  Matrix-variate Gaussian graphical models (GGM) have been widely used for
modeling matrix-variate data. Since the support of sparse precision matrix
represents the conditional independence graph among matrix entries, conducting
support recovery yields valuable information. A commonly used approach is the
penalized log-likelihood method. However, due to the complicated structure of
precision matrices in the form of Kronecker product, the log-likelihood is
non-convex, which presents challenges 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 developed and based on that, we use the popular Benjamini and Hochberg's
procedure to control false discovery rate (FDR) asymptotically. Our method
involves only convex optimization, making it computationally attractive.
Theoretically, our method allows very weak conditions, i.e., even when the
sample size is finite and the dimensions go to infinity, the asymptotic
normality of the test statistics and FDR control can still be guaranteed. We
further provide the power analysis result. The finite sample performance of the
proposed method is illustrated by both simulated and real data analysis.
