Tangential Wasserstein Projections

TL;DR

This paper introduces a novel method for projecting probability measures in the Wasserstein space, enabling efficient analysis of multivariate data with applications in causal inference and object data analysis.

Contribution

It develops a new projection technique in Wasserstein space using tangent cones, offering computational efficiency and applicability to diverse multivariate problems.

Findings

Provides a unique projection solution in regular settings.

Enables generalized synthetic control methods for multivariate data.

Applicable to causal inference and object data analysis.

Abstract

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.