  This paper addresses intervention-based causal representation learning (CRL)
under a general nonparametric latent causal model and an unknown transformation
that maps the latent variables to the observed variables. Linear and general
transformations are investigated. The paper addresses both the identifiability
and achievability aspects. Identifiability refers to determining
algorithm-agnostic conditions that ensure recovering the true latent causal
variables and the latent causal graph underlying them. Achievability refers to
the algorithmic aspects and addresses designing algorithms that achieve
identifiability guarantees. By drawing novel connections between score
functions (i.e., the gradients of the logarithm of density functions) and CRL,
this paper designs a score-based class of algorithms that ensures both
identifiability and achievability. First, the paper focuses on linear
transformations and shows that one stochastic hard intervention per node
suffices to guarantee identifiability. It also provides partial identifiability
guarantees for soft interventions, including identifiability up to ancestors
for general causal models and perfect latent graph recovery for sufficiently
non-linear causal models. Secondly, it focuses on general transformations and
shows that two stochastic hard interventions per node suffice for
identifiability. Notably, one does not need to know which pair of
interventional environments have the same node intervened.
