On Similarity
This work addresses a foundational issue in science and technology for researchers and practitioners needing similarity measures, but it appears incremental as it builds on existing concepts like the Kronecker's delta.
The authors tackled the problem of quantifying similarity between mathematical structures by developing a principled approach based on the Kronecker's delta function, resulting in new indices that generalize to multisets, vectors, and functions, including an interpretation of the Jaccard index.
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.