On Similarity

TL;DR

This paper introduces a new principled approach for quantifying similarity between mathematical structures, generalizing scalar similarity indices to multisets, vectors, and functions, with applications to convolution and correlation.

Contribution

It develops a set of similarity indices based on Kronecker's delta, generalizes them to various mathematical objects, and interprets the Jaccard index within this framework.

Findings

Derived three similarity indices bounded between 0 and 1.

Generalized indices to multisets, vectors, and functions.

Connected similarity indices to convolution and correlation operations.

Abstract

The objective quantification of similarity between two mathematical structures constitutes a recurrent issue in science and technology. In the present work, we developed a principled approach that took the Kronecker's delta function of two scalar values as the prototypical reference for similarity quantification and then derived for more yielding indices, three of which bound between 0 and 1. Generalizations of these indices to take into account the sign of the scalar values were then presented and developed to multisets, vectors, and functions in real spaces. Several important results have been obtained, including the interpretation of the Jaccard index as a yielding implementation of the Kronecker's delta function. When generalized to real functions, the four described similarity indices become respective functionals, which can then be employed to obtain associated operations of convolution and correlation.