Localization Estimator for High Dimensional Tensor Covariance Matrices

TL;DR

This paper introduces a new high-dimensional covariance localization estimator for tensor data, addressing complex covariance structures and decay patterns, with proven statistical properties and practical validation on ocean data.

Contribution

It proposes a novel localization estimator for tensor covariance matrices with established minimax convergence rates and demonstrated effectiveness in real-world applications.

Findings

Estimator achieves optimal convergence rates.

Method effectively captures complex covariance structures.

Successful application to ocean eddy data.

Abstract

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.