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：给定一个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，并附加假设$\Sigma$可表示为若干较小矩阵的克罗内克积之和。在温和条件下，我们首次推导出$\Sigma$与广泛使用的惩罚置换最小二乘估计之间弗罗贝尼乌斯距离的非渐近、无维度依赖的高概率界。由于隐藏结构的存在，所建立的收敛速率快于标准协方差估计问题。