Gaussian and Robust Kronecker Product Covariance Estimation: Existence and Uniqueness

TL;DR

This paper investigates the existence and uniqueness of Gaussian and robust Kronecker product covariance estimators, providing tight conditions on sample size for reliable estimation in high-dimensional settings.

Contribution

It establishes new sufficient conditions for the existence and uniqueness of Kronecker product covariance estimators in both Gaussian and robust frameworks.

Findings

Gaussian case requires p/q+q/p+2 samples for existence and uniqueness

Robust case requires max[p/q, q/p]+1 samples with known mean

Provides theoretical guarantees for high-dimensional covariance estimation

Abstract

We study the Gaussian and robust covariance estimation, assuming the true covariance matrix to be a Kronecker product of two lower dimensional square matrices. In both settings we define the estimators as solutions to the constrained maximum likelihood programs. In the robust case, we consider Tyler's estimator defined as the maximum likelihood estimator of a certain distribution on a sphere. We develop tight sufficient conditions for the existence and uniqueness of the estimates and show that in the Gaussian scenario with the unknown mean, $p/q+q/p + 2$ samples are almost surely enough to guarantee the existence and uniqueness, where $p$ and $q$ are the dimensions of the Kronecker product factors. In the robust case with the known mean, the corresponding sufficient number of samples is $\max[p/q, q/p] + 1$.