Covariance estimation via sparse Kronecker structures

Abstract

The problem of estimating covariance matrices is central to statistical analysis and is extensively addressed when data are vectors. This paper studies a novel Kronecker-structured approach for estimating such matrices when data are matrices and arrays. Focusing on matrix-variate data, we present simple approaches to estimate the row and the column correlation matrices, formulated separately via convex optimization. We also discuss simple thresholding estimators motivated by the recent development in the literature. Non-asymptotic results show that the proposed method greatly outperforms methods that ignore the matrix structure of the data. In particular, our framework allows the dimensionality of data to be arbitrary order even for fixed sample size, and works for flexible distributions beyond normality. Simulations and data analysis further confirm the competitiveness of the method. An extension to general array-data is also outlined.