Expected value of the one-dimensional earth mover’s distance

Abstract

From a combinatorial point of view, we consider the earth mover's distance (EMD) associated with a metric measure space. The specific case considered is deceptively simple: Let the finite set of integers [n] = {1,. .. , n} be regarded as a metric space by restricting the usual Euclidean distance on the real numbers. The EMD is defined on ordered pairs of probability distributions on [n]. We provide an easy method to compute a generating function encoding the values of EMD in its coefficients, which is related to the Segre embedding from projective algebraic geometry. As an application we use the generating function to compute the expected value of EMD in this one-dimensional case. The EMD is then used in clustering analysis for a specific data set.

1-Wasserstein distance on the standard simplex

“…Note, our original random variable X n is equal to X 1,n . In [1] Bourn and Willenbring define the unit normalized EMD on P n as the EMD scaled by 1 n−1 and extend their results to this case. In our notation, this just amounts to taking α = 1 n−1 .…”

“…There are numerous applications of Wasserstein distances on finite sets. In [1] Bourn and Willenbring, use W 1 , which they refer to as the earth mover's distance, or EMD, following the computer science convention, to compare grade distributions -distributions on the finite set {A, A−, B+, . .…”

“…In this paper, we find a closed form for the first and second moments of X n . Additionally, we "solve" the recurrence provided by [1] to verify consistency of results. This provides an interesting link between the discrete combinatorial approach of [1] and our calculus-based approach.…”

“…In [1] a double sequence M p,q , p, q ∈ ‫ގ‬ 0 , is defined as the solution of the recursion (3). Here we "solve" this recursion.…”

1-Wasserstein Distance on the Standard Simplex

“…There are numerous applications of Wasserstein distances on finite sets. In [BW19] Bourn and Willenbring, use W 1 , which they refer to as Earth Movers Distance or EMD following computer science convention, to compare grade distributions -distributions on the finite set {A, A−, B+, . .…”