Statistical Inference for Matrix-variate Gaussian Graphical Models and False Discovery Rate Control

Matrix-variate Gaussian graphical models (GGM) have been widely used for modelling matrix-variate data. Since the supports of sparse row and column precision matrices encode the conditional independence among rows and columns of the data, it is of great interest to conduct support recovery. A commonly used approach is the penalized log-likelihood method. However, due to the complicated structure of the precision matrices of matrix-variate GGMs, the log-likelihood is non-convex, which brings a great challenge for both computation and theoretical analysis. In this paper, we propose an alternative approach by formulating the support recovery problem into a multiple testing problem. A new test statistic is proposed and based on that, we further develop a method to control false discovery rate (FDR) asymptotically. Our method is computationally attractive since it only involves convex optimization. Theoretically, our method allows very weak conditions, i.e., even when the sample size is a constant and the dimensions go to infinity, the asymptotic normality of the test statistics and FDR control can still be guaranteed. The finite sample performance of the proposed method is illustrated by both simulated and real data analysis.