Optimal transport for causal discovery

Recently, approaches based on Functional Causal Models (FCMs) have been proposed to determine causal direction between two variables, by restricting model classes; however, their performance is sensitive to the model assumptions. In this paper, we provide a dynamical-system view of FCMs and propose a new framework for identifying causal direction in the bivariate case. We first interpret FCMs in the bivariate case as an optimal transport problem under proper structural constraints. By exploiting the dynamical interpretation of optimal transport, we then derive the underlying time evolution of static cause-effect pair data under the least action principle. It provides a new dimension for describing static causal discovery tasks, while enjoying more freedom for modeling the quantitative causal influences. In particular, we show that additive noise models correspond to volume-preserving pressureless flows. Consequently, based on their velocity field divergence, we derive a criterion to determine causal direction. With this criterion, we propose a novel optimal transport based algorithm which is robust to the choice of models. Our method demonstrated promising results on both synthetic and real cause-effect pair datasets.