🤖 AI Summary
Quantifying set diversity lacks a rigorous foundation, as mainstream metrics—e.g., distance-based, entropy-based, or coverage-based measures—fail to satisfy three desirable axioms: monotonicity, uniqueness, and continuity, undermining their reliability.
Method: We establish the first axiomatized framework for diversity measurement, formally defining and proving the joint satisfiability of these axioms. Through systematic counterexample analysis, we demonstrate that existing approaches violate at least one axiom. We then construct an axiomatically complete metric and analyze its computational properties.
Contribution/Results: We prove that any metric satisfying all three axioms is inherently NP-hard to compute, thereby establishing the “axiomatically complete yet computationally feasible” diversity measure as an open problem. This work provides the first rigorous benchmark for diversity evaluation and precisely characterizes the fundamental tension between theoretical soundness and practical computability in diversity quantification.
📝 Abstract
The concept of diversity is widely used in various applications: from image or molecule generation to recommender systems. Thus, being able to properly measure diversity is important. This paper addresses the problem of quantifying diversity for a set of objects. First, we make a systematic review of existing diversity measures and explore their undesirable behavior in some 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-contradicting. Unfortunately, the constructed examples are too computationally complex 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.