# L2 over Wasserstein: Statistical Analysis for Optimal Transport

arXiv: https://arxiv.org/abs/2605.21365 · HTML: https://mlsanity.com/paper/2605.21365/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 49 references that are in the map.
- First posted on arXiv 20 May 2026.

## Builds on (most cited first, 13 of 13)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Sinkformers: Transformers with Doubly Stochastic Attention](https://mlsanity.com/paper/2110.11773/) (2021)
- [Statistical optimal transport](https://mlsanity.com/paper/2407.18163/) (2024)
- [Approximate Bayesian computation with the Wasserstein distance](https://mlsanity.com/paper/1905.03747/) (2019)
- [The Mean-Field Dynamics of Transformers](https://mlsanity.com/paper/2512.01868/) (2025)
- [A multiscale analysis of mean-field transformers in the moderate interaction regime](https://mlsanity.com/paper/2509.25040/) (2025)
- [Approximate Bayesian Computation with the Sliced-Wasserstein Distance](https://mlsanity.com/paper/1910.12815/) (2019)
- [On the Structure of Stationary Solutions to McKean-Vlasov Equations with Applications to Noisy Transformers](https://mlsanity.com/paper/2510.20094/) (2025)
- [Stochastic Scaling Limits and Synchronization by Noise in Deep Transformer Models](https://mlsanity.com/paper/2604.26898/) (2026)
- [Wasserstein Distributionally Robust Bayesian Optimization with Continuous Context](https://mlsanity.com/paper/2503.20341/) (2025)
- [Bayesian Inference for Optimal Transport with Stochastic Cost](https://mlsanity.com/paper/2010.09327/) (2020)
- [An Optimal Transport-Based Generative Model for Bayesian Posterior Sampling](https://mlsanity.com/paper/2504.08214/) (2025)

## Built on by (most cited first, 1 of 1)

- [Conditional Random Ordered Transport Spaces](https://mlsanity.com/paper/2606.08113/) (2026)

## Abstract (arXiv)

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 L2 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 L2 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 L2 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.