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 P(P(Ω)). 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 P(P(Ω)). Rather than slicing along the transport geometry of P(Ω), 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 P(P(Ω)), 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 VSW 2 squared 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.