Earth Mover's Distance as a Similarity Measure for Linear Order Statistics and Fuzzy Integrals

This paper focuses on a powerful nonlinear aggregation function, the Choquet integral (ChI). Specifically, we focus on situations where the parameters of the ChI are learned from data. For N inputs, the ChI breaks down into N! underlying linear convex sums (LCSs) with 2N shared variables. Typically, these LCSs are reducible into a drastically smaller number of linear order statistics (LOSs). In the spirit of explainable AI (XAI), our goal is to discover the minimal underlying operator structure of a learned ChI to be conveyed to its users. The challenge is, there does not appear to be widespread research or agreement regarding how to compute similarity within and between measures or integrals. In this paper, we explore the earth mover's distance (EMD), a parametric cross-bin measure, to capture semantic relatedness between LOSs. EMD is used to measure dissimilarity between integrals. In the case of a single ChI, underlying aggregation operator structure is discovered via EMD and clustering. A combination of synthetic and real-world experiments are provided to demonstrate interpretability and reduction of complexity.