Non-additive measures, also known as fuzzy measures, capacities, and monotonic games, are increasingly used in different fields. Applications have been built within computer science and artificial intelligence related to e.g. decision making, image processing, machine learning for both classification, and regression. Tools for measure identification have been built. In short, as non-additive measures are more general than additive ones (i.e., than probabilities), they have better modeling capabilities allowing to model situations and problems that cannot be modelled by the latter. See e.g. the application of non-additive measures and the Choquet integral to model both Ellsberg paradox and Allais paradox. Because of that, there is an increasing need to analyze non-additive measures. The need for distances and similarities to compare them is no exception. Some work has been done for definining $f$-divergence for them. In this work we tackle the problem of definining the transport problem for non-additive measures, which has not been considered up to our knowledge up to now. Distances for pairs of probability distributions based on the optimal transport are extremely used in practical applications, and they are being studied extensively for the mathematical properties. We consider that it is necessary to provide appropriate definitions with a similar flavour, and that generalize the standard ones, for non-additive measures. We provide definitions based on the M\"obius transform, but also based on the $(\max, +)$-transform that we consider that has some advantages. We will discuss in this paper the problems that arise to define the transport problem for non-additive measures, and discuss ways to solve them. In this paper we provide the definitions of the optimal transport problem, and prove some properties.
翻译:不增加的措施,也称为模糊的措施、能力和单调游戏,越来越多地在不同领域使用。应用程序是在计算机科学和人工智能中建立的,涉及决策、图像处理、机器学习,用于分类和回归。测量识别工具已经建立。简言之,由于非增加措施比添加措施(即概率)更为一般,它们具有更好的建模能力,可以模拟无法由后者模拟的情况和问题。例如,应用非增加措施和Choquet作为Elsberg悖论和Allais悖论的模型的组成部分。因此,越来越需要分析非增加措施。需要距离和相似性来比较它们。一些非增加措施比添加措施(即,比概率)更为普通,因此,它们具有更好的建模能力,因此我们解决了非增加措施的运输问题。我们一直没有考虑过到我们的知识,现在还没有考虑过的Choquetrequet 。对于概率分布定义的距离和相似性定义是用来进行最优化分析的。我们用最优化的运算方法来解释。我们用最优化的运法来解释。我们用最优化的方法来研究最优化的运法。我们用最优化的方法来研究最优化的方法去研究。