# Robust Estimation under the Wasserstein Distance

arXiv: https://arxiv.org/abs/2302.01237 · HTML: https://mlsanity.com/paper/2302.01237/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 33 of its 72 references that are in the map.
- First posted on arXiv 2 Feb 2023.

## Builds on (most cited first, 30 of 33)

- [Analyzing and Improving the Image Quality of StyleGAN](https://mlsanity.com/paper/1912.04958/) (2019)
- [Progressive Growing of GANs for Improved Quality, Stability, and Variation](https://mlsanity.com/paper/1710.10196/) (2017)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Optimal Transport for Domain Adaptation](https://mlsanity.com/paper/1507.00504/) (2015)
- [Wasserstein Auto-Encoders](https://mlsanity.com/paper/1711.01558/) (2017)
- [Data-driven Distributionally Robust Optimization Using the Wasserstein Metric: Performance Guarantees and Tractable Reformulations](https://mlsanity.com/paper/1505.05116/) (2015)
- [Robust Estimators in High Dimensions without the Computational Intractability](https://mlsanity.com/paper/1604.06443/) (2016)
- [Optimal transport mapping via input convex neural networks](https://mlsanity.com/paper/1908.10962/) (2019)
- [Distributionally Robust Stochastic Optimization with Wasserstein Distance](https://mlsanity.com/paper/1604.02199/) (2016)
- [Generalization and Equilibrium in Generative Adversarial Nets (GANs)](https://mlsanity.com/paper/1703.00573/) (2017)
- [Optimal Entropy-Transport problems and a new Hellinger-Kantorovich distance between positive measures](https://mlsanity.com/paper/1508.07941/) (2015)
- [Unbalanced minibatch Optimal Transport; applications to Domain Adaptation](https://mlsanity.com/paper/2103.03606/) (2021)
- [Recent Advances in Algorithmic High-Dimensional Robust Statistics](https://mlsanity.com/paper/1911.05911/) (2019)
- [Robust Optimal Transport with Applications in Generative Modeling and Domain Adaptation](https://mlsanity.com/paper/2010.05862/) (2020)
- [Statistical and Topological Properties of Sliced Probability Divergences](https://mlsanity.com/paper/2003.05783/) (2020)
- [Smooth and Sparse Optimal Transport](https://mlsanity.com/paper/1710.06276/) (2017)
- [Outlier-Robust Optimal Transport](https://mlsanity.com/paper/2012.07363/) (2020)
- [Robust and Heavy-Tailed Mean Estimation Made Simple, via Regret Minimization](https://mlsanity.com/paper/2007.15839/) (2020)
- [Resilience: A Criterion for Learning in the Presence of Arbitrary Outliers](https://mlsanity.com/paper/1703.04940/) (2017)
- [Asymptotic Guarantees for Learning Generative Models with the Sliced-Wasserstein Distance](https://mlsanity.com/paper/1906.04516/) (2019)
- [Convergence and Concentration of Empirical Measures under Wasserstein Distance in Unbounded Functional Spaces](https://mlsanity.com/paper/1804.10556/) (2018)
- [Statistical, Robustness, and Computational Guarantees for Sliced Wasserstein Distances](https://mlsanity.com/paper/2210.09160/) (2022)
- [High-Dimensional Robust Mean Estimation in Nearly-Linear Time](https://mlsanity.com/paper/1811.09380/) (2018)
- [On Projection Robust Optimal Transport: Sample Complexity and Model Misspecification](https://mlsanity.com/paper/2006.12301/) (2020)
- [Optimal Transport Based Distributionally Robust Optimization: Structural Properties and Iterative Schemes](https://mlsanity.com/paper/1810.02403/) (2018)
- [On Robust Optimal Transport: Computational Complexity and Barycenter Computation](https://mlsanity.com/paper/2102.06857/) (2021)
- [When OT meets MoM: Robust estimation of Wasserstein Distance](https://mlsanity.com/paper/2006.10325/) (2020)
- [A Framework for Wasserstein-1-Type Metrics](https://mlsanity.com/paper/1701.01945/) (2017)
- [On parameter estimation with the Wasserstein distance](https://mlsanity.com/paper/1701.05146/) (2017)

## Built on by (most cited first, 8 of 8)

- [Outlier-Robust Wasserstein DRO](https://mlsanity.com/paper/2311.05573/) (2023)
- [On Partial Optimal Transport: Revising the Infeasibility of Sinkhorn and Efficient Gradient Methods](https://mlsanity.com/paper/2312.13970/) (2023)
- [Robust Distribution Learning with Local and Global Adversarial Corruptions](https://mlsanity.com/paper/2406.06509/) (2024)
- [Data Distribution Valuation](https://mlsanity.com/paper/2410.04386/) (2024)
- [Robust Alignment via Partial Gromov-Wasserstein Distances](https://mlsanity.com/paper/2506.21507/) (2025)
- [Estimation of Stochastic Optimal Transport Maps](https://mlsanity.com/paper/2512.09499/) (2025)
- [On Robust Cross Domain Alignment](https://mlsanity.com/paper/2412.15861/) (2024)
- [Wasserstein Filtering: A Sample Selection Method for Robust Distribution Learning](https://mlsanity.com/paper/2608.13418/) (2026)

## Abstract (arXiv)

We study the problem of robust distribution estimation under the Wasserstein distance, a popular discrepancy measure between probability distributions rooted in optimal transport (OT) theory. Given n samples from an unknown distribution μ, of which varepsilon n are adversarially corrupted, we seek an estimate for μ with minimal Wasserstein error. To address this task, we draw upon two frameworks from OT and robust statistics: partial OT (POT) and minimum distance estimation (MDE). We prove new structural properties for POT and use them to show that MDE under a partial Wasserstein distance achieves the minimax-optimal robust estimation risk in many settings. Along the way, we derive a novel dual form for POT that adds a sup-norm penalty to the classic Kantorovich dual for standard OT. Since the popular Wasserstein generative adversarial network (WGAN) framework implements Wasserstein MDE via Kantorovich duality, our penalized dual enables large-scale generative modeling with contaminated datasets via an elementary modification to WGAN. Numerical experiments demonstrating the efficacy of our approach in mitigating the impact of adversarial corruptions are provided.