Penalty functions have been a common tool in data aggregation for decades. Unfortunately, although the definition of a penalty function has evolved over the years, the use of penalty functions has been reduced to the aggregation of real numbers. However, in this ‘era of aggregation,’ the need of generalizing the current definition in order to comply with the characteristics of new types of data arises. In this paper, we bring to the attention the notion of betweenness relation and propose to replace the currently required property of quasiconvexity of a penalty function by the compatibility with a betweenness relation. Several construction methods for a penalty function are provided based on the use of a monometric. Interestingly, several prominent data aggregation methods are proved to fit into this new framework for penalty-based data aggregation.

中文翻译：

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论计量学在基于惩罚的数据聚合中的作用


几十年来，惩罚函数一直是数据聚合的常用工具。不幸的是，尽管罚函数的定义多年来一直在发展，但罚函数的使用已简化为实数的聚合。然而，在这个“聚合时代”，需要概括当前的定义，以符合新型数据的特征。在本文中，我们引起了介数关系的概念，并建议通过与介数关系的兼容性来取代目前所需的罚函数拟凸性质。基于单度量的使用，提供了几种罚函数的构造方法。有趣的是，几种著名的数据聚合方法被证明适合这种基于惩罚的数据聚合的新框架。