Matrix Non-Normal Graphical Model

Abstract: Contemporary data often have a matrix form, with intrinsic information embedded in the data structure, and are high-dimensional. The matrix Gaussian graphical model is an important tool for understanding the dependence structure among such data. A central assumption in this model is that the observations follow a matrix normal distribution, which enables characterization of conditional independence via the precision matrix. However, such an assumption can be too stringent in practice. If data are heavy-tailed, normality-based methods may produce misleading results and unstable estimation. Motivated by this challenge, we consider the matrix non-normal graphical model (MANGO). The MANGO model includes the matrix Gaussian graphical model as a special case, but easily adapts to potential heavy tails. Under this general model, we investigate the relationship between conditional dependence and the precision matrices, revealing that non-normality immediately affects our interpretation of the precision matrix. Two sub-models (U-MANGO and H-MANGO) are further studied to accommodate different levels of heavy tails. We develop efficient estimation procedures for these two models that require minimal additional computation over the Gaussian model. Moreover, two hypothesis tests are developed to identify a suitable model for a particular dataset. Numerical results demonstrate the superior performance of our methods.