# Statistical Optimal Transport posed as Learning Kernel Embedding

arXiv: https://arxiv.org/abs/2002.03179 · HTML: https://mlsanity.com/paper/2002.03179/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Feb 2020.

## Built on by (most cited first, 7 of 7)

- [Plugin Estimation of Smooth Optimal Transport Maps](https://mlsanity.com/paper/2107.12364/) (2021)
- [Riemannian block SPD coupling manifold and its application to optimal transport](https://mlsanity.com/paper/2201.12933/) (2022)
- [Submodular Framework for Structured-Sparse Optimal Transport](https://mlsanity.com/paper/2406.04914/) (2024)
- [SeTformer is What You Need for Vision and Language](https://mlsanity.com/paper/2401.03540/) (2024)
- [A Riemannian Approach to Ground Metric Learning for Optimal Transport](https://mlsanity.com/paper/2409.10085/) (2024)
- [Max-sliced Wasserstein concentration and uniform ratio bounds of empirical measures on RKHS](https://mlsanity.com/paper/2405.13153/) (2024)
- [A Riemannian Approach to Low-Rank Optimal Transport](https://mlsanity.com/paper/2606.12120/) (2026)

## Abstract (arXiv)

The objective in statistical Optimal Transport (OT) is to consistently estimate the optimal transport plan/map solely using samples from the given source and target marginal distributions. This work takes the novel approach of posing statistical OT as that of learning the transport plan's kernel mean embedding from sample based estimates of marginal embeddings. The proposed estimator controls overfitting by employing maximum mean discrepancy based regularization, which is complementary to φ-divergence (entropy) based regularization popularly employed in existing estimators. A key result is that, under very mild conditions, ε-optimal recovery of the transport plan as well as the Barycentric-projection based transport map is possible with a sample complexity that is completely dimension-free. Moreover, the implicit smoothing in the kernel mean embeddings enables out-of-sample estimation. An appropriate representer theorem is proved leading to a kernelized convex formulation for the estimator, which can then be potentially used to perform OT even in non-standard domains. Empirical results illustrate the efficacy of the proposed approach.