Slicing Wasserstein Over Wasserstein Via Functional Optimal Transport

• URL: http://arxiv.org/abs/2509.22138v1
• Date: Fri, 26 Sep 2025 09:59:14 GMT
• Title: Slicing Wasserstein Over Wasserstein Via Functional Optimal Transport
• Authors: Moritz Piening, Robert Beinert,
• Abstract summary: Wasserstein distances define a metric between probability measures on arbitrary metric spaces.<n>Existing sliced WoW accelerations rely on parametric meta-measures or the existence of high-order moments.<n>We show that DSW minimization is equivalent to WoW minimization for discretized meta-measures.
• Score: 2.649859884914447
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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