  Zipf's law predicts a power-law relationship between word rank and frequency
in language communication systems and has been widely reported in a variety of
natural language processing applications. However, the emergence of natural
language is often modeled as a function of bias between speaker and listener
interests, which lacks a direct way of relating information-theoretic bias to
Zipfian rank. A function of bias also serves as an unintuitive interpretation
of the communicative effort exchanged between a speaker and a listener. We
counter these shortcomings by proposing a novel integral transform and kernel
for mapping communicative bias functions to corresponding word frequency-rank
representations at any arbitrary phase transition point, resulting in a direct
way to link communicative effort (modeled by speaker/listener bias) to specific
vocabulary used (represented by word rank). We demonstrate the practical
utility of our integral transform by showing how a change from bias to rank
results in greater accuracy and performance at an image classification task for
assigning word labels to images randomly subsampled from CIFAR10. We model this
task as a reinforcement learning game between a speaker and listener and
compare the relative impact of bias and Zipfian word rank on communicative
performance (and accuracy) between the two agents.
