  This paper studies the problem of learning causal structures from
observational data. We reformulate the Structural Equation Model (SEM) with
additive noises in a form parameterized by binary graph adjacency matrix and
show that, if the original SEM is identifiable, then the binary adjacency
matrix can be identified up to super-graphs of the true causal graph under mild
conditions. We then utilize the reformulated SEM to develop a causal structure
learning method that can be efficiently trained using gradient-based
optimization, by leveraging a smooth characterization on acyclicity and the
Gumbel-Softmax approach to approximate the binary adjacency matrix. It is found
that the obtained entries are typically near zero or one and can be easily
thresholded to identify the edges. We conduct experiments on synthetic and real
datasets to validate the effectiveness of the proposed method, and show that it
readily includes different smooth model functions and achieves a much improved
performance on most datasets considered.
