标题： 高斯与鲁棒Kronecker积协方差估计：存在性和唯一性 显示英文标题

标题： Gaussian and Robust Kronecker Product Covariance Estimation: Existence and Uniqueness

摘要： 我们研究了高斯协方差估计和鲁棒协方差估计，假设真实协方差矩阵为两个较低维度的平方矩阵的Kronecker乘积。 在两种情况下，我们将估计值定义为约束最大似然问题的解。 在鲁棒情形下，我们考虑泰勒（Tyler）的估计量，该估计量被定义为球面上某一分布的最大似然估计量。 我们给出了估计值存在且唯一的充分条件，并证明在高斯情形下，当均值未知时，样本数量几乎肯定足以保证存在性和唯一性，其中 $p/q+q/p + 2$ 是样本数量， $p$和 $q$分别是Kronecker乘积因子的维度。 在鲁棒情形下，当均值已知时，相应的样本数量为 $\max[p/q, q/p] + 1$。 显示英文摘要

摘要： 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 $\max[p/q, q/p] + 1$.