English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In this note we investigate properties of metric projection operators ontoclosed and geodesically convex proper subsets of Wasserstein spaces$(\mathcal{P}_p(\mathbf{R}^d),W_p).$ In our study we focus on the particularsubset of probability measures having densities uniformly bounded by a givenconstant. When $d=1$, $(\mathcal{P}_2(\mathbf{R}),W_2)$ is isometricallyisomorphic to a flat space with a Hilbertian structure, and so thecorresponding projection operators are nonexpansive. We prove a general "weaknonexpansiveness" property in arbitrary dimension which provides (among otherthings) a direct proof of nonexpansiveness when $d=1$. When $d>1$, the space$(\mathcal{P}_2(\mathbf{R}^d),W_2)$ is non-negatively curved in the sense ofAlexandrov. So, the question of nonexpansiveness of projection operators on$(\mathcal{P}_p(\mathbf{R}^d),W_p)$ is more subtle. We show the failure of thisproperty for $p\in (1,p(d))$ and we give a quantitative asymptotic estimate on$p(d)>1$ as $d \to \infty$. This result provides a heuristic argument for thenon-negative curvature of $(\mathcal{P}_p(\mathbf{R}^d),W_p)$ when $p\in(1,p(d))$. Further geometric properties of independent interest are alsodiscussed.

On nonexpansiveness of metric projection operators on Wasserstein spaces