A note on the triangle inequality for the Jaccard distance

Two simple proofs of the triangle inequality for the Jaccard distance in terms of nonnegative, monotone, submodular functions are given and discussed. The Jaccard index [8] is a classical similarity measure on sets with a lot of practical applications in information retrieval, data mining, machine learning, and many more (cf., e.g., [7]). Measuring the relative size of the overlap of two finite sets A and B, the Jaccard index J and the associated Jaccard distance Jδ are formally defined as: J(A,B) =def |A ∩B| |A ∪B| , Jδ(A,B) =def 1− J(A,B) = 1− |A ∩B| |A ∪B| = |A△B| |A ∪B| where J(∅, ∅) =def 1. The Jaccard distance Jδ is known to fulfill all properties of a metric, most notably, the triangle inequality—a fact that has been observed many times, e.g., via metric transforms [12, 13, 4], embeddings in vector spaces (e.g., [15, 11, 4]), minwise independent permutations [1], or sometimes cumbersome arithmetics [10, 3]. A very simple, elementary proof of the triangle inequality was given in [5] using an appropriate partitioning of sets. Here, we give two more simple, direct proofs of the triangle inequality. One proof comes without any set difference or disjointness of sets. It is based only on the fundamental equation |A ∪ B| + |A ∩ B| = |A| + |B|. As such, the proof is generic and leads to (sub)modular versions of the Jaccard distance (as defined below). The second proof unfolds a subtle difference between the two possible versions. Though the original motivation was to give a proof of the triangle inequality as simple as possible, the link with submodular functions is interesting in itself (as also recently suggested in [6]). Let X be a finite, non-empty ground set. A set function f : P(X) → R is said to be submodular on X if f(A∪B)+f(A∩B) ≤ f(A)+f(B) for all A,B ⊆ X. If all inequalities are equations then f is called modular on X. It is known that f is submodular on X if and only if the following condition holds (cf., e.g., [14]): f(A ∪ {x}) − f(A) ≥ f(B ∪ {x}) − f(B) for all A ⊆ B ⊆ X, x ∈ B (1) A set function f is monotone if f(A) ≤ f(B) for all A ⊆ B ⊆ X; f is nonnegative if f(A) ≥ 0 for all A ⊆ X. Each nonnegative, monotone, modular function f on X can be