  Identifying latent variables and causal structures from observational data is
essential to many real-world applications involving biological data, medical
data, and unstructured data such as images and languages. However, this task
can be highly challenging, especially when observed variables are generated by
causally related latent variables and the relationships are nonlinear. In this
work, we investigate the identification problem for nonlinear latent
hierarchical causal models in which observed variables are generated by a set
of causally related latent variables, and some latent variables may not have
observed children.
  We show that the identifiability of causal structures and latent variables
(up to invertible transformations) can be achieved under mild assumptions: on
causal structures, we allow for multiple paths between any pair of variables in
the graph, which relaxes latent tree assumptions in prior work; on structural
functions, we permit general nonlinearity and multi-dimensional continuous
variables, alleviating existing work's parametric assumptions. Specifically, we
first develop an identification criterion in the form of novel identifiability
guarantees for an elementary latent variable model. Leveraging this criterion,
we show that both causal structures and latent variables of the hierarchical
model can be identified asymptotically by explicitly constructing an estimation
procedure. To the best of our knowledge, our work is the first to establish
identifiability guarantees for both causal structures and latent variables in
nonlinear latent hierarchical models.
