  In causal models, a given mechanism is assumed to be invariant to changes of
other mechanisms. While this principle has been utilized for inference in
settings where the causal variables are observed, theoretical insights when the
variables of interest are latent are largely missing. We assay the connection
between invariance and causal representation learning by establishing
impossibility results which show that invariance alone is insufficient to
identify latent causal variables. Together with practical considerations, we
use these theoretical findings to highlight the need for additional constraints
in order to identify representations by exploiting invariance.
