  Estimation of Markov Random Field and covariance models from high-dimensional
data represents a canonical problem that has received a lot of attention in the
literature. A key assumption, widely employed, is that of {\em sparsity} of the
underlying model. In this paper, we study the problem of estimating such models
exhibiting a more intricate structure comprising simultaneously of {\em sparse,
structured sparse} and {\em dense} components. Such structures naturally arise
in several scientific fields, including molecular biology, finance, and
political science. We introduce a general framework based on a novel structured
norm that enables us to estimate such complex structures from high-dimensional
data. The resulting optimization problem is convex and we introduce a
linearized multi-block alternating direction method of multipliers (ADMM)
algorithm to solve it efficiently. We illustrate the superior performance of
the proposed framework on a number of synthetic data sets generated from both
random and structured networks. Further, we apply the method to a number of
real data sets and discuss the results.
