Title: Slicing Wasserstein over Wasserstein via Functional Optimal Transport
Primary area: other topics in machine learning (i.e., none of the above)
Keywords: Optimal Transport, Sliced Wasserstein, Dataset Distances, Wasserstein, Function Spaces, Infinite-dimensional

Abstract: Wasserstein distances define a metric between probability measures on arbitrary metric spaces, 
including *meta-measures* (measures over measures).
The resulting *Wasserstein over Wasserstein* (WoW) distance is a powerful, but computationally costly tool for comparing datasets or distributions over images and shapes.
Existing sliced WoW accelerations rely on parametric meta-measures or the existence of high-order moments, leading to numerical instability. As an alternative, we propose to leverage the isometry between the 1d Wasserstein space and the quantile functions in the function space $L_2([0,1])$.
For this purpose, we introduce a general sliced Wasserstein framework for arbitrary Banach spaces. 
Due to the 1d Wasserstein isometry, 
this framework defines a sliced distance between 1d meta-measures via infinite-dimensional $L_2$-projections, 
parametrized by Gaussian processes. 
Combining this 1d construction with classical integration over the Euclidean unit sphere yields the *double-sliced Wasserstein* (DSW) metric for general meta-measures. We show that DSW minimization is equivalent to WoW minimization for discretized meta-measures, while avoiding unstable higher-order moments and  computational savings. Numerical experiments on datasets, shapes, and images validate DSW as a scalable substitute for the WoW distance.

TL;DR: We propose double-sliced Wasserstein: a scalable and stable optimal transport metric between measures over measures.