Given a sample of i.i.d. high-dimensional centered random vectors, we consider a problem of estimation of their covariance matrix $\Sigma$ with an additional assumption that $\Sigma$ can be represented as a sum of a few Kronecker products of smaller matrices. Under mild conditions, we derive the first non-asymptotic dimension-free high-probability bound on the Frobenius distance between $\Sigma$ and a widely used penalized permuted least squares estimate. Because of the hidden structure, the established rate of convergence is faster than in the standard covariance estimation problem.

翻译：给定独立同分布的高维中心化随机向量样本，我们考虑其协方差矩阵 $\Sigma$ 的估计问题，并附加 $\Sigma$ 可表示为若干较小矩阵的克罗内克积之和的假设。在温和条件下，我们推导出首个非渐近无维数高概率界，该界描述了 $\Sigma$ 与广泛使用的惩罚置换最小二乘估计量之间的弗罗贝尼乌斯距离。由于隐含的结构，所建立的收敛速度快于标准协方差估计问题。