  Causal representation learning seeks to uncover latent, high-level causal
representations from low-level observed data. It is particularly good at
predictions under unseen distribution shifts, because these shifts can
generally be interpreted as consequences of interventions. Hence leveraging
{seen} distribution shifts becomes a natural strategy to help identifying
causal representations, which in turn benefits predictions where distributions
are previously {unseen}. Determining the types (or conditions) of such
distribution shifts that do contribute to the identifiability of causal
representations is critical. This work establishes a {sufficient} and
{necessary} condition characterizing the types of distribution shifts for
identifiability in the context of latent additive noise models. Furthermore, we
present partial identifiability results when only a portion of distribution
shifts meets the condition. In addition, we extend our findings to latent
post-nonlinear causal models. We translate our findings into a practical
algorithm, allowing for the acquisition of reliable latent causal
representations. Our algorithm, guided by our underlying theory, has
demonstrated outstanding performance across a diverse range of synthetic and
real-world datasets. The empirical observations align closely with the
theoretical findings, affirming the robustness and effectiveness of our
approach.
