On nonexpansiveness of metric projection operators on Wasserstein spaces

TL;DR

This paper studies the nonexpansiveness of metric projection operators in Wasserstein spaces, proving nonexpansiveness in one dimension and showing its failure in higher dimensions, while also exploring curvature properties of these spaces.

Contribution

It provides a direct proof of nonexpansiveness in one-dimensional Wasserstein spaces and demonstrates the failure of this property in higher dimensions, along with analyzing curvature characteristics.

Findings

Projection operators are nonexpansive in one-dimensional Wasserstein spaces.

Nonexpansiveness fails in higher dimensions for certain regimes of p.

Wasserstein spaces exhibit positive curvature properties and are nowhere locally Busemann NPC spaces.

Abstract

In this paper we investigate properties of metric projections onto specific closed and geodesically convex proper subsets of Wasserstein spaces $(\mathcal{P}_p(\mathbf{R}^d),W_p).$ When $d=1$, as $(\mathcal{P}_2(\mathbf{R}),W_2)$ is isometrically isomorphic to a flat space with a Hilbertian structure, the corresponding projection operators are expected to be nonexpansive. We give a direct proof of this fact, relying on intrinsic analysis, which also implies nonexpansiveness in certain special cases in higher dimensions. When $d>1$, we show the failure of this property in two regimes: when $p>1$ is either small enough or large enough. Finally, we prove some positive curvature properties of Wasserstein spaces $(\mathcal{P}_p(\mathbf{R}^d),W_p)$ when $d\ge 2$ and $p\in(1,+\infty)$ are arbitrary: we show that Wasserstein spaces are nowhere locally Busemann NPC spaces, and they nowhere locally satisfy the so-called projection criterion. As a corollary of the former, they have nonnegative upper Alexandrov curvature, in a precise sense that we define here. In our analysis a particular subset of probability measures having densities uniformly bounded above by a given constant plays a special role.