[Submitted on 14 Oct 2026]
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.
Submission history
From: Elaheh Akbari [view email]
[v1] Wed, 14 Oct 2026 18:03:09 UTC (8,802 KB)
