Learning Latent Variable Gaussian Graphical Models
This work addresses the challenge of modeling non-sparse data in high-dimensional applications like biology and finance, though it appears incremental as it builds on existing LVGGM theory.
The paper tackles the problem of real-world data not fitting sparse Gaussian graphical models by focusing on latent variable Gaussian graphical models (LVGGM), where the model is conditionally sparse given latent variables, and derives novel parameter estimation error bounds under mild high-dimensional conditions.
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.