  Causal disentanglement seeks a representation of data involving latent
variables that relate to one another via a causal model. A representation is
identifiable if both the latent model and the transformation from latent to
observed variables are unique. In this paper, we study observed variables that
are a linear transformation of a linear latent causal model. Data from
interventions are necessary for identifiability: if one latent variable is
missing an intervention, we show that there exist distinct models that cannot
be distinguished. Conversely, we show that a single intervention on each latent
variable is sufficient for identifiability. Our proof uses a generalization of
the RQ decomposition of a matrix that replaces the usual orthogonal and upper
triangular conditions with analogues depending on a partial order on the rows
of the matrix, with partial order determined by a latent causal model. We
corroborate our theoretical results with a method for causal disentanglement
that accurately recovers a latent causal model.
