  We establish conditions under which latent causal graphs are
nonparametrically identifiable and can be reconstructed from unknown
interventions in the latent space. Our primary focus is the identification of
the latent structure in measurement models without parametric assumptions such
as linearity or Gaussianity. Moreover, we do not assume the number of hidden
variables is known, and we show that at most one unknown intervention per
hidden variable is needed. This extends a recent line of work on learning
causal representations from observations and interventions. The proofs are
constructive and introduce two new graphical concepts -- imaginary subsets and
isolated edges -- that may be useful in their own right. As a matter of
independent interest, the proofs also involve a novel characterization of the
limits of edge orientations within the equivalence class of DAGs induced by
unknown interventions. These are the first results to characterize the
conditions under which causal representations are identifiable without making
any parametric assumptions in a general setting with unknown interventions and
without faithfulness.
