  Causal representation learning seeks to extract high-level latent factors
from low-level sensory data. Most existing methods rely on observational data
and structural assumptions (e.g., conditional independence) to identify the
latent factors. However, interventional data is prevalent across applications.
Can interventional data facilitate causal representation learning? We explore
this question in this paper. The key observation is that interventional data
often carries geometric signatures of the latent factors' support (i.e. what
values each latent can possibly take). For example, when the latent factors are
causally connected, interventions can break the dependency between the
intervened latents' support and their ancestors'. Leveraging this fact, we
prove that the latent causal factors can be identified up to permutation and
scaling given data from perfect $do$ interventions. Moreover, we can achieve
block affine identification, namely the estimated latent factors are only
entangled with a few other latents if we have access to data from imperfect
interventions. These results highlight the unique power of interventional data
in causal representation learning; they can enable provable identification of
latent factors without any assumptions about their distributions or dependency
structure.
