Vertical Sliced Wasserstein Distances for Meta-Measures

Abstract:Many modern learning problems require comparing probability distributions whose individual observations are themselves probability measures, naturally giving rise to meta-measures in $\mathcal{P}(\mathcal{P}(\Omega))$. However, optimal transport distances on this space can be prohibitively expensive to evaluate. We introduce the vertical sliced Wasserstein ($\mathrm{VSW}$) distance, a scalable metric on $\mathcal{P}(\mathcal{P}(\Omega))$. Rather than slicing along the transport geometry of $\mathcal{P}(\Omega)$, $\mathrm{VSW}$ projects each inner measure using continuous test functions and compares the resulting one-dimensional distributions with one-dimensional Wasserstein distance. We prove that $\mathrm{VSW}$ is a valid metric on $\mathcal{P}(\mathcal{P}(\Omega))$, metrizes weak convergence, and admits dimension-independent statistical convergence guarantees. Empirically, $\mathrm{VSW}$ provides high accuracy and efficiency compared with existing distances for meta-measures. We further derive a practical particle gradient for the empirical $\mathrm{VSW}_2^2$ objective and demonstrate its use for gradient-based transport of meta-measures. $\mathrm{VSW}$ provides a simple and scalable alternative for comparing and optimizing distributions of distributions.

Machine Learning (cs.LG)

arXiv:2610.20743 [cs.LG]

(or arXiv:2610.20743v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2610.20743 Focus to learn more arXiv-issued DOI via DataCite

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