On Learning Multi-Attribute Graphs With Non-Convex Penalties

Most existing methods for graph estimation are based on single-attribute models where one associates a scalar random variable with each node. In multi-attribute graphical models, each node represents a random vector. In this paper we investigate multi-attribute graph learning using a non-convex penalized log-likelihood objective function. We consider both convex (sparse-group lasso) and sparse-group non-convex (log-sum and smoothly clipped absolute deviation (SCAD)) penalty functions. An alternating direction method of multipliers (ADMM) approach coupled with local linear approximation to non-convex penalties is presented for optimization of the objective function. We provide sufficient conditions including some irrepresentability conditions, in a high-dimensional setting for consistency, convexity and graph recovery. In a synthetic data example the sparse-group log-sum penalized objective function significantly outperforms the lasso penalized as well as SCAD penalized objective functions with F<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf>-score and Hamming distance as performance metrics.