  Targeted and uniform interventions to a system are crucial for unveiling
causal relationships. While several methods have been developed to leverage
interventional data for causal structure learning, their practical application
in real-world scenarios often remains challenging. Recent benchmark studies
have highlighted these difficulties, even when large numbers of single-variable
intervention samples are available. In this work, we demonstrate, both
theoretically and empirically, that such datasets contain a wealth of causal
information that can be effectively extracted under realistic assumptions about
the data distribution. More specifically, we introduce the notion of
interventional faithfulness, which relies on comparisons between the marginal
distributions of each variable across observational and interventional
settings, and we introduce a score on causal orders. Under this assumption, we
are able to prove strong theoretical guarantees on the optimum of our score
that also hold for large-scale settings. To empirically verify our theory, we
introduce Intersort, an algorithm designed to infer the causal order from
datasets containing large numbers of single-variable interventions by
approximately optimizing our score. Intersort outperforms baselines (GIES, PC
and EASE) on almost all simulated data settings replicating common benchmarks
in the field. Our proposed novel approach to modeling interventional datasets
thus offers a promising avenue for advancing causal inference, highlighting
significant potential for further enhancements under realistic assumptions.
