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[2509.22138] Slicing Wasserstein Over Wasserstein Via Functional Optimal Transport
Summary
This article presents a novel approach to the Wasserstein over Wasserstein (WoW) distance by introducing the double-sliced Wasserstein (DSW) metric, which enhances computational efficiency while maintaining accuracy in comparing datasets and distributions.
Why It Matters
The proposed DSW metric offers a significant advancement in the field of optimal transport by addressing the computational challenges associated with the WoW distance. This innovation is crucial for applications in machine learning, particularly in data analysis and shape comparison, where efficient metrics are essential for scalability and performance.
Key Takeaways
- The DSW metric provides a scalable alternative to the computationally intensive WoW distance.
- It leverages the isometry between 1D Wasserstein space and quantile functions to enhance stability.
- Numerical experiments validate DSW's effectiveness in various applications, including images and shapes.
- This framework can be applied to arbitrary Banach spaces, broadening its applicability.
- The approach avoids reliance on high-order moments, reducing numerical instability.
Computer Science > Machine Learning arXiv:2509.22138 (cs) [Submitted on 26 Sep 2025 (v1), last revised 19 Feb 2026 (this version, v2)] Title:Slicing Wasserstein Over Wasserstein Via Functional Optimal Transport Authors:Moritz Piening, Robert Beinert View a PDF of the paper titled Slicing Wasserstein Over Wasserstein Via Functional Optimal Transport, by Moritz Piening and Robert Beinert View PDF HTML (experimental) 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 ...
