# Outlier-Robust Optimal Transport: Duality, Structure, and Statistical Analysis

arXiv: https://arxiv.org/abs/2111.01361 · HTML: https://mlsanity.com/paper/2111.01361/

## Key facts

- Cited by 27 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 54 references that are in the map.
- First posted on arXiv 2 Nov 2021.

## Builds on (most cited first, 20 of 20)

- [Analyzing and Improving the Image Quality of StyleGAN](https://mlsanity.com/paper/1912.04958/) (2019)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [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)
- [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)
- [Robust Optimal Transport with Applications in Generative Modeling and Domain Adaptation](https://mlsanity.com/paper/2010.05862/) (2020)
- [Robust Covariance and Scatter Matrix Estimation under Huber's Contamination Model](https://mlsanity.com/paper/1506.00691/) (2015)
- [Outlier-Robust Optimal Transport](https://mlsanity.com/paper/2012.07363/) (2020)
- [Generalized Resilience and Robust Statistics](https://mlsanity.com/paper/1909.08755/) (2019)
- [Optimal Transport Based Distributionally Robust Optimization: Structural Properties and Iterative Schemes](https://mlsanity.com/paper/1810.02403/) (2018)
- [When OT meets MoM: Robust estimation of Wasserstein Distance](https://mlsanity.com/paper/2006.10325/) (2020)
- [Attribute Privacy: Framework and Mechanisms](https://mlsanity.com/paper/2009.04013/) (2020)
- [Pufferfish Privacy Mechanisms for Correlated Data](https://mlsanity.com/paper/1603.03977/) (2016)
- [A Framework for Wasserstein-1-Type Metrics](https://mlsanity.com/paper/1701.01945/) (2017)
- [Fast block-coordinate Frank-Wolfe algorithm for semi-relaxed optimal transport](https://mlsanity.com/paper/2103.05857/) (2021)

## Built on by (most cited first, 27 of 27)

- [Statistical, Robustness, and Computational Guarantees for Sliced Wasserstein Distances](https://mlsanity.com/paper/2210.09160/) (2022)
- [Outlier-Robust Wasserstein DRO](https://mlsanity.com/paper/2311.05573/) (2023)
- [Extremal Domain Translation with Neural Optimal Transport](https://mlsanity.com/paper/2301.12874/) (2023)
- [Scalable Unbalanced Sobolev Transport for Measures on a Graph](https://mlsanity.com/paper/2302.12498/) (2023)
- [Quantum Pufferfish Privacy: A Flexible Privacy Framework for Quantum Systems](https://mlsanity.com/paper/2306.13054/) (2023)
- [Generalized Sobolev Transport for Probability Measures on a Graph](https://mlsanity.com/paper/2402.04516/) (2024)
- [Robust Barycenter Estimation using Semi-Unbalanced Neural Optimal Transport](https://mlsanity.com/paper/2410.03974/) (2024)
- [Optimal Transport for Measures with Noisy Tree Metric](https://mlsanity.com/paper/2310.13653/) (2023)
- [A New Robust Partial p-Wasserstein-Based Metric for Comparing Distributions](https://mlsanity.com/paper/2405.03664/) (2024)
- [Tree-Sliced Wasserstein Distance: A Geometric Perspective](https://mlsanity.com/paper/2406.13725/) (2024)
- [Addressing GAN Training Instabilities via Tunable Classification Losses](https://mlsanity.com/paper/2310.18291/) (2023)
- [Data Distribution Valuation](https://mlsanity.com/paper/2410.04386/) (2024)
- [Distance-Based Tree-Sliced Wasserstein Distance](https://mlsanity.com/paper/2503.11050/) (2025)
- [Approximate Algorithms For k-Sparse Wasserstein Barycenter With Outliers](https://mlsanity.com/paper/2404.13401/) (2024)
- [Tree-Sliced Wasserstein Distance with Nonlinear Projection](https://mlsanity.com/paper/2505.00968/) (2025)
- [Inference via robust optimal transportation: theory and methods](https://mlsanity.com/paper/2301.06297/) (2023)
- [(αD,αG)-GANs: Addressing GAN Training Instabilities via Dual Objectives](https://mlsanity.com/paper/2302.14320/) (2023)
- [On the Convergence of Semi-Relaxed Sinkhorn with Marginal Constraint and OT Distance Gaps](https://mlsanity.com/paper/2205.13846/) (2022)
- [On Robust Wasserstein Barycenter: The Model and Algorithm](https://mlsanity.com/paper/2312.15762/) (2023)
- [Robust Alignment via Partial Gromov-Wasserstein Distances](https://mlsanity.com/paper/2506.21507/) (2025)
- [Distributionally Robust Optimization with Adversarial Data Contamination](https://mlsanity.com/paper/2507.10718/) (2025)
- [On making optimal transport robust to all outliers](https://mlsanity.com/paper/2206.11988/) (2022)
- [On the Shape of Brainscores for Large Language Models (LLMs)](https://mlsanity.com/paper/2405.06725/) (2024)
- [Partial Distribution Alignment via Adaptive Optimal Transport](https://mlsanity.com/paper/2503.05087/) (2025)
- [E-ROBOT: a dimension-free method for robust statistics and machine learning via Schr"odinger bridge](https://mlsanity.com/paper/2509.11532/) (2025)
- [The Observable Wasserstein Distance](https://mlsanity.com/paper/2605.09916/) (2026)
- [Partial Optimal Transport on the Circle for All Transported Masses in O(N log N)](https://mlsanity.com/paper/2608.23910/) (2026)

## Abstract (arXiv)

The Wasserstein distance, rooted in optimal transport (OT) theory, is a popular discrepancy measure between probability distributions with various applications to statistics and machine learning. Despite their rich structure and demonstrated utility, Wasserstein distances are sensitive to outliers in the considered distributions, which hinders applicability in practice. We propose a new outlier-robust Wasserstein distance mathsfWp^varepsilon which allows for varepsilon outlier mass to be removed from each contaminated distribution. Under standard moment assumptions, mathsfWp^varepsilon is shown to achieve strong robust estimation guarantees under the Huber varepsilon-contamination model. Our formulation of this robust distance amounts to a highly regular optimization problem that lends itself better for analysis compared to previously considered frameworks. Leveraging this, we conduct a thorough theoretical study of mathsfWp^varepsilon, encompassing robustness guarantees, characterization of optimal perturbations, regularity, duality, and statistical estimation. In particular, by decoupling the optimization variables, we arrive at a simple dual form for mathsfWp^varepsilon that can be implemented via an elementary modification to standard, duality-based OT solvers. We illustrate the virtues of our framework via applications to generative modeling with contaminated datasets.