# Inference for Empirical Wasserstein Distances on Finite Spaces

arXiv: https://arxiv.org/abs/1610.03287 · HTML: https://mlsanity.com/paper/1610.03287/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Oct 2016.

## Built on by (most cited first, 21 of 21)

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- [Likelihood estimation of sparse topic distributions in topic models and its applications to Wasserstein document distance calculations](https://mlsanity.com/paper/2107.05766/) (2021)
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- [A Specialized Semismooth Newton Method for Kernel-Based Optimal Transport](https://mlsanity.com/paper/2310.14087/) (2023)
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- [Statistical Inference for Optimal Transport Maps: Recent Advances and Perspectives](https://mlsanity.com/paper/2506.19025/) (2025)
- [Estimation of Stationary Optimal Transport Plans](https://mlsanity.com/paper/2107.11858/) (2021)
- [Tighter Learning Guarantees on Digital Computers via Concentration of Measure on Finite Spaces](https://mlsanity.com/paper/2402.05576/) (2024)
- [Statistical Guarantees for Reasoning Probes on Looped Boolean Circuits](https://mlsanity.com/paper/2602.03970/) (2026)
- [Statistical Inference for Adversarial Training: Central Limit Theorems via Optimal Transport](https://mlsanity.com/paper/2609.22240/) (2026)

## Abstract (arXiv)

The Wasserstein distance is an attractive tool for data analysis but statistical inference is hindered by the lack of distributional limits. To overcome this obstacle, for probability measures supported on finitely many points, we derive the asymptotic distribution of empirical Wasserstein distances as the optimal value of a linear program with random objective function. This facilitates statistical inference (e.g. confidence intervals for sample based Wasserstein distances) in large generality. Our proof is based on directional Hadamard differentiability. Failure of the classical bootstrap and alternatives are discussed. The utility of the distributional results is illustrated on two data sets.