# Optimal Transport for ε-Contaminated Credal Sets: To the Memory of Sayan Mukherjee

arXiv: https://arxiv.org/abs/2410.03267 · HTML: https://mlsanity.com/paper/2410.03267/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 69 references that are in the map.
- First posted on arXiv 4 Oct 2024.

## Builds on (most cited first, 19 of 19)

- [Aleatoric and Epistemic Uncertainty in Machine Learning: An Introduction to Concepts and Methods](https://mlsanity.com/paper/1910.09457/) (2019)
- [Data-driven Distributionally Robust Optimization Using the Wasserstein Metric: Performance Guarantees and Tractable Reformulations](https://mlsanity.com/paper/1505.05116/) (2015)
- [On Wasserstein Two Sample Testing and Related Families of Nonparametric Tests](https://mlsanity.com/paper/1509.02237/) (2015)
- [Finite-Sample Guarantees for Wasserstein Distributionally Robust Optimization: Breaking the Curse of Dimensionality](https://mlsanity.com/paper/2009.04382/) (2020)
- [Credal Bayesian Deep Learning](https://mlsanity.com/paper/2302.09656/) (2023)
- [Is the Volume of a Credal Set a Good Measure for Epistemic Uncertainty?](https://mlsanity.com/paper/2306.09586/) (2023)
- [Optimal Testing for Properties of Distributions](https://mlsanity.com/paper/1507.05952/) (2015)
- [Second-Order Uncertainty Quantification: A Distance-Based Approach](https://mlsanity.com/paper/2312.00995/) (2023)
- [Conformal prediction under ambiguous ground truth](https://mlsanity.com/paper/2307.09302/) (2023)
- [Credal Learning Theory](https://mlsanity.com/paper/2402.00957/) (2024)
- [Robust Hypothesis Testing Using Wasserstein Uncertainty Sets](https://mlsanity.com/paper/1805.10611/) (2018)
- [Optimal transport natural gradient for statistical manifolds with continuous sample space](https://mlsanity.com/paper/1805.08380/) (2018)
- [Judicious Judgment Meets Unsettling Updating: Dilation, Sure Loss, and Simpson's Paradox](https://mlsanity.com/paper/1712.08946/) (2017)
- [On the Calibration of Probabilistic Classifier Sets](https://mlsanity.com/paper/2205.10082/) (2022)
- [DC4L: Distribution Shift Recovery via Data-Driven Control for Deep Learning Models](https://mlsanity.com/paper/2302.10341/) (2023)
- [On the Robustness of Kernel Goodness-of-Fit Tests](https://mlsanity.com/paper/2408.05854/) (2024)
- [Discrete Optimal Transport with Independent Marginals is #P-Hard](https://mlsanity.com/paper/2203.01161/) (2022)
- [Imprecise Markov Semigroups and their Ergodicity](https://mlsanity.com/paper/2405.00081/) (2024)
- [The transport problem for non-additive measures](https://mlsanity.com/paper/2211.12150/) (2022)

## Abstract (arXiv)

We present generalized versions of Monge's and Kantorovich's optimal transport problems with the probabilities being transported replaced by lower probabilities. We show that, when the lower probabilities are the lower envelopes of ε-contaminated sets, then our version of Monge's, and a restricted version of our Kantorovich's problems, coincide with their respective classical versions. We also give sufficient conditions for the existence of our version of Kantorovich's optimal plan, and for the two problems to be equivalent. As a byproduct, we show that for ε-contaminations the lower probability versions of Monge's and Kantorovich's optimal transport problems need not coincide. The applications of our results to Machine Learning and Artificial Intelligence are also discussed.