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.

翻译：多样性概念广泛应用于各类应用中：从图像或分子生成到推荐系统。因此，能够准确度量多样性至关重要。本文致力于解决对象集合的多样性量化问题。首先，我们对现有多样性度量方法进行了系统性梳理，并探讨了它们在某些情况下的不良表现。基于此综述，我们提出了可靠多样性度量应具备的三个理想性质（公理）：单调性、唯一性和连续性。我们证明现有度量方法均不同时具备这三条性质，因此不适合用于多样性量化。随后，我们构建了两个满足所有理想性质的度量示例，从而证明该公理体系不存在自相矛盾。遗憾的是，所构建的示例计算复杂度过高而难以实际应用，因此我们提出一个开放性问题：如何构造一个满足所有列举性质且能在实践中计算的多样性度量方法。