  Learning the directed acyclic graph (DAG) structure of a Bayesian network
from observational data is a notoriously difficult problem for which many
hardness results are known. In this paper we propose a provably polynomial-time
algorithm for learning sparse Gaussian Bayesian networks with equal noise
variance --- a class of Bayesian networks for which the DAG structure can be
uniquely identified from observational data --- under high-dimensional
settings. We show that $O(k^4 \log p)$ number of samples suffices for our
method to recover the true DAG structure with high probability, where $p$ is
the number of variables and $k$ is the maximum Markov blanket size. We obtain
our theoretical guarantees under a condition called Restricted Strong Adjacency
Faithfulness, which is strictly weaker than strong faithfulness --- a condition
that other methods based on conditional independence testing need for their
success. The sample complexity of our method matches the information-theoretic
limits in terms of the dependence on $p$. We show that our method out-performs
existing state-of-the-art methods for learning Gaussian Bayesian networks in
terms of recovering the true DAG structure while being comparable in speed to
heuristic methods.
