Gaussian and Robust Kronecker Product Covariance Estimation: Existence and Uniqueness

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 $\max[p/q, q/p] + 1$.