This paper considers covariance matrix estimation of tensor data under high dimensionality. A multi-bandable covariance class is established to accommodate the need for complex covariance structures of multi-layer lattices and general covariance decay patterns. We propose a high dimensional covariance localization estimator for tensor data, which regulates the sample covariance matrix through a localization function. The statistical properties of the proposed estimator are studied by deriving the minimax rates of convergence under the spectral and the Frobenius norms. Numerical experiments and real data analysis on ocean eddy data are carried out to illustrate the utility of the proposed method in practice.

翻译：本文研究高维张量数据的协方差矩阵估计问题。针对多层格点结构的复杂协方差特性及一般协方差衰减模式，建立了多波段协方差矩阵类。我们提出一种适用于张量数据的高维协方差局部化估计器，该估计器通过局部化函数对样本协方差矩阵进行正则化处理。通过推导谱范数与Frobenius范数下的极小极大收敛速率，研究了所提估计器的统计性质。通过数值实验及海洋涡旋数据的实证分析，验证了该方法在实际应用中的有效性。