Latent Graphical Model Selection: Efficient Methods for Locally Tree-like Graphs

Graphical model selection refers to the problem of estimating the unknown graph structure given observations at the nodes in the model. We consider a challenging instance of this problem when some of the nodes are latent or hidden. We characterize conditions for tractable graph estimation and develop efficient methods with provable guarantees. We consider the class of Ising models Markov on locally tree-like graphs, which are in the regime of correlation decay. We propose an efficient method for graph estimation, and establish its structural consistency when the number of samples $n$ scales as $n = \Omega(\theta_{\min}^{-\delta \eta(\eta+1)-2}\log p)$, where $\theta_{\min}$ is the minimum edge potential, $\delta$ is the depth (i.e., distance from a hidden node to the nearest observed nodes), and $\eta$ is a parameter which depends on the minimum and maximum node and edge potentials in the Ising model. The proposed method is practical to implement and provides flexibility to control the number of latent variables and the cycle lengths in the output graph. We also present necessary conditions for graph estimation by any method and show that our method nearly matches the lower bound on sample requirements.