Fuzzy Integral = Contextual Linear Order Statistic

TL;DR

This paper demonstrates that the fuzzy integral can be represented by contextual linear order statistics, enabling scalable, interpretable models with improved measure acquisition, validated through synthetic and real-world data.

Contribution

It introduces a novel representation of the fuzzy integral using contextual linear order statistics, enhancing scalability and interpretability.

Findings

Representation of fuzzy integral via LOS is effective.

Clustering improves measure sampling and model generalization.

Validated on synthetic and real-world datasets.

Abstract

The fuzzy integral is a powerful parametric nonlin-ear function with utility in a wide range of applications, from information fusion to classification, regression, decision making,interpolation, metrics, morphology, and beyond. While the fuzzy integral is in general a nonlinear operator, herein we show that it can be represented by a set of contextual linear order statistics(LOS). These operators can be obtained via sampling the fuzzy measure and clustering is used to produce a partitioning of the underlying space of linear convex sums. Benefits of our approach include scalability, improved integral/measure acquisition, generalizability, and explainable/interpretable models. Our methods are both demonstrated on controlled synthetic experiments, and also analyzed and validated with real-world benchmark data sets.