将最优传输拓展到ε-污染模糊集，揭示其与经典理论的联系与差异。
Optimal Transport for $ε$-Contaminated Credal Sets: To the Memory of Sayan Mukherjee
- 用下概率替代传统概率，构建模糊集上的最优传输模型
- 在ε-污染情形下，部分版本与经典最优传输等价
- 为机器学习中的不确定性建模提供理论支持
我们提出了Monge和Kantorovich最优传输问题的广义版本，其中被传输的概率被下概率所取代。当这些下概率是ε-污染集的下包络时，我们的Monge问题以及受限版Kantorovich问题与经典的对应版本一致。我们还给出了Kantorovich最优计划存在性的充分条件，以及两个问题等价的条件。作为副产品，我们发现对于ε-污染情形，Monge和Kantorovich的下概率版本未必等价。我们的结果在机器学习与人工智能中的应用也被讨论。
原文摘要 · Abstract (English)
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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