  We consider two variables that are related to each other by an invertible
function. While it has previously been shown that the dependence structure of
the noise can provide hints to determine which of the two variables is the
cause, we presently show that even in the deterministic (noise-free) case,
there are asymmetries that can be exploited for causal inference. Our method is
based on the idea that if the function and the probability density of the cause
are chosen independently, then the distribution of the effect will, in a
certain sense, depend on the function. We provide a theoretical analysis of
this method, showing that it also works in the low noise regime, and link it to
information geometry. We report strong empirical results on various real-world
data sets from different domains.
