Learning Counterfactual Dynamics via Wasserstein Parallel Transport

Music City Center 

In many real-world applications, ranging from computer vision and machine learning to causal inference, researchers want to understand how the dynamics of a system would behave if it were placed in a different environment. We address this question by presenting an approach for learning such local counterfactual dynamics across systems described via probability measures using parallel transport in the Wasserstein space. To perform this parallel transport, standard approximations like Schild's ladder are computationally too costly and require strong regularity conditions on the measures, which are not met in most real-world settings. We therefore provide a new approximation scheme for parallel transport in the Wasserstein space based on Jacobi fields. This approach circumvents the shortcomings of Schild's ladder and can be efficiently implemented by solving a continuity equation. In the simple setting of empirical measures on

Euclidean space, it generalizes domain adaptation via optimal transport by also incorporating dynamics. We demonstrate the utility of the approach through applications in deformation transfer in computer vision, dynamic domain adaptation for smooth measures, and external validity in causal inference.

joint work with Ka Yan Chen, Wonjun Choi, and Marcelo Ortiz

Optimal Transport

Nonparametric inference

Semiparametric inference