🤖 AI Summary
This work systematically addresses statistical uncertainty in probability measures within the optimal transport framework. By extending classical optimal transport theory to the setting of random probability measures, it establishes—for the first time—the Riemannian geometric structure of the L²-type Wasserstein space, characterizing its metric and geodesics, and introduces a stochastic flow whose sample paths follow Wasserstein gradient flows. This approach unifies statistical inference and generative modeling, extends Schwartz’s posterior consistency theorem to the Wasserstein topology, and provides convergence guarantees for principled inference under random sampling. Furthermore, it offers a theoretical foundation compatible with stochastic token sampling in Transformer architectures.
📝 Abstract
Optimal transport provides an inherently geometric and highly structured framework for studying spaces of probability measures, supplying a rich theoretical toolkit for contemporary statistics, machine learning, and generative modelling. In applications, however, the measures of interest are almost never known precisely, calling for a theory of optimal transport that accounts for statistical uncertainty. We construct such a framework, lifting the classical theory to the setting of random probability measures. We introduce the $L^2$ over Wasserstein space establishing that it inherits the formal Riemannian structure of the Wasserstein space by characterising distances and geodesic geometry. The structure induces random flows with Wasserstein gradient flow sample paths, making it the natural extension of the Wasserstein space which allows for random gradient flow dynamics. We ensemble statistical convergence results of the optimal transport machinery using the empirical measure within the $L^2$ over Wasserstein framework. Moreover, in the setting of Bayesian non-parametrics, we refine Schwartz's consistency theorem to the Wasserstein topology and deduce posterior convergence of the same machinery in the $L^2$ over Wasserstein space. We demonstrate that the growing theory of random token sampling for transformer models using self-attention flow paths can be embedded into the our framework. The results provide a unified treatment of random optimal transport and its consequences for principled inference and generative modelling under the statistical uncertainty of random sampling.