  Covariance estimation for matrix-valued data has received an increasing
interest in applications. Unlike previous works that rely heavily on matrix
normal distribution assumption and the requirement of fixed matrix size, we
propose a class of distribution-free regularized covariance estimation methods
for high-dimensional matrix data under a separability condition and a bandable
covariance structure. Under these conditions, the original covariance matrix is
decomposed into a Kronecker product of two bandable small covariance matrices
representing the variability over row and column directions. We formulate a
unified framework for estimating bandable covariance, and introduce an
efficient algorithm based on rank one unconstrained Kronecker product
approximation. The convergence rates of the proposed estimators are
established, and the derived minimax lower bound shows our proposed estimator
is rate-optimal under certain divergence regimes of matrix size. We further
introduce a class of robust covariance estimators and provide theoretical
guarantees to deal with heavy-tailed data. We demonstrate the superior
finite-sample performance of our methods using simulations and real
applications from a gridded temperature anomalies dataset and a S&P 500 stock
data analysis.
