  Discovering the causal relationship via recovering the directed acyclic graph
(DAG) structure from the observed data is a well-known challenging
combinatorial problem. When there are latent variables, the problem becomes
even more difficult. In this paper, we first propose a DAG structure recovering
algorithm, which is based on the Cholesky factorization of the covariance
matrix of the observed data. The algorithm is fast and easy to implement and
has theoretical grantees for exact recovery. On synthetic and real-world
datasets, the algorithm is significantly faster than previous methods and
achieves the state-of-the-art performance. Furthermore, under the equal error
variances assumption, we incorporate an optimization procedure into the
Cholesky factorization based algorithm to handle the DAG recovering problem
with latent variables. Numerical simulations show that the modified "Cholesky +
optimization" algorithm is able to recover the ground truth graph in most cases
and outperforms existing algorithms.
