  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 {the numbers equivalent}. Unfortunately, numbers equivalent
heterogeneity measures for non-categorical data require {a priori} (A)
categorical partitioning and (B) pairwise distance measurement on the
observable data space, thereby precluding application to problems with
ill-defined categories 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 requires neither {a priori} binning nor definition of a
distance function on the observable space. We show that RRH can generalize
existing biodiversity and economic equality indices. Compared with existing
indices on a beta-mixture distribution, we show that RRH responds more
appropriately to changes in mixture component separation and weighting.
Finally, we demonstrate the measurement of RRH in a set of natural images, with
respect to abstract representations learned by a deep neural network. The RRH
approach will further enable heterogeneity measurement in disciplines whose
data do not easily conform to the assumptions of existing indices.
