  Causal representation learning aims to unveil latent high-level causal
representations from observed low-level data. One of its primary tasks is to
provide reliable assurance of identifying these latent causal models, known as
identifiability. A recent breakthrough explores identifiability by leveraging
the change of causal influences among latent causal variables across multiple
environments \citep{liu2022identifying}. However, this progress rests on the
assumption that the causal relationships among latent causal variables adhere
strictly to linear Gaussian models. In this paper, we extend the scope of
latent causal models to involve nonlinear causal relationships, represented by
polynomial models, and general noise distributions conforming to the
exponential family. Additionally, we investigate the necessity of imposing
changes on all causal parameters and present partial identifiability results
when part of them remains unchanged. Further, we propose a novel empirical
estimation method, grounded in our theoretical finding, that enables learning
consistent latent causal representations. Our experimental results, obtained
from both synthetic and real-world data, validate our theoretical contributions
concerning identifiability and consistency.
