Metric Comparisons of Relations

Abstract

This paper defines a new pseudometric for binary relations between finite sets that measures consensus among subsets. The main results are (1) a concise restatement of this pseudometric with an intuitively appealing interpretation via a full and faithful functor from the category of relations to a category of relation multisets and (2) that the pseudometric can be bounded without an expensive search of possible mappings, based solely on the dimensions of the relations themselves. Additionally, (3) an algorithm is described to calculate this bound with time and memory complexity at worst proportional to the product of those dimensions: $\mathcal{O}(m \times n).$. The tools developed in this paper should find broad application in exploratory data analysis. We provide one such application by briefly exploring ad hoc consensus specifications for the well-known PDF file format.