Asymptotics of empirical eigenvalues for large separable covariance matrices

TL;DR

This paper analyzes the asymptotic behavior of eigenvalues of large separable covariance matrices in Gaussian processes, revealing their dependence on the Kolmogorov constant and correlation parameters, and introduces a nonlinear shrinkage estimator for top eigenvalues.

Contribution

It provides a semi-closed-form expression for the limiting spectral distribution of large separable covariance matrices using free harmonic analysis techniques.

Findings

Derived explicit LSD depending on Kolmogorov constant and correlations

Developed a nonlinear shrinkage estimator for top eigenvalues

Numerical validation shows estimator effectiveness

Abstract

We investigate the asymptotics of eigenvalues of sample covariance matrices associated with a class of non-independent Gaussian processes (separable and temporally stationary) under the Kolmogorov asymptotic regime. The limiting spectral distribution (LSD) is shown to depend explicitly on the Kolmogorov constant (a fixed limiting ratio of the sample size to the dimensionality) and parameters representing the spatio- and temporal- correlations. The Cauchy, M- and N-transforms from free harmonic analysis play key roles to this LSD calculation problem. The free multiplication law of free random variables is employed to give a semi-closed-form expression (only the final step is numerical based) of the LSD for the spatio-covariance matrix being a diagonally dominant Wigner matrix and temporal-covariance matrix an exponential off-diagonal decay (Toeplitz) matrix. Furthermore, we also derive a nonlinear shrinkage estimator for the top eigenvalues associated with a low rank matrix (Hermitian) from its noisy measurements. Numerical studies about the effectiveness of the estimator are also presented.