Tangential Wasserstein Projections
This provides a new tool for researchers and practitioners in fields like causal inference and object data analysis, though it appears incremental as it builds on existing Wasserstein space concepts.
The paper tackles the problem of projecting between sets of probability measures by developing a notion of tangential Wasserstein projections based on geometric properties of the 2-Wasserstein space, resulting in a computationally efficient method with unique solutions in regular settings. An application to causal inference generalizes synthetic controls to multivariate data with individual-level heterogeneity and enables joint estimation of optimal weights across time periods.
We develop a notion of projections between sets of probability measures using the geometric properties of the 2-Wasserstein space. It is designed for general multivariate probability measures, is computationally efficient to implement, and provides a unique solution in regular settings. The idea is to work on regular tangent cones of the Wasserstein space using generalized geodesics. Its structure and computational properties make the method applicable in a variety of settings, from causal inference to the analysis of object data. An application to estimating causal effects yields a generalization of the notion of synthetic controls to multivariate data with individual-level heterogeneity, as well as a way to estimate optimal weights jointly over all time periods.