Comparing frameworks for similarity-aware generalized entropy (sentropy)

Shannon entropy is not the only entropy that is relevant to machine-learning datasets, nor possibly even the most important one. Traditional entropies such as Shannon entropy capture information represented by elements' frequencies, but not the richer information encoded by their similarities and differences. Capturing the latter requires similarity-sensitive entropy: "sentropy." Sentropy can be measured using either the recently developed Leinster-Cobbold-Reeve framework (LCR) or the newer Vendi score (VS). This raises the practical question of which one to use: LCR or VS. Here we address this question theoretically and numerically, using 53 large and well-known imaging and tabular datasets. We find that LCR and VS values can differ by orders of magnitude and are complementary, except in limiting cases. We show that both LCR and VS results depend on how similarities are scaled and introduce the notion of "half distance" to parametrize this dependence. We prove that VS provides an upper bound on LCR for all non-negative values of the Rényi-Hill order parameter, as well as for negative values in the special case that the similarity matrix is full rank. We conclude that VS is preferable only when a dataset's elements can be usefully interpreted as linear combinations of a more fundamental set of "ur-elements" or when the system that the dataset describes has a quantum mechanical character. In the broader case where one simply wishes to capture the rich information encoded by elements' similarities and differences as well as their frequencies, we propose that LCR should be favored; nevertheless, for certain half distances, the two methods can complement each other.