  This paper studies the causal representation learning problem when the latent
causal variables are observed indirectly through an unknown linear
transformation. The objectives are: (i) recovering the unknown linear
transformation (up to scaling) and (ii) determining the directed acyclic graph
(DAG) underlying the latent variables. Sufficient conditions for DAG recovery
are established, and it is shown that a large class of non-linear models in the
latent space (e.g., causal mechanisms parameterized by two-layer neural
networks) satisfy these conditions. These sufficient conditions ensure that the
effect of an intervention can be detected correctly from changes in the score.
Capitalizing on this property, recovering a valid transformation is facilitated
by the following key property: any valid transformation renders latent
variables' score function to necessarily have the minimal variations across
different interventional environments. This property is leveraged for perfect
recovery of the latent DAG structure using only \emph{soft} interventions. For
the special case of stochastic \emph{hard} interventions, with an additional
hypothesis testing step, one can also uniquely recover the linear
transformation up to scaling and a valid causal ordering.
