Structured regularization covariance estimation in tensor-valued data analysis

Covariance estimation poses a crucial challenge in high-dimensional data analysis, especially when traditional methods (e.g., sample covariance) are inaccurate, particularly with small sample sizes. A promising solution is to exploit inherent data structures such as low-rankness, sparsity, or smoothness. For tensor data (multi-dimensional arrays), structured regularization aids in dimensionality reduction. This paper introduces novel regularization methods for tensor covariance estimation, specifically applying banded and tapering structures to the covariance matrix. We use Kronecker Product Canonical Polyadic (KPCP) decomposition to approximate large matrices via the Kronecker product of smaller matrices. A split resampling scheme is employed to select parameters for the KPCP decomposition from noisy data. This leads to two methods: KPCP-TB-R (Triply Banded-Resampling) and KPCP-TT-R (Triply Tapering-Resampling). Additionally, sparse (thresholding) and multi-structured regularization approaches are introduced for comparison. The effectiveness and robustness of the proposed methods are validated through extensive simulations and applied to monthly export trade volume data.