Matrix-valued data, where the sampling unit is a matrix consisting of rows and columns of measurements, are emerging in numerous scientific and business applications. Matrix Gaussian graphical models are a useful tool to characterize the conditional dependence structure of rows and columns. We employ non-convex penalization to tackle the estimation of multiple graphs from matrix-valued data under a matrix normal distribution. We propose a highly efficient non-convex optimization algorithm that can scale up for graphs with hundreds of nodes. We establish the asymptotic properties of the estimator, which requires less stringent conditions and has a sharper probability error bound than existing results. We demonstrate the efficacy of our proposed method through both simulations and real functional magnetic resonance imaging analyses.
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