Learning Latent Variable Gaussian Graphical Models

TL;DR

This paper introduces a novel approach to learning latent variable Gaussian graphical models (LVGGM), which handle non-sparse data by modeling the inverse covariance as a low-rank plus sparse structure, with new error bounds for high-dimensional estimation.

Contribution

The paper develops a regularized maximum likelihood method for LVGGM and derives new high-dimensional parameter estimation error bounds, expanding the theoretical understanding of these models.

Findings

Derived novel error bounds for LVGGM estimation.

Extended theoretical analysis to high-dimensional settings.

Opened new avenues for statistical inference using LVGGM.

Abstract

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.