Mathematical properties of soft cardinality: Enhancing Jaccard, Dice and cosine similarity measures with element-wise distance

The soft cardinality function generalizes the concept of counting measure of the classic cardinality of sets. This function provides an intuitive measure of the amount of elements in a collection (i.e. a set or a bag) exploiting the similarities among them. Although soft cardinality was first proposed in an ad-hoc way, it has been successfully used in various tasks in the field of natural language processing. In this paper, a formal definition of soft cardinality is proposed together with an analysis of its boundaries, monotonicity property and a method for constructing similarity functions. Additionally, an empirical evaluation of the model was carried out using synthetic data.