English

$L^2$ over Wasserstein: Statistical Analysis for Optimal Transport

Statistics Theory 2026-05-21 v1 Machine Learning Statistics Theory

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 L2L^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 L2L^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 L2L^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.

Keywords

Cite

@article{arxiv.2605.21365,
  title  = {$L^2$ over Wasserstein: Statistical Analysis for Optimal Transport},
  author = {Riccardo Passeggeri and Rohan M. Shenoy and Pengcheng Ye},
  journal= {arXiv preprint arXiv:2605.21365},
  year   = {2026}
}

Comments

49 pages. Comments are welcome

R2 v1 2026-07-22T07:24:21.362Z