  Learning causal relationships from empirical observations is a central task
in scientific research. A common method is to employ structural causal models
that postulate noisy functional relations among a set of interacting variables.
To ensure unique identifiability of causal directions, researchers consider
restricted subclasses of structural causal models. Post-nonlinear (PNL) causal
models constitute one of the most flexible options for such restricted
subclasses, containing in particular the popular additive noise models as a
further subclass. However, learning PNL models is not well studied beyond the
bivariate case. The existing methods learn non-linear functional relations by
minimizing residual dependencies and subsequently test independence from
residuals to determine causal orientations. However, these methods can be prone
to overfitting and, thus, difficult to tune appropriately in practice. As an
alternative, we propose a new approach for PNL causal discovery that uses
rank-based methods to estimate the functional parameters. This new approach
exploits natural invariances of PNL models and disentangles the estimation of
the non-linear functions from the independence tests used to find causal
orientations. We prove consistency of our method and validate our results in
numerical experiments.
