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

arXiv 2410.03267 ↗First posted 4 Oct 2024stat.ML cs.LG math.PRcited by 0

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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.

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References · 19

  1. 2019Aleatoric and Epistemic Uncertainty in Machine Learning: An Introduction to Concepts and Methodscited by 903
  2. 2015Data-driven Distributionally Robust Optimization Using the Wasserstein Metric: Performance Guarantees and Tractable Reformulationscited by 285
  3. 2015On Wasserstein Two Sample Testing and Related Families of Nonparametric Testscited by 165
  4. 2020Finite-Sample Guarantees for Wasserstein Distributionally Robust Optimization: Breaking the Curse of Dimensionalitycited by 49
  5. 2023Credal Bayesian Deep Learningcited by 45
  6. 2023Is the Volume of a Credal Set a Good Measure for Epistemic Uncertainty?cited by 40
  7. 2015Optimal Testing for Properties of Distributionscited by 36
  8. 2023Second-Order Uncertainty Quantification: A Distance-Based Approachcited by 35
  9. 2023Conformal prediction under ambiguous ground truthcited by 31
  10. 2024Credal Learning Theorycited by 30
  11. 2018Robust Hypothesis Testing Using Wasserstein Uncertainty Setscited by 28
  12. 2018Optimal transport natural gradient for statistical manifolds with continuous sample spacecited by 14
7 more
  1. 2017Judicious Judgment Meets Unsettling Updating: Dilation, Sure Loss, and Simpson's Paradoxcited by 8
  2. 2022On the Calibration of Probabilistic Classifier Setscited by 8
  3. 2023DC4L: Distribution Shift Recovery via Data-Driven Control for Deep Learning Modelscited by 7
  4. 2024On the Robustness of Kernel Goodness-of-Fit Testscited by 7
  5. 2022Discrete Optimal Transport with Independent Marginals is #P-Hardcited by 3
  6. 2024Imprecise Markov Semigroups and their Ergodicitycited by 3
  7. 2022The transport problem for non-additive measurescited by 1

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