  This paper focuses on causal representation learning (CRL) under a general
nonparametric latent causal model and a general transformation model that maps
the latent data to the observational data. It establishes identifiability and
achievability results using two hard uncoupled interventions per node in the
latent causal graph. Notably, one does not know which pair of intervention
environments have the same node intervened (hence, uncoupled). For
identifiability, the paper establishes that perfect recovery of the latent
causal model and variables is guaranteed under uncoupled interventions. For
achievability, an algorithm is designed that uses observational and
interventional data and recovers the latent causal model and variables with
provable guarantees. This algorithm leverages score variations across different
environments to estimate the inverse of the transformer and, subsequently, the
latent variables. The analysis, additionally, recovers the identifiability
result for two hard coupled interventions, that is when metadata about the pair
of environments that have the same node intervened is known. This paper also
shows that when observational data is available, additional faithfulness
assumptions that are adopted by the existing literature are unnecessary.
