# Learning Unfaithful $K$-separable Gaussian Graphical Models

De Wen Soh, Sekhar Tatikonda.

Year: 2019, Volume: 20, Issue: 109, Pages: 1−30

#### Abstract

The global Markov property for Gaussian graphical models ensures graph separation implies conditional independence. Specifically if a node set $S$ graph separates nodes $u$ and $v$ then $X_u$ is conditionally independent of $X_v$ given $X_S$. The opposite direction need not be true, that is, $X_u \perp X_v \mid X_S$ need not imply $S$ is a node separator of $u$ and $v$. When it does, the relation $X_u \perp X_v \mid X_S$ is called faithful. In this paper we provide a characterization of faithful relations and then provide an algorithm to test faithfulness based only on knowledge of other conditional relations of the form $X_i \perp X_j \mid X_S$. We study two classes of separable Gaussian graphical models, namely, weakly $K$-separable and strongly $K$-separable Gaussian graphical models. Using the above test for faithfulness, we introduce algorithms to learn the topologies of weakly $K$-separable and strongly $K$-separable Gaussian graphical models with $\Omega(K\log p)$ sample complexity. For strongly $K$-separable Gaussian graphical models, we additionally provide a method with error bounds for learning the off-diagonal precision matrix entries.