Vertical Sliced Wasserstein Distances for Meta-Measures

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* (VSW) distance, a scalable metric on $\mathcal{P}(\mathcal{P}(\Omega ))$. Rather than slicing along the transport geometry of $\mathcal{P}(\Omega )$, VSW projects each inner measure using continuous test functions and compares the resulting one-dimensional distributions with one-dimensional Wasserstein distance. We prove that VSW is a valid metric on $\mathcal{P}(\mathcal{P}(\Omega ))$, metrizes weak convergence, and admits dimension-independent statistical convergence guarantees. Empirically, 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. VSW provides a simple and scalable alternative for comparing and optimizing distributions of distributions.