On the role of monometrics in penalty-based data aggregation

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.