Optimal Transport for $\epsilon$-Contaminated Credal Sets: To the Memory of Sayan Mukherjee

TL;DR

This paper extends classical optimal transport problems to lower probabilities within $\,\epsilon$-contaminated credal sets, establishing conditions for equivalence and existence of solutions, with implications for machine learning and AI.

Contribution

It introduces generalized Monge and Kantorovich problems with lower probabilities and explores their properties, including conditions for their equivalence and solution existence.

Findings

Classical and generalized problems coincide under certain conditions.

Sufficient conditions for the existence of Kantorovich's optimal plan.

Lower probability versions may not always coincide for $\,\epsilon$-contaminations.

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.