Slicing Wasserstein Over Wasserstein Via Functional Optimal Transport
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| author | Piening, Moritz Beinert, Robert |
| author_facet | Piening, Moritz Beinert, Robert |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_22138 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Slicing Wasserstein Over Wasserstein Via Functional Optimal Transport Piening, Moritz Beinert, Robert Machine Learning Metric Geometry Optimization and Control 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. |
| title | Slicing Wasserstein Over Wasserstein Via Functional Optimal Transport |
| topic | Machine Learning Metric Geometry Optimization and Control |
| url | https://arxiv.org/abs/2509.22138 |