed populations
• Eigenvectors of Sample Covariance Matrices: Universality of global fluctuations
• The local circular law II: the edge case
• The Spectral Norm of Random Inner-Product Kernel Matrices
• Principal components of spiked covariance matrices in the supercritical regime
• Isotropic self-consistent equations for mean-field random matrices
• The Local Circular Law III: General Case
• Convergence Rate of Empirical Spectral Distribution of Random Matrices from Linear Codes
• Eigenvalue variance bounds for Wigner and covariance random matrices
• Sparse Recovery from Extreme Eigenvalues Deviation Inequalities
• Tracy-Widom limit for the largest eigenvalue of high-dimensional covariance matrices in elliptical distributions
• Spectral goodness-of-fit tests for complete and partial network data
• A spectral approach to Hebbian-like neural networks
• Top eigenpair statistics of diluted Wishart matrices
• The Tracy-Widom law for the Largest Eigenvalue of F Type Matrix
• Tracy-Widom law for the extreme eigenvalues of sample correlation matrices
• Universality for eigenvalue algorithms on sample covariance matrices
• Goodness-of-fit Test for Latent Block Models
• On estimation in the reduced-rank regression with a large number of responses and predictors
• How do kernel-based sensor fusion algorithms behave under high dimensional noise?
• Universality of the least singular value for sparse random matrices
• Spiked sample covariance matrices with possibly multiple bulk components
• Tracy-Widom limit for Kendall's tau
• A unified matrix model including both CCA and F matrices in multivariate analysis: the largest eigenvalue and its applications