Covariance Matrix Estimation from Linearly-Correlated Gaussian Samples

TL;DR

This paper develops a non-asymptotic analysis for estimating covariance matrices from correlated Gaussian samples, showing that with certain conditions, the classical sample complexity results still hold.

Contribution

It extends classical covariance estimation results to correlated Gaussian samples, especially Toeplitz-shaped covariance matrices, with new theoretical error bounds.

Findings

Error bounds depend on sample size, dimension, and shape parameter.

O(n) samples suffice for Toeplitz-correlated samples.

Simulations confirm theoretical predictions.

Abstract

Covariance matrix estimation concerns the problem of estimating the covariance matrix from a collection of samples, which is of extreme importance in many applications. Classical results have shown that $O(n)$ samples are sufficient to accurately estimate the covariance matrix from $n$-dimensional independent Gaussian samples. However, in many practical applications, the received signal samples might be correlated, which makes the classical analysis inapplicable. In this paper, we develop a non-asymptotic analysis for the covariance matrix estimation from correlated Gaussian samples. Our theoretical results show that the error bounds are determined by the signal dimension $n$, the sample size $m$, and the shape parameter of the distribution of the correlated sample covariance matrix. Particularly, when the shape parameter is a class of Toeplitz matrices (which is of great practical interest), $O(n)$ samples are also sufficient to faithfully estimate the covariance matrix from correlated samples. Simulations are provided to verify the correctness of the theoretical results.