  Recently directed acyclic graph (DAG) structure learning is formulated as a
constrained continuous optimization problem with continuous acyclicity
constraints and was solved iteratively through subproblem optimization. To
further improve efficiency, we propose a novel learning framework to model and
learn the weighted adjacency matrices in the DAG space directly. Specifically,
we first show that the set of weighted adjacency matrices of DAGs are
equivalent to the set of weighted gradients of graph potential functions, and
one may perform structure learning by searching in this equivalent set of DAGs.
To instantiate this idea, we propose a new algorithm, DAG-NoCurl, which solves
the optimization problem efficiently with a two-step procedure: 1) first we
find an initial cyclic solution to the optimization problem, and 2) then we
employ the Hodge decomposition of graphs and learn an acyclic graph by
projecting the cyclic graph to the gradient of a potential function.
Experimental studies on benchmark datasets demonstrate that our method provides
comparable accuracy but better efficiency than baseline DAG structure learning
methods on both linear and generalized structural equation models, often by
more than one order of magnitude.
