Optimal transport for $ε$-contaminated credal sets

Michele Caprio
Proceedings of the Fourteenth International Symposium on Imprecise Probabilities: Theories and Applications, PMLR 290:33-46, 2025.

Abstract

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 $\epsilon$-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 $\epsilon$-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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v290-caprio25a, title = {Optimal transport for $ε$-contaminated credal sets}, author = {Caprio, Michele}, booktitle = {Proceedings of the Fourteenth International Symposium on Imprecise Probabilities: Theories and Applications}, pages = {33--46}, year = {2025}, editor = {Destercke, Sébastien and Erreygers, Alexander and Nendel, Max and Riedel, Frank and Troffaes, Matthias C. M.}, volume = {290}, series = {Proceedings of Machine Learning Research}, month = {15--18 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v290/main/assets/caprio25a/caprio25a.pdf}, url = {https://proceedings.mlr.press/v290/caprio25a.html}, abstract = {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 $\epsilon$-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 $\epsilon$-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.} }
Endnote
%0 Conference Paper %T Optimal transport for $ε$-contaminated credal sets %A Michele Caprio %B Proceedings of the Fourteenth International Symposium on Imprecise Probabilities: Theories and Applications %C Proceedings of Machine Learning Research %D 2025 %E Sébastien Destercke %E Alexander Erreygers %E Max Nendel %E Frank Riedel %E Matthias C. M. Troffaes %F pmlr-v290-caprio25a %I PMLR %P 33--46 %U https://proceedings.mlr.press/v290/caprio25a.html %V 290 %X 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 $\epsilon$-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 $\epsilon$-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.
APA
Caprio, M.. (2025). Optimal transport for $ε$-contaminated credal sets. Proceedings of the Fourteenth International Symposium on Imprecise Probabilities: Theories and Applications, in Proceedings of Machine Learning Research 290:33-46 Available from https://proceedings.mlr.press/v290/caprio25a.html.

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