  Discovery of causal relationships from observational data is an important
problem in many areas. Several recent results have established the
identifiability of causal DAGs with non-Gaussian and/or nonlinear structural
equation models (SEMs). In this paper, we focus on nonlinear SEMs defined by
non-invertible functions, which exist in many data domains, and propose a novel
test for non-invertible bivariate causal models. We further develop a method to
incorporate this test in structure learning of DAGs that contain both linear
and nonlinear causal relations. By extensive numerical comparisons, we show
that our algorithms outperform existing DAG learning methods in identifying
causal graphical structures. We illustrate the practical application of our
method in learning causal networks for combinatorial binding of transcription
factors from ChIP-Seq data.
