  We study the problem of learning causal representations from unknown, latent
interventions in a general setting, where the latent distribution is Gaussian
but the mixing function is completely general. We prove strong identifiability
results given unknown single-node interventions, i.e., without having access to
the intervention targets. This generalizes prior works which have focused on
weaker classes, such as linear maps or paired counterfactual data. This is also
the first instance of causal identifiability from non-paired interventions for
deep neural network embeddings. Our proof relies on carefully uncovering the
high-dimensional geometric structure present in the data distribution after a
non-linear density transformation, which we capture by analyzing quadratic
forms of precision matrices of the latent distributions. Finally, we propose a
contrastive algorithm to identify the latent variables in practice and evaluate
its performance on various tasks.
