  The goal of causal representation learning is to find a representation of
data that consists of causally related latent variables. We consider a setup
where one has access to data from multiple domains that potentially share a
causal representation. Crucially, observations in different domains are assumed
to be unpaired, that is, we only observe the marginal distribution in each
domain but not their joint distribution. In this paper, we give sufficient
conditions for identifiability of the joint distribution and the shared causal
graph in a linear setup. Identifiability holds if we can uniquely recover the
joint distribution and the shared causal representation from the marginal
distributions in each domain. We transform our identifiability results into a
practical method to recover the shared latent causal graph.
