Representational R\'enyi heterogeneity

TL;DR

This paper introduces representational R\'enyi heterogeneity (RRH), a novel method for measuring heterogeneity in data by transforming observable data into a latent space, avoiding the need for predefined categories or distance metrics.

Contribution

The paper proposes RRH, a new heterogeneity measure that operates on learned latent representations, extending applicability to non-categorical and complex data.

Findings

RRH generalizes existing biodiversity and economic indices.

RRH responds more appropriately to changes in mixture components.

RRH effectively measures heterogeneity in natural images using deep neural network representations.

Abstract

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.