Tensor-Based Gaussian Graphical Model

Exploring the relationships between variables of data has attracted a lot of attentions in the fields like machine learning, signal processing, neuroscience, and social network. It enables us to better understand the data before choosing models for particular tasks. Gaussian graphical model provides a natural and straightforward way to reveal the relationships of variables in data, where the zero entries in precision matrices represent conditional independence between corresponding variables and vice versa. However, traditional vector- or matrix-variate-based methods are not suitable for higher-order data analysis due to the possible structural information loss, and this is where tensor-variate-based Gaussian graphical model comes in. In this chapter, we discuss the development of Gaussian graphical models from vector-variate-based methods, matrix-variate-based methods to tensor-variate-based methods and mainly focus on the last one. Detailed assumptions, probability density functions, optimization models based on maximum likelihood estimation, and some typical algorithms are given. In addition, we illustrate the applications of tensor Gaussian graphical models in environmental prediction and mice aging study and give some numerically experimental results.