  Causal abstraction (CA) theory establishes formal criteria for relating
multiple structural causal models (SCMs) at different levels of granularity by
defining maps between them. These maps have significant relevance for
real-world challenges such as synthesizing causal evidence from multiple
experimental environments, learning causally consistent representations at
different resolutions, and linking interventions across multiple SCMs. In this
work, we propose COTA, the first method to learn abstraction maps from
observational and interventional data without assuming complete knowledge of
the underlying SCMs. In particular, we introduce a multi-marginal Optimal
Transport (OT) formulation that enforces do-calculus causal constraints,
together with a cost function that relies on interventional information. We
extensively evaluate COTA on synthetic and real world problems, and showcase
its advantages over non-causal, independent and aggregated COTA formulations.
Finally, we demonstrate the efficiency of our method as a data augmentation
tool by comparing it against the state-of-the-art CA learning framework, which
assumes fully specified SCMs, on a real-world downstream task.
