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Learning Latent Variable Gaussian Graphical Models
Zhaoshi Meng, Brian Eriksson, Alfred O. Hero
Gaussian graphical models (GGM) have been widely used in many high-dimensional applications ranging from biological and financial data to recommender systems. Sparsity in GGM plays a central role both statistically and computationally. Unfortunately, real-world data often does not fit well to sparse graphical models. In this paper, we focus on a family of latent variable Gaussian graphical models (LVGGM), where the model is conditionally sparse given latent variables, but marginally non-sparse. In LVGGM, the inverse covariance matrix has a low-rank plus sparse structure, and can be learned in a regularized maximum likelihood framework. We derive novel parameter estimation error bounds for LVGGM under mild conditions in the high-dimensional setting. These results complement the existing theory on the structural learning, and open up new possibilities of using LVGGM for statistical inference.