Measuring Diversity: Axioms and Challenges

TL;DR

This paper reviews existing diversity measures, proposes axioms for ideal measures, and constructs examples that satisfy these axioms but are computationally impractical, highlighting open problems in diversity quantification.

Contribution

It formulates axioms for reliable diversity measures, demonstrates the existence of measures satisfying these axioms, and identifies computational challenges.

Findings

Existing measures lack all three desirable properties

Constructed measures satisfy axioms but are NP-hard to compute

Open problem: practical diversity measures with all axioms

Abstract

This paper addresses the problem of quantifying diversity for a set of objects. First, we conduct a systematic review of existing diversity measures and explore their undesirable behavior in certain cases. Based on this review, we formulate three desirable properties (axioms) of a reliable diversity measure: monotonicity, uniqueness, and continuity. We show that none of the existing measures has all three properties and thus these measures are not suitable for quantifying diversity. Then, we construct two examples of measures that have all the desirable properties, thus proving that the list of axioms is not self-contradictory. Unfortunately, the constructed examples are too computationally expensive (NP-hard) for practical use. Thus, we pose an open problem of constructing a diversity measure that has all the listed properties and can be computed in practice or proving that all such measures are NP-hard to compute.