Representational Rényi heterogeneity

A discrete system's heterogeneity is measured by the R\'enyi heterogeneity family of indices (also known as Hill numbers or Hannah-Kay indices), whose units are known as the numbers equivalent, and whose scaling properties are consistent and intuitive. Unfortunately, numbers equivalent heterogeneity measures for non-categorical data require a priori (A) categorical partitioning and (B) pairwise distance measurement on the space of observable data. This precludes their application to problems in disciplines where categories are ill-defined or where semantically relevant features must be learned as abstractions from some data. We thus introduce representational R\'enyi heterogeneity (RRH), which transforms an observable domain onto a latent space upon which the R\'enyi heterogeneity is both tractable and semantically relevant. This method does not require a priori binning nor definition of a distance function on the observable space. Compared with existing state-of-the-art indices on a beta-mixture distribution, we show that RRH more accurately detects the number of distinct mixture components. We also show that RRH can measure heterogeneity in natural images whose semantically relevant features must be abstracted using deep generative models. We further show that RRH can uniquely capture heterogeneity caused by distinct components in mixture distributions. Our novel approach will enable measurement of heterogeneity in disciplines where a priori categorical partitions of observable data are not possible, or where semantically relevant features must be inferred using latent variable models.