Sliced Wasserstein distance between probability measures on Hilbert spaces

TL;DR

This paper extends the sliced Wasserstein distance to infinite-dimensional Hilbert spaces, exploring its properties, relation to measure convergence, and empirical approximation methods.

Contribution

It introduces a novel definition of sliced Wasserstein distance in Hilbert spaces and analyzes its theoretical properties and approximation techniques.

Findings

Established the relation between sliced Wasserstein distance and narrow convergence.

Derived methods for empirical measure approximation.

Extended the concept of sliced Wasserstein to infinite-dimensional settings.

Abstract

The sliced Wasserstein distance as well as its variants have been widely considered in comparing probability measures defined on $\mathbb R^d$. Here we derive the notion of sliced Wasserstein distance for measures on an infinite dimensional separable Hilbert spaces, depict the relation between sliced Wasserstein distance and narrow convergence of measures and quantize the approximation via empirical measures.