Wasserstein projections in the convex order: regularity and characterization in the quadratic Gaussian case

TL;DR

This paper investigates the properties of Wasserstein projections in the convex order, focusing on regularity, continuity, and explicit characterizations in the quadratic Gaussian case, including uniqueness and covariance structure analysis.

Contribution

It provides new insights into the regularity and characterization of Wasserstein projections in the convex order, especially for Gaussian measures and in higher dimensions.

Findings

Continuity of Wasserstein projections when unique

Gaussian projections are characterized explicitly

Existence and uniqueness of Gaussian projections in higher dimensions

Abstract

In this paper, we first show continuity of both Wasserstein projections in the convex order when they are unique. We also check that, in arbitrary dimension $d$, the quadratic Wasserstein projection of a probability measure $\mu$ on the set of probability measures dominated by $\nu$ in the convex order is non-expansive in $\mu$ and H\"older continuous with exponent $1/2$ in $\nu$. When $\mu$ and $\nu$ are Gaussian, we check that this projection is Gaussian and also consider the quadratic Wasserstein projection on the set of probability measures $\nu$ dominating $\mu$ in the convex order. In the case when $d\ge 2$ and $\nu$ is not absolutely continuous with respect to the Lebesgue measure where uniqueness of the latter projection was not known, we check that there is always a unique Gaussian projection and characterize when non Gaussian projections with the same covariance matrix also exist. Still for Gaussian distributions, we characterize the covariance matrices of the two projections. It turns out that there exists an orthogonal transformation of space under which the computations are similar to the easy case when the covariance matrices of $\mu$ and $\nu$ are diagonal.