  We consider the problem of regularized maximum likelihood estimation for the
structure and parameters of a high-dimensional, sparse directed acyclic
graphical (DAG) model with Gaussian distribution, or equivalently, of a
Gaussian structural equation model. We show that the $\ell_0$-penalized maximum
likelihood estimator of a DAG has about the same number of edges as the
minimal-edge I-MAP (a DAG with minimal number of edges representing the
distribution), and that it converges in Frobenius norm. We allow the number of
nodes p to be much larger than sample size n but assume a sparsity condition
and that any representation of the true DAG has at least a fixed proportion of
its nonzero edge weights above the noise level. Our results do not rely on the
faithfulness assumption nor on the restrictive strong faithfulness condition
which are required for methods based on conditional independence testing such
as the PC-algorithm.
